Cécile

Justin Hadrien

Foreword & Acknowledgements

The academic world, into which the PhD thesis stands as the rst step, has appeared to me like an adventurous hike. It begins with the same excitement for the unknown, keeps fed by the same pleasure of discoveries, requires a strong and lasting commitment, but also hides long days of loneliness, losses and setbacks. No matter how near lies the aim nor how much have been achieved, the same endlessness of the nal kilometers remains. One of the greatest challenges of this trail appears to me these days: the need to mourn the unreachable perfection and the desire to embrace the whole everything, learning to stop and nish. After all, every su ering the road could have unveiled quickly fades in front of the joy to reach the end and the strong motivation to sail for new horizons it brings.

Through more than four years of life and work, my days have been not only made of pure thoughts but also shared with many people. First and foremost I would like to acknowledge here their crucial role for me by some words, despite these remain far from what they deserve.

The appeal for this abstract adventure blossomed for about a decade, through lectures where the enthusiasm of my former teachers turned out to be a lighthouse for those long years, and a motivation to bring mathematics into life in my turn, by research and teaching, loving to discover and share. I would not have taken this path without the rst fundamental steps I owe to Sébastien Moulin, Patrick Rauch, Philippe Patte, Alain Pommellet, Filippo Santambrogio, Régis de la Bretèche and Étienne Fouvry, whose advices and support led me to uncover this beautiful experience.

A PhD thesis is also a human adventure, which one of the faces, perhaps the harshest one, is loneliness. Its quiet heroes are those who everyday brought life and colors to o ces and co ee rooms, those with whom I shared time and discussions, those who made it a living experience. I wish to sincerely stress how my workmates from Paris, Lausanne and Göttingen have been important to me, in particular Irène, Guillaume, Daniel, Tom, Ramon, Ian, Hunter, Andy, Dino and Jitendra whose comradeship has been considerable. The "heroic days" with Pierre and Antoine deserves a speci c mention as a friendly and motivating way of study. v vi On the sea of formal conferences and lectures, rare possibilities of expression without pressure and tribunes for sharing what we learn and like exist, and I am happy to have taken part in the workgroup of Analytic Number Theory, organized by Étienne Fouvry then Lucile Devin in Paris, as well as the seminars organized by Harald Helfgott in Göttingen. They stayed during the four last years a great opportunity to breathe new air and renew curiosity. Those who bear the charge of organizing them serve in a crucial way the community and endure an always underestimated amount of work. While acknowledging the role of community, here should be the right place to thank the various interesting threads and sometimes exceptional contributions on MathOver ow, where the will of sharing and helping is praiseworthy. I am sincerely grateful to all of them, as well as to the participating folk for bringing them into life.

After some years of lone exploration, I discovered the joys and excitement of sharing research projects. Ian and Hunter were the rst with whom a new joint adventure began in Lausanne. Both our common interests and our di erent cultures led to settle new exciting projects, nowadays pursued with Ian. I experienced the same enthusiasm during this last year in Göttingen, through passionate discussions with Andy and Jitendra where so many projects hatched. I would like to sincerely thank them for having made intense all those moments, both intellectually and humanly, for much more can be achieved together. I deep in my heart hope these collaborations and friendships will last for long.

Nothing would have been done without the warm environments and institutions that hosted me day after day, the atmosphere of the Université Paris 13, the peacefulness of the École Polytechnique Fédérale de Lausanne and the loveliness of the Mathematisches Institut in Göttingen. Particular emphasis is due to the secretaries who handle a huge load of work with an signi cant dedication: Isabelle, Yolande, Marcia and Sabrina. Various grants made possible wonderful months abroad, and I am grateful to the con dence of the ANR PerCoLaTor, the Fondation Sciences Mathématiques de Paris (FSMP), the Deutscher Akademischer Austausdienst (DAAD) and the École doctorale Galilée. Many of these travels have been achieved thanks to generous accommodation by friends all over the world: I would like to acknowledge here these open doors from Cécile, David, Farrell, Hunter, Hsueh-Yung, Ian, Jitendra, Olivier, Pascale, Ramon and Sarah.

Mentors are as fundamental as the soil is for seeds to grow, for they are the ones who show the directions when settling on new lands, and strongly a ect our ways of thinking and research interests. I am deeply thankful to Philippe Michel and Valentin Blomer for their invitations in Lausanne and Göttingen, where they provided me with all the infrastructures and environment I dreamed of. Their constant presence have been a inexhaustible source of motivation as well as valuable opportunities of enlightening discussions. All these e orts made my experiences abroad as part of my happiest moments, and I sincerely look forward having further opportunities to work with them on our joint interests. Their faithful support furthermore paved my way towards my forthcoming years in China, and I would like to testify to them my deepest gratitude. vii Above all mathematicians and mentors, I am in nitely indebted to my advisor Farrell Brumley, for having led me through the frontiers of this beautiful yet tremendously wide world of research. I began to learn with him more than four years ago, and since then he never stopped being an inexhaustible well of knowledge, motivation and support for mathematical as well as human matters. He trusted in me for handling the aims of this thesis and stayed strongly present and implied in their resolution. The many discussions we shared and the con dence he endowed me with blossomed into what this thesis became.

I am warmly grateful to Gergely Harcos and Philippe Michel for having made me the honor or reviewing this thesis. In spite of the time and e ort they spent on carefully reading my manuscript, they can be sure they contributed to make it better. Furthermore, I am honored to have them among my defense committee as well as Valentin Blomer, Gaëtan Chenevier and Jacques Tilouine.

Beyond the too often closed thick doors of the academic world, friends turn to be a much stronger source of motivation than they could imagine. I never said it enough, and grasp this opportunity to underline how important they are to me, and how much they have taken part in shaping what I am today, for they are my closest friends: Adrien, Amiel, Bruno, Camille, Cécile Carrère, Cécile Houdard, Clément, Élie, Hadrien, Jill-Jênn, Justin, Laura, Luc, Marc, Maria, Michaël, Morgane, Paul Pegon, Paul Simon and Raphaël. I shared many peaceful moments and exciting discussions with Bénédicte with whom the meals in Göttingen acquired a deeper avor; and the others rare moments of German evasion have been spent playing Go with the Göttingen club with Colin, Detlef, Gerd, Johann, Leon and Tim. The GICS and all its members have a speci c status in this list, as the parallel adventure and challenging companion it has been to vulgarize science as well as climbing towards its summits. This thesis would not have reached its term without a vital human presence through terrible months of sorrow and despair, when everything appeared tasteless and hopeless. Among those who shared this pain and helped me through, Luc Faucher and Farrell showed me a constant and benevolent support, leading me to begin to build a new home and life on what was no more than shattered ruins. Beyond them, many friends have shown a decisive empathy, and I dedicate without any doubt this thesis to the ve heroes whose hands and hearts stayed opened and bore me during my worst days, and to whom I owe everything: Cécile, Hadrien, Justin, Luc and Maria.

A thesis also has a sneaky e ect of colonizing one's life, and even if it was sometimes by will and passion, I kept in mind all along the years those who always shared part of this weight when I was at times overwhelmed, at others absent. I owe this quiet trust to my mother, father and brother for whom I never had enough time, but who I love.

Contents

List of Abbreviations and Symbols

A ring of adeles of F A(G), A(Q)

universal family and truncated one 2, A(q, σ R , δ, Ω) subset of the universal family of xed parameters

A(q, σ R , δ, Ω; ϕ) counting measures restricted to A, B(q, σ R , δ, Ω) α π (p) Satake parameters π p B division quaternion algebra B(q, σ R , δ, Ω; ϕ) counting measures restricted to B, B(q, σ R , δ, Ω) B(q, σ R , δ, Ω) subset of
(π, ϕ), D(ϕ, Q) one-level density of L(s, π ), of A(Q) δ F constant 2(1 + [F : Q]) -1 δ discrete archimedean parameters (δ, M) ϵ δ,
G(F ), G(F ), G(A) groups of points of G over F , F , A G local group G(F ) G dual of G [G] automorphic quotient G(F )\G(A) 1 h P n , h Weil height h δ,Ω,ϕ ρ archimedean spectral selecting function H (G), H (G )
Hecke algebra of G, G JL, JL global and local Jacquet-Langlands correspondence spec , geom spectral and geometrical parts of the trace formula temp , comp tempered and complementary parts of spec K maximal compact subgroup of G K 0 (q), K 0,p (p r ) congruence subgroup, local congruence subgroups K X 1 (p r ), K(p r ) ltrations for U and GSp L(s, π ), ξ (s, π )

L-function associated to π and completed one

L ∞ Levi subgroups of GL(2, F R ∞ ) λ π (p) Hecke eigenvalue of π for T p Λ (ν),σ q,σ R sum of Hecke eigenvalues for T p ν over A Q (q, σ R ) m(π, q) spectral multiplicities µ, ν limit measure for equidistribution, µ Pl Plancherel measure on G µ ST Sato-Tate measure µ Q , ν Q counting measures of A(Q) N (Q)
universal family and truncated one

ν 1,Q , ν ell,Q , ν Ξ,Q , ν comp,Q
splitting parts of the identity contribution

ν {(p),(e1),(e2)} 1,Q identity contributions from the trace formula N q norm of q O ring of integers of F O γ (Φ) orbital integral Ω(X ), Ω δ (X )
truncated archimedean universal family p prime ideal, nite place of F p m (Q) number of L(s, π ) vanishing at 1 2 with order m π , π global automorphic representation and component

P (ν) (π, ϕ)
one-level density of order ν density determining the type of symmetry X S , X S prime-to-S part and S-part of X X (s, π ) completing factor of the L-function L(s, π )

P (ν ),σ q,σ R one-level density of order ν along A(q, σ R ) P (ν ) Q one-level density of order ν along A(Q) q, d, m, ... ideals of F R q residue characteristic of F R set of rami ed places of B ρ π , γ π zeros of L(s, π ) S places
ξ σ normalized matrix coe cient for σ Ξ(π R , ϕ) characters contribution ζ , ζ
Dedekind zeta functions associated to F , residue

Chapter 1

On Arithmetic Statistics

Automorphic forms are central objects in modern number theory. Despite their ubiquity, they remain mysterious and their behavior is far from understood. Embedding them in wider families has a smoothing e ect and leads to results on average: these are the aims of arithmetic statistics and motivates the recent interest towards automorphic forms in families. Among families, some are more natural and carry powerful results, and a particular emphasis has been granted to the universal family consisting of all the automorphic representations on a given group. This chapter is dedicated to present new results in this philosophy.

In the case of the universal family of quaternion algebras, the growth law of the truncated family with respect to a suitable notion of size is stated. Further statistics lie in the equidistribution result of the global universal family with respect to a geometrically signi cant measure. It leads to answering the Sato-Tate conjecture for this family, concerning the local measures. Finally, the distribution of low-lying zeros of L-functions is explored, and the density conjecture partially veri ed.

Other reductive groups on which these arithmetic statistics problems make sense are mentioned as an invitation to investigate universal families on a larger scale. A glimpse towards such a program of research is provided by exploring the counting law for some unitary and symplectic groups of low ranks.

Automorphic world 1.Universal family of quaternion algebras

Automorphic forms and their L-functions appear to be the central objects of modern number theory since the fertile conjectures formulated by Langlands [80] and the powerful applications of their many avatars, such as elliptic curves [35] or modular forms [108]. In spite of their ubiquity, they remain quite mysterious objects. Embedding them in larger families has a smoothing e ect: singular behavior and unreachable objects lose their weights and this allows to establish regularity results concerning the family as well as results on average, supplying to the lack of pointwise knowledge on every particular object information on typical ones. This is the spirit of arithmetic statistics.

The recent years unleashed a wide enthusiasm toward the study of families of automorphic forms and their associated L-functions. Understanding what makes a family relevant for this philosophy is a critical issue. General attempts to de ne a suitable notion of family of automorphic forms have been made in the recent years [109, 110, 77], with a particular emphasis towards the universal family of a group, consisting of all its cuspidal automorphic representations.

Given such a family F , the rst natural question relative to it as a whole concerns its size. For in nite families, a truncation to a nite set makes sense of the problem. Assuming F Q to be a nite subfamily of F indexed by a positive parameter Q, such that F Q grow sto F when Q goes to in nity, the question is to determine the asymptotic behavior for the size of F Q .

The general linear group is the fundamental groundwork for automorphic representations, and the Langlands philosophy considers it as an ambiant group for more general reductive groups. The case of GL(2) is the rst non-commutative one, yet far from totally understood. One way to explore some of its features is to consider its inner forms: they are the groups of units of quaternion algebras, groundwork of the present thesis. Let a quaternion algebra B over a number eld F and introduce G = PB × = Z (B)\B × . Consider A(G) the universal family of G, that is the set of all automorphic in nite-dimensional representations of in L 2 (G(F )\G(A)). Following Sarnak [109], a deep understanding of A(G) is of fundamental importance in the theory of automorphic forms.

As an analogy and a guide for the methods, turn for a moment to a more usual setting: the one of general linear groups. The universal family A(G) embeds, via the Jacquet-Langlands correspondence, as a subfamily of the universal family A(PGL(2)), composed of all the cuspidal automorphic representations of PGL(2). In the latter context, even in the broader setting of cusp forms on general linear groups, Iwaniec and Sarnak [65] have de ned a good notion of size, given by the analytic conductor. It is a positive real number c(π ) de ned from the functional equation satis ed by the nite part L-function L(s, π ) associated to π ∈ A(PGL(2)), which takes the form

L(1 -s, π ) = ε π X (s, π )L(s, π ), (1.1)
where ε π is the root number of π . The completing factor X (s, π ) takes value ε π at the central point 1 2 , and the additive analytic conductor is de ned to be c(π ) = X ( 1 2 , π ) following the presentation of Conrey et al. [29]. Further considerations on the analytic conductor as well as discussions concerning its di erent avatars are carried on in Section 2.2. The function X (s, π ) involves the usual arithmetic conductor as well as archimedean gamma factors, so that the analytic conductor encapsulates the complexity of π . It allows to truncate the universal family of PGL(2), and hence the one of G, to a nite set [16]. The truncated universal family may then be introduced as

A(Q) = {π ∈ A(G) : c(π ) Q }, Q 1. (1.2)
The present work is a statistical exploration of this family in various aspects, such as asymptotic growth, equidistribution with respect to a geometric signi cant measure, behavior of the associated local components, and some statistics on the zeros of the associated L-functions.

Analogy with the height on algebraic varieties

The counting problem admits an interesting analogy with the well-known question of counting rational points of bounded height on a smooth projective variety over a number eld. The absolute Weil height is the proper notion of size in this setting and is de ned by

h P n (x) = max 0 i n |x i | 1/[F :Q] , x = (x i ) 0 i n ∈ P n (F ). (1.3)
where the product runs over the places of F and does not depend on the choice of homogeneous coordinates. Given any projective variety V over F endowed with a xed embedding ι into the projective space P n (F ), a height function on V can be de ned by pulling back the Weil height on P n (F ), setting

h V (x) = h P n (ι(x)), x ∈ V . (1.4)
The most natural setting for considering such generalized questions is the one of Fano varieties, where there are precise conjectures due to Batyrev, Manin and Peyre [6, 99]. On those grounds, Northcott [95] proved the niteness of the set of points of bounded height for the projective spaces, re ned by Schanuel [112] in an asymptotic counting law.

Theorem 1 (Schanuel). There exists C n > 0 such that for any Q 1,

# {x ∈ P n (F ) : h(x) Q } = C n Q n+1 + O (Q log Q) if n = 1and F = Q; O Q n-1/[F :Q] otherwise. (1.5)
In recent years, Sarnak has repeatedly emphasized the analogy between the Schanuel theorem on counting rational points on projective spaces and the problem of counting automorphic cusp forms on GL(n), so that the natural questions on algebraic varieties carry to the theory of automorphic forms and serve as a guideline for the methods.

Theorem A: Counting law

Analogously to rational points on algebraic varieties, the rst natural question concerning a family of automorphic forms is to determine its size. The case of quaternion algebras can be embedded in GL(2) so that, following the analogy with algebraic varieties, the notion of analytic conductor used in this main theorem is inspired by the procedure (1.4) for heights: given the by now standard notion of analytic conductor for GL(2), the analytic conductor for quaternion algebra is provided by pulling it back via the associated identity map between their dual groups. This canonical notion is consistent with the one de ned by the attached functional equation. The rst result of this thesis provides an asymptotic formula for the cardinality Theorem A (Counting law for quaternion algebras). Let R be the nite set of places where B rami es. There exists C > 0 such that for any Q 1,

N (Q) = #A(Q), Q 1
N (Q) = CQ 2 +          O Q 1+ε if F = Q and B totally de nite; O Q 2-δ F if F Q and B totally de nite; O Q 2 log Q if B is not totally de nite.
(1.7)

The constant C > 0 is de ned explicitly in (1.10), and

δ F = 2(1 + [F : Q]) -1 .
Remarks. The form of this asymptotic growth appeals some comments.

(i) There is a stricking similarity between the error term in Theorem A and that of the classical result of Schanuel in Theorem 1 on the number of rational points of bounded height in projective spaces. His result, when specialized to F = Q, also has an error term that picks up an additional power of log. (ii) The presence of a power savings error term in the totally de nite case, i.e. when every archimedean place is rami ed, is noteworthy. This feature is lost without this assumption, like the corresponding result [17] for GL(2), where only a logarithmic savings is obtained. The reason for this di erence lies in the passage from smooth to sharp counting at archimedean places, see Section 3.2.5. (iii) The assumption that B is a division quaternion algebra induces an automorphic compact quotient, hence avoiding the technical considerations due to the continuous part of the automorphic spectrum, see Section 3.1.

(iv) The center has been removed for technical purposes and to avoid to deal with the central terms in the Selberg trace formula. All the methods are expected to carry on to a setting considering the center without considerable adaptation.

The precise knowledge of the constant C unveils a lot of information, and its geometrical interpretation has considerable importance as in the conjectures of Peyre. An explicit and meaningful formulation of the constant is given below, in the context of the equidistribution properties of A(G), and shows striking similarities with the ones computed for algebraic varieties [22].

Equidistribution results

Theorem B: Equidistribution

Beyond estimating the size of the universal family lies the question of the geometrical distribution of the automorphic representations of G. A good formulation of the problem is to nd a measure with respect to which the universal family equidistributes, what is carried on in this section after giving a glance at the topological and measurable structure the universal family is endowed with.

Each local unitary dual group G is endowed with the Fell topology and the product G is then given the product topology. Introduce the measure µ on G that assigns to every basic open set X = X , i.e. where X is an open set of G and X = G for all but nitely many , the positive real number

µ(X ) = ∫ X d π c(π ) 2 , (1.8)
where the regularized integral is de ned as

ζ (1) ζ (1) -1 ∫ X d π c(π ) 2 .
(1.9)

Here ζ is the local zeta function associated to F , the notation ζ (1) stands for the residue of the Dedekind zeta function of F at 1, and d π is the Plancherel measure on G , introduced and normalized according to the convention in Section 2.3.2.

Remarks. This integral is not as disturbing as it seems for the following reasons.

(i) The Plancherel measure is supported on the tempered dual; since tempered representations are generic, the conductors appearing in the integral are well-de ned for the sets actually arising in what follows, see Section 2.2.

(ii) It is by no mean obvious that the integral (1.8) actually converges. It is the case and Section 3.2.4 contains the explicit computations of the local factors ensuring the convergence as well as motivating the regularization.

The measure µ has nite total mass µ . All the de nitions are now in place to uncover the expression of the leading constant in Theorem A, namely

C = 1 2 vol(G(F )\G(A)) µ , (1.10)
where the measure giving the volume of the automorphic quotient G(F )\G(A) is normalized as in Section 2.1.1. The main result is the following one.

Theorem B (Equidistribution for quaternion algebras). The universal family of G equidistributes with respect to the measure µ, in the following sense. For every relatively quasi-compact open set X of G with boundary of measure zero,

#{π ∈ A(Q) : π ∈ X } N (Q) -→ µ µ (X ), as Q → ∞.
(1.11)

Sato-Tate conjectures

Once global equidistribution results reached, the behavior of the local components at a xed place p can be investigated, following the general conjectures of Shin and Templier [118]. This is the aim of the so-called Sato-Tate conjectures, that have their own outside motivations worth reviewing.

An origin lying in elliptic curves

Pursuing the fruitful analogy with algebraic varieties gives ground to formulate statistical problems for automorphic forms. The simplest yet already rich case is the one of an elliptic curve E de ned over Q. It can be de ned by the equation 2 = x 3 +ax +b, assuming for the sake of this motivating groundwork that a, b ∈ Z, so that its reduction E p modulo p remains smooth for almost every p, i.e. E p is an elliptic curve on F p .

Of great interest is to study elements of the curve E p (F p ). Getting rid of the x 3 factor gives a toy model in which the number of those points would be N p ( 2 = ax +b) = p+1, up to adding a point at in nity and provided a is nonzero. More generally, for an elliptic curve E, the number of its rational points modulo p can be written in the form Conjecture 1 (Sato-Tate for elliptic curves). For a non CM elliptic curve E, the numbers a p (E), resp. θ p (E), equidistribute in [-2, 2], resp. [0, π ], with respect to the half-circle measure

N p (E) = #E p (F p ) = p + 1 -a p (E), (
µ ST = 1 π 1 - x 2 4 d x, resp. μST = 2 π sin 2 ϕ d ϕ.
(1.13)

The Sato-Tate conjecture for elliptic curves has been proven in 2006 by Clozel, Harris, Shepherd-Barron and Taylor [20] under the hypothesis that the j-invariant j(E) is not an algebraic integer, in particular implying that E is non CM. In the exceptional case of CM curves, half of the a p (E) vanish and the limit distribution is also known.

Automorphic Sato-Tate conjecture

The work of Taylor and Wiles [126] unveiled that a semistable elliptic curve E over Q corresponds to a modular cusp form on GL(2) of weight 2 with integer coe cients. More generally, for modular cusp forms of weight k, the Ramanujan conjecture, known for GL(2) by the work of Deligne [33], gives the analogous of the Hasse bound and states that its Fourier coe cients |a p (f )| are bounded by 2 √ p k-1 . This naturally leads to lift the Sato-Tate problem in this modular setting.

Conjecture 2 (Modular Sato-Tate). For f a normalized non-CM modular cusp form on

SL 2 (Z), the √ p -(k-1) a p (f ) equidistribute in [-2, 2] with respect to the measure µ ST .
The aforementioned result of Clozel, Harris, Shepherd-Barron and Taylor proves that this conjecture holds in the case of modular forms of weight 2 with integer coefcients. In 2011, Barrett-Lamb, Geraghty, Harris and Taylor [5] generalized it to any weight k 2.

Towards families of automorphic representations: Corollary C

Automorphic forms give rise to automorphic representations, providing a motivation to consider Sato-Tate conjectures in this even wider setting. An automorphic representation π ∈ A(GL(n)) decomposes as a restricted tensor product π = ⊗ π of local factors. Since only a nite number of the π 's are rami ed, for a xed π and large enough almost all the local component π can be identi ed with its Satake parameters

π (α 1 (π ), . . . , α n (π )) ∈ T c /W , (1.14)
where T c is the standard complex split torus of GL(n) and W is its associated Weyl group. These local factors living in a priori di erent dual groups hence have an interpretation on a common ground, allowing to consider the Sato-Tate problem which consists in determining the distribution of the local components π .

Besides considering a xed automorphic form for which information is barely reached under strong hypothesis as above, it is possible to provide results by averaging on a whole family of automorphic forms F . This allows not only to study the varying coefcients associated to a xed object in function of as in the previous instances of the conjectures, what is called "horizontal" statistics, but also to x a parameter and study the coe cients α i (π ) for π varying in F . These give rise to the "vertical" conjectures. Such families have been considered for instance in the works of Bruggeman Sarnak, Shin and Templier [110] surveyed the evolution of recent de nitions of general families of automorphic forms as well as their attached Sato-Tate analogues, and formulated the Sato-Tate conjecture for families.

Conjecture 3 (Sarnak-Shin-Templier). The family F , when ordered by the analytic conductor, is equidistributed in G with respect to a measure µ(F ) such that (i) it is a probability measure supported on the tempered spectrum, so does its projections µ p (F ) on the local duals G p ; (ii) the log-average over p exists and de nes the Sato-Tate measure, more precisely there exists a measure µ ST (F ) such that

1 Q Np Q log(Np)µ p (F ) |T -→ µ ST (F ), as Q → ∞.
(1.15) Shin and Templier [118] recently proved a precise quantitative version of this conjecture in a fairly broad setting for families of automorphic representations with discrete series at in nity. Besides, there is no result of this type when both parameters and objects vary, only horizontal or vertical conjectures have been proven so far.

Coming back to the universal family of quaternion algebras, once the global equidistribution result stated in Theorem B, the Sato-Tate conjecture questions the behavior of the projections µ p of the limit measure on the local components G p when the norm of p grows. On the common ground where all the representations in the support of the Plancherel measures of G p live, given by the tempered Satake parameters space T c /W , the Sato-Tate question acquires a precise meaning and local representations are equidistributed with respect with the half-circle measure.

Corollary C (Sato-Tate for quaternion algebras). For all ϕ ∈ C(T c /W ),

∫ T c /W ϕ(x) d µ p (x) -→ ∫ T c /W ϕ(x) d µ ST (x), as Np -→ ∞, (1.16)
where the measures µ p are explicit given by (1.9).

Low-lying zeros of L-functions 1.3.1 Importance of zeros of L-functions

The L-functions, among which the Riemann ζ function is the most celebrated representative, are ubiquitous in number theory and provide an analytic way of grasping properties of arithmetic objects. Their zeros, even if they remain mostly mysterious, carry information concerning the distribution of prime numbers, and more generally the nature of the object to which they are attached, thus justifying the tremendous e orts and interest towards the Riemann hypothesis.

Indeed, the so-called explicit formulas link distributions of the zeros to quantities of arithmetic nature. A motivation for statistical studies on zeros of L-functions is provided by Mazur [87], who notices that explicit formulas are generically of the form

π (x) = MT + ET + OT,
(1.17)

where π (x) is a relevant statistic on prime numbers; MT stands for the main term coming from particular zeros of the L-function; ET is a sum over trivial zeros that constitutes an error term; and OT is an oscillating term coming from the other zeros. This last term is expected to contribute as an error term, yet is often tough to estimate and requires a su ciently precise knowledge of the behavior of the zeros in order to use compensations. For instance, assuming the Riemann hypothesis, every nontrivial zero of the Riemann zeta function lies on the critical line Re(s) = 1 2 , improving the remainder in the prime number theorem as follows:

π (x) = li(x) + O xe -α √ log x , without RH; π (x) = li(x) + O √ x log x , with RH.
Statistics on zeros of L-functions hence lead to a priori nontrivial results towards the arithmetic of the underlying objects, providing a strong motivation to their study.

Pair correlations

Analogy between matrices and L-functions

The theory of random matrices [88] is a glass through which understand the eld of statistics on zeros of L-functions. The eigenangles of random matrices behave strikingly like these zeros and, since well more explored, will serve as a guide for the Lfunction world. Let A ∈ M n (F ) be a diagonalizable unitary matrix, and consider its eigenvalues λ (j)

A = e iθ (j) A ordered such that 0 θ (1)

A • • • θ (n) A < 2π .
The mean spacing between neighboring eigenangles is 2π N . In order to renormalize it to one, set

θ (j) A := N 2π θ (j) A , 1 j n. (1.18)
Similarly, associate to an L-function L(s, π ) its nontrivial zeros ρ (j) π = 1 2 + iγ (j) π , with a priori γ (j) π ∈ C without assuming the Riemann hypothesis, and ordered so that the

γ (i) π satisfy • • • Rγ (-1) π 0 Rγ (1) π Rγ (2) π • • • . The mean spacing between neighboring zeros [65] is m log c(π ) 2π
where m is the degree of the L-function. Renormalize them to 1 introducing

γ (j) π := log c(π ) 2π γ (j) π , j ∈ Z. (1.19)
The strong similarity between both settings leads to motivate statistical questions on zeros of L-functions by existing statistical results for eigenangles of matrices.

Pair correlation for matrices

In the 50s, Wigner investigated random matrices in order to modelize atomical phenomena. These are matrices of the gaussian unitary ensemble, denoted GUE(N ), i.e. the set of unitary matrices of size N with independent random gaussian coe cients. A particular way to grasp the behavior of their associated eigenangles is to study the distribution of the spacings [69] between them, given by

R A [a, b] = 1 N j k : θ (j) A -θ (k) A ∈ [a, b] , A ∈ GUE(N ). (1.20)
This statistics is called the pair correlation of the matrix A. The pair correlation of a family of matrices is naturally de ned as the average of the pair correlations over the family, that is to say in the case of the whole gaussian unitary ensemble

R GUE(N ) [a, b] = ∫ GUE(N ) R A [a, b] d A.
(1.21)

Dyson determined in 1962 the correlation density of GUE in the following result.

Theorem 2 (Dyson).

There is a measure r GUE such that

R GUE(N ) [a, b] -→ N →∞ R GUE [a, b] = R GUE [a, b] = ∫ b a r GUE (x) d x, (1.22) moreover r GUE (x) = 1 -sin πx πx 2 .
Katz and Sarnak [68] proved more generally in 1997 that the spacings between the eigenvalues of random matrices belonging to fairly general families, viz. the compact symmetric irreducible Lie group, are all governed by the GUE distribution. Introduce more precisely the classical groups G(N ) among the groups of unitary matrices, orthogonal matrices, or symplectic matrices of size N and independent gaussian coe cients.

Theorem 3 (Katz-Sarnak). For every family G(N ) of classical group, in the L 1 -sense,

∫ G(N ) R A [a, b] d A -→ N →∞ R GUE [a, b],
a, b ∈ R.

(1.23)

Firk and Miller [46] gave arguments for the ubiquity of the GUE density in statistical modelisations in physics. The results summarized here suggest, following faithfully the fruitful analogy between matrices and L-functions, that the same universality holds for statistics on L-functions.

Pair correlation for L-functions

Much later, Montgomery [94] rst explored the analogous distribution law of spacings between zeros of L-functions. In the particular case of the Riemann ζ function, he strikingly noticed that the pair correlation between zeros is the same than the one Dyson obtained for eigenangles of random unitary matrices.

Theorem 4 (Montgomery, 1974). For ϕ ∈ S(R) such that supp( φ) ⊆ (-1, 1), the pair correlation of the zeros of the Riemann zeta function is given by

1 N 1 j k N ϕ( γj -γk ) -→ N →∞ ∫ R ϕ(x)r GUE (x)dx . (1.24)
Many computations led by Odlyzko [96] for other L-functions then brought strong evidence that this statistical behavior of the zeros of L-functions seem to match the analogous statistics for eigenangles of random matrices in the gaussian unitary ensemble, leading to expect the Montgomery result to be a general property of zeros of L-functions. This universal behavior is known as the Montgomery-Odlyzko law. Results in this direction ourished from then on, culminating with Rudnick and Sarnak [105] who proved in 1995 that the same universal distribution holds for pair correlations of a generic L-function on GL(n). The following statement is restricted to GL(2) and suits the purposes of the present motivational background without having to introduce extra technical condition, yet morally holds for every general linear groups.

Theorem 5 (Rudnick-Sarnak, 1996). Let π be a cuspidal automorphic representation of GL(2, Q). Let ϕ be an even Schwartz function such that supp( ϕ) ⊆ (-1, 1). Then

1 N 1 j k N ϕ( γj -γk ) -→ N →∞ ∫ R ϕ(x)r GUE (x)dx . (1.25)
The result proved by Rudnick and Sarnak is the rst towards the analogous conjecture for every L-functions. Indeed, the Langlands functoriality conjectures [26] postulate that every L-function comes from an L-function attached to a cuspidal automorphic representations on a general linear group. The result of Rudnick and Sarnak positively answers these "standard" cases, even for higher level correlations.

One-level densities

The universality of the GUE law for pair correlations is surprising. Indeed, the previous results are disappointing for they are blind to the di erences between the classical groups and it can be expected, in contrast with the Montgomery-Odlyzko law, that other statistics will be able to distinguish among them. A second disappointment with correlations is that they are blind to many modi cations on the zeros, and in particular do not give any importance to zeros usually of arithmetic signi cance, like the central point.

One-level density for matrices

The correlation statistics considered until now are global statistics, taking into account all eigenangles, since they consider only the distribution of spacings between them. Katz and Sarnak broke this universality, turning their interest towards statistics concentrated on small eingenangles.

De nition 1. The one-level density attached to A is the distribution de ned by, for ϕ be an even Schwartz function on R and A ∈ M n (R),

D(A, ϕ) := 1 j N ϕ θ (j)
A .

(1.26)

Here ϕ is a quickly decreasing test function which is no more supposed to be a function of the di erences as for pair correlations. This time, large eigenangles are essentially cut o , and hence D(A, ϕ) is a weighted average of the small eigenangles.

De nition 2. Let ϕ be an even Schwartz function on R. The one-level density of a family F endowed with a probability measure is

D(F , ϕ) = ∫ F D(A, ϕ) d A.
(1.27)

In the matrice setting, Katz and Sarnak [68] proved that the average density over a family di ers depending on the group considered, breaking the embarrassing universality of GUE.

Theorem 6 (Katz-Sarnak). For the classical groups G(N ), for every real Schwartz function ϕ of compactly supported Fourier transform,

D(G(N ), ϕ) -→ N →∞ ∫ R W G (x)ϕ(x) d x, (1.28)
where dA is a normalized Haar measure on G(N ), and the densities functions on R are de ned by

W U (x) = 1 W Sp (x) = 1 - sin 2πx 2πx W SO(even) (x) = 1 + sin 2πx 2πx W SO(odd) (x) = 1 - sin 2πx 2πx + δ 0 (x) W O (x) = 1 2 W SO(even) (x) + W SO(odd) (x) = 1 + 1 2 δ 0 (x)
This function W G is the one-density function for G. The fact that the limit is no more universal but depends on the family associated to a classical group, yet also falls in nitely many cases, gives rise to the notion of type of symmetry of a family of Lfunctions. The computations of this limit detecting in some sense which of the classical group govern the behavior of the zeros.

Remark. The result of Katz and Sarnak holds for classical groups, yet it remains to know whether or not it remains true for more general families of groups, what would turn the classical groups as universal representant of the di erent symmetries governing the eigenangles.

One-level densities for families of L-functions

Following the enlightening analogy with random matrices, it can be expected that the one-level density of the zeros attached to every reasonable family of L-functions behaves as the eigenangles of classical random matrices groups, and in particular that the behavior of low-lying zeros of L-functions is modeled by the classical groups. The socalled density conjecture postulates this universality, more precisely that every reasonable family of matrices or L-functions will match one of these cases.

De nition 3. Let ϕ be an even Schwartz function on R and π an automorphic representation. The one-level density attached to A is

D(π, ϕ) := γ (j) π ϕ γ (j) π , (1.29)
Remark. Without assuming the Riemann hypothesis, the above de nition has to be broadened, for the γ (j) π need not be real. Motivated by the density conjecture, "Schwartz function on R" has to be understood as a function of Schwartz class on R and of compactly supported Fourier transform. This ensures the existence of an analytic continuation to the whole C making sense of the expression above.

De nition 4. Let ϕ be an even Schwartz function on R. The one-level density of a nite family F is

D(F , ϕ) = 1 |F | π ∈F D(π, ϕ).
(1.30)

The rst result in this direction is given by Özlük and Snyder [127] in 1993 for Lfunctions attached to Dirichlet characters. Since then, a wide literature has been published concerning the statistical behavior of low-lying zeros of families of L-functions [38,58,64,84,104]. This led Katz and Sarnak [69] to formulate the so-called density conjecture stating the same universality of the types of symmetry arising for group of matrices.

Conjecture 4 (Density conjecture). Let F be a family of automorphic representations in the sense of Sarnak and F Q a nite truncation increasing to F when Q grows. Then for all even Schwartz function on R with compactly supported Fourier transform, there is one classical group type G such that

D(F Q , ϕ) -→ Q→∞ ∫ R ϕ(x)W G (x) d x .
(1.31)

The family F is said to have the type of symmetry of G.

Remark. For families of L-functions associated to algebraic varieties over function elds, the type of symmetry is determined by the monodromy of the family, see [68], shredding light on the reason why zeros of L-functions are governed by random matrix groups. However, no such analogue is known on number elds.

Theorem D: Type of symmetry

Low-lying zeros

Considering the statistics on low-lying zeros of L-functions attached to the universal family of quaternion algebras, the one-level density (1.30) of the truncated family is

D(A(Q), ϕ) = 1 N (Q) π ∈A(Q) D(π, ϕ).
(1.32)

The problem is to determine whether or not the quantity D(A(Q), ϕ) admits a limit and unveils the associated type of symmetry according to the density conjecture. The following statement partially determines the type of symmetry of quaternion algebras and ful lling the expectations of the density conjecture.

Theorem D. For every even and Schwartz function ϕ with Fourier transform compactly supported in (-2/3, 2/3), 1

N (Q) π ∈A(Q) D(π, ϕ) -→ Q∞ ϕ(0) + 1 2 ϕ(0) = ∫ R ϕ(x)W O (x) d x .
(1.33)

In particular, the type of symmetry of inner forms of PGL(2) is one of the orthogonal types of symmetry.

An important caveat ought to be mentioned concerning the orthogonal types of symmetry. The density conjecture postulates results for Schwartz function with compactly supported Fourier transform, yet with no constraint on the support. Assuming this conjecture, proving the convergence for a narrower class of allowed Fourier supports may determine uniquely the postulated type of symmetry. However this is not the case for all supports, and an uncertainty remains in the case or supports trapped in (-1, 1). Explicitly, the Plancherel formula yields

∫ R ϕ(x)W (x) d x = ∫ R ϕ(x) W (x) d x,
(1.34) and looking at the Fourier transforms of the densities, introducing η the characteristic function of [-1, 1], direct computations leads to

W U (x) = δ 0 (x) W Sp (x) = δ 0 (x) - 1 2 η(x) W SO(even) (x) = δ 0 (x) + 1 2 η(x) W SO(odd) (x) = δ 0 (x) - 1 2 η(x) + 1 W O (x) = δ 0 (x) + 1 2
Unfortunately, the three orthogonal types of symmetry, viz. W O (x), W SO(even) (x) and W SO(odd) (x) are indistinguishable in (-1, 1). Theorem D hence only partially determines the type of symmetry of the universal family of quaternion algebras. Further directions are mentioned in Chapter 4.

Non-vanishing of L-functions

Statistics on the distribution of low-lying zeros of L-functions are known to lead to results concerning vanishing at the central point, following the ideas of Iwaniec, Luo and Sarnak [64]. Introduce the proportion of automorphic representations with vanishing at the central point with order m, that is

p m (Q) = 1 N (Q) # π ∈ A(Q) : ord s=1/2 L(s, π ) = m , m ∈ N. (1.35)
Theorem D yields densities of vanishing of the associated L-functions. Indeed, for every T ϕ such that Theorem D holds for functions whose Fourier transform is suppor-

ted in (-T ϕ ,T ϕ ), lim inf Q→∞ p 0 (Q) 1 2 - 1 T ϕ , (1.36) lim inf Q→∞ m 0 mp m (Q) 1 2 + 1 T ϕ . (1.37)
Unfortunately, it yields nontrivial results on the density of non-vanishing at the central point only if T ϕ is allowed to be large enough, namely in this case larger than two. The second density result above is interesting for every T ϕ > 0. This, in addition of verifying the whole density conjecture, is a strong motivation to strengthen the bounds on the support of the Fourier transform in Theorem D.

Other ground groups

Addressing arithmetic statistics problems for di erent groups than inner forms of GL(2) leads to determine what is critical for the use of the same counting method and what is speci c to the GL(2) setting. Any result in this direction provides clues towards more general conjectures concerning both the growth rate and the form of the constant, that are fundamental in the vein of the analogous program of Batyrev, Manin and Peyre for algebraic varieties. Some unitary and symplectic groups of low ranks can be explored.

The main aim of this opening towards di erent settings is to identify essential assumptions, state precise conjectures and focus on the di erences between these cases and the one of general linear groups, thus it is natural to use some freedom on the assumptions to make space for comments and comparisons avoid to sink in unnecessary technicalities. This also motivates to address the counting problem instead of the more general equidistribution question as it could be expected in the light of the previous sections on quaternion algebras: there is no need to say these problems are by no means irrelevant.

Unitary groups

Let E be a quadratic totally imaginary extension of the totally real eld F , q an hermitian form on E 3 , and U the unitary group associated to q, that is to say the subgroup of GL(3) preserving q. More precisely, it is the subgroup of GL(3, F ) de ned by

U = ∈ GL(3, F ) : ∀x, ∈ E 3 , q( x, ) = q(x, ) . (1.38)
As for quaternion algebras, the classi cation of the group of points over local elds is known. Essentially, U (F ) is isomorphic to GL(3, F ) at half of the non-archimedean places, and isomorphic to the unique quasi-split unitary group in three variables U (F ) at the other half of the places. There is a nite number of places, where di erent behavior can arise. Assume U is totally de nite, that is to say isomorphic to the compact unitary group O(3, R) at archimedean places. This in particular ensures that the automorphic quotient U (F )\U (A) is compact.

There is a notion of size on such a unitary group U , given by the same procedure as for quaternion algebras: pulling back the standard notion of analytic conductor along a suitable embedding, provided in this case by deep works of Flicker [47], in a general linear group provides an analytic conductor on U , still denoted by c(π ) where π is an automorphic representation of U . This allows to consider the truncated universal family

A U (Q) = {π ∈ A(U ) : c(π ) Q }, Q 1. (1.39)
Introducing the measures on the local groups and the local dual groups as normalized in Section 5.1.1, it is possible to formulate a conjecture for the counting law for the cardinality N U (Q) of the truncated universal family of U .

Outline of the story 1.5.1 Universal family decomposition

Chapter 2 is mainly devoted to introducing the universal family A(G) for quaternion algebras, made of all its in nite-dimensional automorphic representations. This is the main groundwork on which this thesis settles and only the last chapter deals with different families. The analytic conductor c(π ) is a convenient notion of size for representations, introduced in Section 2.2 by pulling back the standard notion of conductor on GL(2). The truncated universal family is the nite set

A(Q) = {π ∈ A(G) : c(π ) Q }, Q 0. (1.40)
The counting law Theorem A and the equidistribution Theorem B for the universal family can be reformulated in terms of convergence of a measure ν Q , representing the distribution of automorphic representations of the truncated family A(Q), to a measure ν . More precisely,

ν Q ( ϕ) = π ∈A(Q) ϕ(π ).
(1.41)

The sought convergence is reduced, by the mean of a theorem of density, to a convergence with respect to a well-behaved class of functions ϕ ∈ F ( G S ) acting on a nite set of places S, as explained in Section 2.3.1. The aim then turns to proving that, as Q grows to in nity,

ν Q ( ϕ) -→ ν ( ϕ), ϕ ∈ F ( G S ).
(1.42)

The universal family A(G) decomposes into harmonic subfamilies de ned by adding constraints on the spectral data attached to representations, as explained in Section 2.4. This decomposition is led by the di erent classi cations of representations that exists depending on the place, and by the di erent ways to grasp the conductor. Introducing R the set of places where B rami es, the representation is decomposed as follows, and each component is to be treated in a speci c way.

π = π R ⊗ π R f ⊗ π R ∞ .
(1.43)

Explicitly, the discrete spectral data consists in the arithmetic conductor q of the split nite part of π , the isomorphism class of the rami ed part π R and the discrete series δ for a certain Levi subgroup, described by a certain set D, to which the archimedean split part for π belongs. This data only partially classi es the representations in the universal family. There is a continuous set of parameters achieving the description of representations at split archimedean places π R ∞ , so that they are essentially parametrized by δ and ν , denoting π δ,ν the representation attached to such parameters. Continuous archimedean parameters cannot be precisely selected due to regularity constraints in the methods, hence they are allowed to vary in a restricted set of the form

Ω δ (X ) = {π ∈ G R ∞ : c(π ) X and π π δ, }, δ ∈ D, X > 0. (1.44)
For xed discrete spectral data q, π R and δ , consider the subfamily of automorphic representations with such spectral parameters, that is

A(q, σ R , δ, Ω δ (X )) = {π ∈ A(G) : π R σ R , c(π R f ) = q, π R ∞ ∈ Ω δ (X )}, (1.45)
so that the truncated universal family A(Q) decomposes into such subsets of restricted spectral data, when the data vary. The counting measure decomposes accordingly into

ν Q ( ϕ) = N q Q q∧R,S=1 σ R ∈ G R c(σ R ) Q/N q δ ∈D π ϕ(π ), (1.46)
where the last sum runs over

π in A(q, σ R , δ, Ω δ (Q/N qc(σ R )))
. This is the content of Section 2.4.2. It is then natural to consider the innermost sum running over more general sets of continuous spectral parameters Ω, namely A(q, σ R , δ, Ω). A quantity more amenable to trace formula methods has extra weights given by the spectral multiplicities, that is to say

B(q, σ R , δ, Ω; ϕ) = π dim π K 0 (q) f ϕ(π ), (1.47)
where the sum runs over π in A(q, σ R , δ, Ω). The theory of local newforms for GL(2) allows to recast the counting measure ν Q in terms of these quantities, as explained in Lemma 1, so that it is enough to study these sums over representations of xed spectral data. More precisely,

ν Q ( ϕ) = N q Q q∧R,S=1 σ R ∈ G R c(σ R ) Q/N q δ ∈D d|q λ 2 q d B(d, σ R , δ, Ω δ (Q/N qc(σ R )); ϕ).
(1.48)

Spectral count

The heart of the proof is to interpret the quantity B(q, σ R , δ, Ω; ϕ) as a spectral side of the Selberg trace formula for a suitable test function. Lemma 4 achieves this goal up to some error terms, which precise control is fundamental.

In contrast with the discrete spectral data that can be exactly xed by test functions admissible for the trace formulas, it is necessary to approximate the characteristic function for the continuous set of spectral parameters Ω by an admissible function. Such functions are obtained by the mean of Paley-Wiener type theorems, that provides functions blowing up outside the tempered spectrum. The better the approximation for the tempered spectral parameters, the worse this blow up on the complementary part, feature encoded in a parameter ρ > 0. For this reason, only the tempered part of the spectral continuous parameters Ω temp is e ciently approximated by the trace formula. In other words, there is a splitting between tempered and complementary part of the archimedean spectrum, each part receiving di erent treatment, namely B(q, σ R , δ, Ω; ϕ) = B(q, σ R , δ, Ω temp ; ϕ) + B(q, σ R , δ, Ω comp ; ϕ).

(1.49)

The smoothing step closely follows the work or Brumley and Milićević [17], and the tempered part B(q, σ R , δ, Ω temp ) is approximated by the tempered spectral part of the trace formula applied to an explicit test function Φ, depending on the equidistribution function ϕ; the discrete spectral data q, σ R , and δ ; the the set of continuous parameters Ω from now on assumed to be tempered; and on the approximation parameter ρ. This is the aim of Lemma 2.70, of the form

B(q, σ R , δ, Ω temp ) = temp (Φ) + O(∂ ρ B(q, σ R , δ, Ω)), (1.50)
where ∂ ρ B(q, σ R , δ, Ω) is an error term corresponding to the fact that the test function at archimedean split places is a smoothed version of the characteristic function of Ω. More precisely for spectral parameters farther than ρ from the boundary of Ω, thus in some sense strongly inside or strongly outside Ω, the approximation is good enough and the induced error term is of good quality. However, for spectral parameters lying in the transition zone around the boundary of Ω, the approximation is of lower quality and is the origin of this error term as well as a justi cation for its notation. This is precised by Lemma 3, and hence the quantity ∂ ρ B(q, σ R , δ, Ω) appears as a smoothed version of B(q, σ R , δ, ∂ ρ Ω), the number of representations with spectral parameters lying in the boundary ∂ ρ Ω. This number is bounded by another tempered count, amenable to the very same methods than what follows, and hence contributing to the same error terms.

The complementary part of the spectrum is responsible for another error term. Indeed, in order to make use of the trace formula, the whole spectral part relative to Φ should be taken into account, and this one is

spec (Φ) = temp (Φ) + comp (Φ) + char (Φ), (1.51)
where char (Φ) corresponds to unwelcome global characters selected by the test function Φ, and where comp (Φ) is the contribution coming from non-tempered representations whose continuous spectral parameters have tempered part lying in Ω. The character contribution is shown to contribute as an error term by directly estimating the number of such characters by an analogous strategy, in Lemma 12, with the Selberg trace formula replaced by the Poisson summation formula. The complementary part of the spectrum is exponentially weighted due to the behavior of the chosen test function at these places, and is bounded by a certain quantity B comp (q, σ R , δ, ρ, Ω) in Lemma 13.

At last, this leads to

B(q, σ R , δ, Ω; ϕ) = spec (Φ) + O(∂ ρ B(q, σ R , δ, Ω)) + O(B comp (q, σ R , δ, ρ, Ω)).
(1.52)

Lemma 9 provides a bound for the complementary and smoothing terms, essentially bounding them by the counting measure for di erent subsets than Ω, and hence amenable to the same methods.

Geometric side

In Section 3.1, the Selberg trace formula comes into play and translates the spectral term appearing in (1.52) into a geometrical sum, running over G(F )-conjugacy classes, of weighted orbital integrals. It is of the form spec (Φ) = geom (Φ).

(1.53)

The main contribution to this expansion, as often expected in applications of the trace formula, comes from the term of the geometrical expansion corresponding to the identity, leading to split the geometric side into geom (Φ) = 1 (Φ) + ell (Φ).

(1.54)

The identity term can be explicitly computed by means of the Plancherel formula and direct computations. However, there is still an error term coming from the approximation in the test function at archiemdean places, bounded similarly than the one obtained above due to the smoothing (1.50), so that The critical ingredient in the estimations of this main term are sizes φ(q) of the congruence subgroups that de ne the conductor. The computations carried out in Section3.2 consist in summing over all the discrete spectral data, and yields the growth rate of Q 2 announced in Theorem A and the limit distribution measure announced in Theorem B.

Geometrical error terms are the ones given by nontrivial orbital integrals arising in the Selberg trace formula, encapsulated in ell (Φ), which is of the form

ell (Φ) := {γ } 1 vol G γ (F )\G γ (A) ∫ G γ (A)\G(A) Φ x -1 γ x d x, (1.56)
where the indexation runs through conjugacy classes {γ } in G(F ). Since Φ is compactly supported and G(F ) is discrete, the sum is nite. The number of elements appearing in this sum depends on the parameter ρ governing the exponential type of the test function at in nity, and growth exponentially with ρ, as stated in Lemma 14. Bounds on orbital integrals are consequences of bounds on their local components, using di erent methods depending on the behavior of the place and the choice of the corresponding local test function, and depend essentially in q as stated in Proposition 21. At last, a speci c choice of ρ depending on q is enough to ensure the negligibility of these error terms compared to the identity term, nishing the proof of Theorem B.

Further consequences

The Sato-Tate corollary, stated in Corollary C, follows from the knowledge of the equidistribution measure and known results for the spherical spectrum of GL(2). Indeed, since quaternion algebra are locally almost everywhere equal to GL(2), a result of Serre already establishes the Sato-Tate conjecture in this case when restricted to unrami ed representation. Rami ed contribution to the equidistribution measure is shown to be negligible through explicit computations carried out in Section 3.5.

Estimating densities of low-lying zeros turns to be the main topic of Chapter 4. The explicit formula restates the problem into a question concerning sums of the Satake parameters, rewritten as sums of associated Hecke eigenvalues of the form, for a Schwarz function ϕ,

D(π, ϕ) = ϕ(0)- 2 log c(π ) ∞ ν=1 p α ν π (p) + β ν π (p) ϕ ν log Np log c(π ) log Np Np ν/2 +O 1 log c(π ) .
High order contributions, more precisely those corresponding to ν 3, are not a problem by bounds on Satake parameters provided by Blomer-Brumley in this setting where Ramanujan is not known to hold, and are shown not to contribute to the type of symmetry.

Lower order terms are less well controlled, even assuming the Ramanujan conjecture. A fundamental step comes from the structure of the L-functions attached to the automorphic representations of A(G), relating the Satake parameters α ν π (p) and β ν π (p) with the Hecke eigenvalues λ π (p ν ). The heart of the proof is to deal with the resulting sums p λ π (p ν ), ν ∈ {1, 2}.

(1.57)

Here the averaging over the family is necessary, feature already underlined in the story of densities and correlations. The Selberg trace formula addresses the problem of low order terms with similar methods than for Theorem B: twisting the already built test function by the suitable Hecke opeator weights the spectral side of the trace formula by the desired Hecke eigenvalues. Since only unrami ed Hecke operators are considered, this allows to treat the sum (1.57) restricted to unrami ed representation at the chosen prime. Similar considerations than in the case of the equidistribution result leads to estimating the averaged inner sum by

π ∈A(d S ,σ R ,δ,Ω) π unrami ed λ π (p ν ) = spec (Φ) + (Remainder).
(1.58) for a suitable test function Φ, and a remainder similar to the one obtained for the equidistribution. There is no identity contribution because of the Hecke operators, the elliptic contribution is hence critical. Orbital integrals are to be precisely estimated in Section 4.4.1, and they are the seed of the limitations for the support of the Fourier transform.

Concerning the rami ed part, the already stated Blomer-Brumley bound reduces the problem to the counting law for the harmonic subfamilies A(q, σ R , δ, Ω), already established on the way towards Theorem B, and enough to prove the rami ed contributes as an error term and achieving the proof of Theorem D.

Non-vanishing results are obtained as a consequence of the type of symmetry, through Plancherel formulas and functional optimization provided for one-level densities of orthogonal type by Iwaniec, Luo and Sarnak [64] in the setting of holomorphic cusp forms on GL(2).

The conclusive Chapter 5 is devoted to openings towards two classes of other ground groups: totally de nite unitary groups in three variables and inner forms of the symplectic group of degree 4 that have compact automorphic quotient. These extra assumptions are a orded in order to make these di erent groups amenable to the same successful methods used for quaternion algebras, and to underline new challenges arising from these new settings as well as to formulate conjecture and provide rst results towards arithmetic statistics on the universal family of these groups. The main di erences lie into the lack on functoriality of the conductor and the necessity to use a theory of local newform for non-split places: that was not the case for quaternion algebras since they were only nitely many.

Universal Family for Quaternion Algebras

Establishing arithmetic statistics on the family of all automorphic representations of a group G requires to understand as sharply as possible the structure of the spectrum. It is necessary to order the spectrum introducing a notion of size, to be able to truncate it into a nite set and give a meaning to the sought statistics. Moreover, a choice of parameters indexing the spectrum has fundamental impact on the treatment of the problem, for it induces a decomposition of the spectrum in more or less handable subfamilies of xed parameters. This chapter settles the groundwork, introducing the automorphic representations and providing the chosen notion of size: the analytic conductor.

The whole strategy for estimating arithmetic statistics on automorphic objects consists in interpreting the sough quantity in terms of a spectral side of the trace formula. A fundamental step is to be able select these new sets as spectral sides: suitable selecting functions are constructed in the last sections.

Odds and ends

Modern analytic number theory makes large use of methods coming from representation theory, measure theory and harmonic analysis, requiring to endow the considered algebraic objects with the relevant structures. Once done, the automorphic groundwork mentioned in the introduction is detailed in this section, in order to properly state the problems.

Number theoretic landscape

Number and local elds

The aim of number theory is to explore the structure and the properties of number elds, that is to say nite extensions of Q. Consider such a number eld F , and let its ring of integers be denoted by O.

The choice of an absolute value on F allows one to embed it in the corresponding completion, endowing F with a well-behaved analytic structure. For a given absolute value , the completion of F with respect to is denoted F . It is a locally compact space, hence a local eld.

Di erent absolute values can give rise to isomorphic completions, case in which they are said to be equivalent. In order to get rid of this redundancy, the suitable notion is the one of equivalence class of absolute values, called place and still denoted by . The places of F are classi ed into nite places and archimedean places. Finite places give rise to non-archimedean local elds, and are parametrized by prime ideals of O; the remaining ones are archimedean places, parametrized by conjugacy classes of complex embeddings. The di erent completions of F are indexed by its places. From now on only places will be mentioned, all the corresponding notions being de ned as the ones related to any absolute value belonging to the place.

Finite places are traditionally denoted by gothic letters p, q, r, etc. These are exclusively used for ideals of O, and p speci cally for prime ideals. The norm of an ideal q is denoted N q, it is the cardinality of the eld O/q, also called residue characteristic. For a nite place p, let w p a uniformizer of p, i.e. a generator of p.

Adeles

Non-equivalent places give rise to non-isomorphic completions, so there is no way to complete F in a canonical fashion. However, F embeds in a locally compact ring taking in account all its completions. This procedure is provided by the ring of adeles. It is the breakthrough of Tate's thesis [119], allowing to enjoy properties similar to completeness without having to arbitrarily choose one place, so that no information attached to the base number eld F should be lost. This opens the path to harmonic analysis on number elds.

De nition 5. The ring of adeles A of F is the restricted product of the completions of F with respect to their associated rings of integers. More precisely, A = (x ) ∈ F : for almost every , x ∈ O .

(2.1)

Every adele can then be written as an element of the full product F . Similarly, some subsets X of the adeles can be decomposed as a product X . The X 's are called the local components of X . Given a set of places S, introduce the S-part and the prime-to-S part of a subset X , de ned respectively by

X S = ∈S X and X S = S X . (2.2)
In particular, the sets admitting a local decomposition as above decomposes as X = X S X S . For the case of a singleton X = {x }, the notation is lightened to x = x S x S . The whole idea of the ring of adeles, besides grasping all the completions at once, is to keep the possibility to enjoy the completeness of each local eld. Naturally this decomposition holds as it stands for any partition of the places in more than two sets, and allows to lift a local behavior at S to a global one relative to the ring of adeles.

The ring of adeles is endowed with the restricted product topology. It is de ned by the basis of open sets

U S × S O , (2.3)
where S runs through nite sets of places and U S is an open set of F S endowed with the product topology. This topology makes A a locally compact topological ring. There is a discrete and cocompact embedding of F in A, the so-called diagonal embedding

F -→ A x -→ (x) (2.4)
Remark. An important fact justifying the choice of the restricted product instead of the full product is that the product of local rings F would have failed to be locally compact. Moreover, since the product topology is stronger than the product one, this choice leads to strengthen density properties and are suitable to the purposes of proving equidistribution results.

Theorem 7 (Strong Approximation). Suppose that G is simply connected, in the sense that the topological space G(C) is simply connected, and that G (F S ) is noncompact for every simple factor G of G over F . Let K S be a compact open subgroup of G(A S ). Then

G(A) = G(F )G(F S )K S .
(2.5)

Moreover, if G (F S ) is noncompact only for every simple quotient G of G over F , then the following set of double cosets is nite:

G(F )\G(A)/G(F S )K S .
(2.6)

This property formalizes the very spirit of the adeles as announced above: embedding F in a better world without loosing its identity. The decomposition (2.5) holds a way back to the F -points of G from properties of its adeles points, justifying the adeles setting as the ground on which modern theoretic problems are handled.

Haar measures

Besides the completeness assets provided by the adelic setting, groups and representations involved in the exploration of the automorphic world are also handled by measure theory tools. This paragraph settles the minimal setting to precisely formulate the problems and state the results, further details on measures on the dual groups are provided in Section 2.3.

Prior to any choice of measure on groups or representations, it is needed to endow underlying global and local elds with measures. A right (resp. left) Haar measure is a positive Radon measure invariant by the right action of G, that is to say d (x ) = d x (resp. d ( x) = d x) for every in G. It is unique up to multiplication by a scalar. A group for which left and right measures are the same is called unimodular. This is the for instance the case for compact groups, discrete groups, abelian groups, connected reductive groups or semisimple Lie groups. For a non-unimodular group with a right Haar measure d , the modular quasicharacter is δ G (h) = d (h )/ d measures the failure of G to be unimodular. Turning to locally compact groups G, it is necessary to endow the group of points G(F ), G(A) and G(F ) with a compatible notion of measure with the one chosen on the base local rings. Following Hahn and Getz's presentation [55] automorphically intended, let n be the dimension of G over F . There is a unique nonzero top-dimensional left-invariant di erential form ω ∈ n g up to scalar in F × . From this di erential form follows through localizing a Radon measure on G given by

On archimedean elds, the

C c (G ) -→ C f -→ ∫ G f d |ω |
This provides a left Haar measure on G since ω is already a left Haar measure on G . Precise construction of those measures are provided in Knapp [74] for the archimedean case and in Oesterlé [97] for the non-archimedean one. For each , x the Haar measure normalized so that the maximal compact subgroup K is given measure one. The group of adelic points is then endowed with the product measure. Since G(F ) embeds discretely in G(A), these choices of measures induce a quotient measure on G(F )\G(A).

Automorphic world

Automorphic representations

Let G be an a ne algebraic group over a global eld F . The group G(A) of adelic points is locally compact and hence admits a Haar measure. Following the discussion above, the quotient G(F )\G(A) is endowed with a Haar measure giving it nite volume if and only if G(A) = G(A) 1 , motivating the introduction of the automorphic quotient

[G] = G(F )\G(A) 1 .
(2.7) This is the base ground on which automorphic theory takes place. The group G acts on the associated Hilbert space L 2 ([G]) by the right regular action

• ϕ = ϕ(x ), ∈ G, ϕ ∈ L 2 ([G]).
(2.8)

De nition 6. An automorphic representation of G is an irreducible unitary representation π of G(A) that is isomorphic to a subquotient of the right regular action on L 2 ([G]). The set of all the in nite-dimensional automorphic representations of G is denoted A(G) and called the universal family of G.

Remarks. This de nitions might seem non-standard, leading to the following remarks.

(i) This de nition does not require to introduce the admissibility condition, as is the standard way to do so [52], and is more suited for e cient introductory purposes. Both notions coincide [55].

(ii) The choice of excluding nite-dimensional representations from the universal family, that is to say characters in the case of general linear groups, is made so that it embeds as a subfamily of the cuspidal automorphic representations of GL(n). For statistical purposes, this choice causes no trouble, for the number of characters of bounded conductor is negligible compared to the size of the universal family.

Automorphic representations are one of the most celebrated objects of modern number theory, and the present thesis addresses arithmetic statistics problems relative to them in di erent settings.

Tensor product theorem

Despite automorphic representations are rather mysterious objects, some of them are particularly easier to grasp. Given a maximal compact subgroup K of G , a unitary representation of G is unrami ed with respect to K if it admits non-zero K -xed vectors, rami ed otherwise.

In the same way as the ring of adeles, objects relative to local elds are expected to be easier than global objects, and there is hope that a local-global principle allows to work on local components instead of on the whole global object. Fortunately, this is the case for automorphic representations by the following structure result.

Theorem 8 (Flath). Every automorphic representation π ∈ A(G) decomposes as a restricted product π = ⊗ π where π is an irreducible unitary representation of G and is unrami ed for almost every place .

Quaternion algebras

After having introduced what automorphic forms are, the time has come to choose the landscape they live on. This section is devoted to do so, presenting the quaternion algebras and their structure before turning back to the associated automorphic quotient. The settings of some unitary and symplectic groups are mentioned in Chapter 5.

First de nitions

A quaternion algebra over a eld F is a central simple algebra B of dimension 4 over F . Dickson proved in the early 1900s [125, Chap. IX, Theorem 1] that this de nition generalizes the usual Hamiltonian quaternions and admits a familiar representation by generator and relations.

Proposition 1. If the characteristic of F is not 2, then B admits a basis (1, i, j, k) over F such that for some a and b ∈ F × ,

i 2 = a, j 2 = b, ij = k = -ji.
(2.9) Such a quaternion algebra is denoted D F (a, b), and is entirely determined by a and b up to isomorphism. It can be embedded in a set of matrices over an extension of degree at most two.

Proposition 2. If α is a root of X 2a in an extension of F , then the following is an F -algebra embedding into F (α)-matrices.

D F (a, b) -→ M 2 (F (α)) t + xi + j + zk -→ t + xα b( + zα) -zα t -xα
A detailed account of the theory of quaternion algebra is established from an algebraic point of view in Vignéras' work [121], as well as quite comprehensively covered in Voight's book [122].

Structure of quaternion algebras

Let B be a quaternion algebra over a global eld F . For a given place , denote by B the group of points B(F ). The local group B is isomorphic to either M 2 (F ), case in which is split, or to a division quaternion algebra, case in which is rami ed. For a given local eld F , there is a unique division quaternion algebra over F up to isomorphism. Quaternion algebra are classi ed up to isomorphism by their rami cation places [121].

Proposition 3. Let B be a quaternion algebra over F , and R its set of rami cation places. Then R is a nite set of even cardinality. Furthermore, it determines B up to isomorphism.

This structure theorem states a strong local similarity with GL(2), for completions of a given quaternion algebra are almost everywhere the group of points of GL(2). This fact appeals two comments. First, while a nite number of places will need a treatment speci c to the division quaternion algebras setting, most of them will borrow methods and results from the GL(2) setting. Second, results obtained for global quaternion algebras are expected to bear information concerning GL(2) automorphic forms.

From now on, consider a quaternion algebra B over F , and write R for the places of F where B is not split. Introduce G = Z \B × , where Z denotes the center of B × .

Inner forms

The groups of units of quaternion algebras is not merely similar to GL(2) at some places, but are its inner forms. More precisely, quaternion algebras is isomorphic to M(2) over an algebraic closure of F and the underlying isomorphism is given by a conjugation. Moreover, this construction exhausts all the inner forms.

Automorphic quotient

Automorphic representations live in the automorphic quotient G(F )\G(A). In the case of division quaternion algebras, a fundamental property holds and allows the use of results unavailable for GL(2).

Analytic conductors

Proposition 5. The automorphic quotient B(F )\B(A) is compact modulo the center.

Proof. This is the Hey theorem, quoted in Voight [122,38.4.3].

Jacquet-Langlands correspondence

The Jacquet-Langlands correspondence [66] provides an embedding of the representations of a division quaternion algebra into representations of GL(2). It is stated here in the centerless setting.

Theorem 9 (Local Jacquet-Langlands). Let be a place where B rami es. There is a bijection between irreducible smooth representations of G and irreducible discrete series representations of PGL(2, F ).

Theorem 10 (Global Jacquet-Langlands). There is a unique bijection between in nite dimensional automorphic representations of G(A) and irreducible cuspidal automorphic representations of PGL(2, A) that is compatible with the local Jacquet-Langlands correspondence.

Analytic conductors

Once automorphic representations introduced, it is necessary to make sense of the counting problem. In order to determine the actual size of the universal family and some sharper statistical properties, as densities or equidistribution, it is needed to truncate it to a nite set. This section explores the notion of size provided by the analytic conductor.

The analytic conductor is an intrinsic notion of size grasping the complexity of automorphic representations. There are di erent standard constructions, either based on representation theoretic properties or using speci c invariants attached to automorphic L-functions. While some appear as more natural, others turn to suit more e ciently the purposes and methods of arithmetic statistics questions. Besides introducing some of the frequent de nitions of the analytic conductor appearing in the literature, their consistency and soundness are discussed in the following paragraphs.

Conductor arising from functional equations

Let turn back for a moment to a more usual setting: the universal family A(G) embeds, via the Jacquet-Langlands correspondence, see Theorem 10, as a subfamily of the universal family A(PGL(2)), composed of all the cuspidal automorphic representations of PGL(2). In this latter context, even in the broader setting of cusp forms on general linear groups, Iwaniec and Sarnak [65] have de ned a good notion of size, given by the analytic conductor. It is a real number c(π ) de ned from the functional equation satis ed by the nite part L-function L(s, π ) associated to a generic π ∈ A(PGL(2)), which takes the form

L(1 -s, π ) = ε π X (s, π )L(s, π ), (2.10)
where ε π is the root number of π and takes value 1 or -1, since π is self-dual. The quantity |X (s, π )| takes value 1 at the central point 1 2 , leading to the de nition of the analytic conductor following Conrey et al. [30].

De nition 7. Let π be a generic automorphic representation on PGL(2). With the completing factor introduced in (2.10), the L-analytic conductor of π is

c(π ) = X 1 2 , π .
(2.11)

The functional equation relates the value of an L-function at a point s with its value at the symmetric point 1s. The easiest case is naturally the symmetric one, i.e. when X (s, π ) = ε π , corresponding to a conductor equal to zero. This leads to interpreting X (s, π ) as a measure of the failure of L(s, π ) to be symmetric. The function X (s, π ) is built from the factors necessary to complete L(s, π ) to get a symmetrical functional equation, and involves the usual arithmetic conductor as well as archimedean gamma factors, so that the analytic conductor encapsulates the complexity of π .

It allows to truncate the universal family of PGL(2) into

A(Q) = {π ∈ A(GL(2)) : c(π ) Q }, Q 1. (2.12)
This set is known to be discrete and nite by the work of Brumley [16]. Even if the analytic conductor is a relevant notion of size satisfying the needed niteness property, it is a rather impenetrable quantity, as is the completing factor X (s, π ).

Opening the ε-factor

The de nition of the analytic conductor provided above is general and e cient in its formulation, yet it is far from easily reachable in practice. As announced in the introduction, despite the niteness of the truncated family, there is no reason for it to be more handable than the whole universal family, for this analytic conductor appears as a mysterious parameter and introducing it may seem to be an unnecessary complication. This is far from the case: the precise knowledge of the structure of the automorphic L-functions and their associated functional equations leads to a more explicit formulation of the conductor in terms of di erent spectral parameters, that turn out to be well suited for trace formula treatment.

The de nition of the analytic conductor is summarized by Conrey, Farmer, Keating, Rubinstein and Snaith [29]. The Selberg class is an axiomatic de nition of what an Lfunction should be in general. In particular, there is a quantity called the ε-factor which 2.2. Analytic conductors is a way to make L(s, π ) entire, for π ∈ A(GL(2)). It is of the form

ε(s, π ) = Q s π |∞ Γ (s + µ π ( )). (2.13)
where Q π is a positive real number and runs through the archimedean places of F . Moreover, Γ is equal to Γ R (s) = π -s/2 Γ(s/2) in case of a real place, and to Γ R (s)Γ R (s + 1) in case of a complex place. This ε-factor is such that the completed function

ξ (s, π ) = ε(s, π )L(s, π ), s ∈ C, (2.14)
is entire and satis es the symmetric functional equation

ξ (s, π ) = ε π ξ (1 -s, π ), s ∈ C. (2.15)
Following the presentation of Iwaniec and Sarnak [65], the analytic conductor is de ned as follows from the data of the ε-factor.

De nition 8. Let π be a generic automorphic representation on PGL(2) and introduce the associated ε-factor (2.13). The ε-conductor of π is

c ε (π ) = Q π (1 + |µ π ( )|) . (2.16)
These de nitions of the analytic conductor appeal quite a few caveats concerning their compatibility. Both notions c ε and c de ned above does not coincide, as straightforward computations show. Indeed, comparing the functional equation satis ed by ξ (s, π ) with the de nition of X (s, π ) in the completion (2.10), it follows that

X (s, π ) = ε(1 -s, π ) ε(s, π ) .
(2.17)

The explicit de nition of the gamma factors (2.13) yields

log(ε) (s) = log Q π + log |∞ Γ Γ (s + µ π ( )) , (2.18)
so that both de nitions are equivalent up to the approximation of replacing the digamma function Γ /Γ by the logarithm. This is not an equality yet a standard approximation for small values of s, in particular around the central point, provided by the Stirling formula. At last, both de nitions di er by constants and normalizing factors. The so-called arithmetic part Q π of the conductor is a well-de ned and non ambiguous notion, it is always present as it stands in any de nition of the analytic conductor. However this is not the case for its archimedean part. It has to be underlined that the notion of archimedean conductor is a working de nition intended to reasonably grasp the complexity of an L-function to govern some of its statistical behavior. Some attempts have recently been made, for instance in unpublished works due to Paul Nelson or Peter Humphries, to nd a more canonical way to de ne the archimedean conductor, motivated by theories of local newforms, yet for now they fail to match the expected behavior of the analytic conductor.

The fundamental property of the conductor is the niteness of the truncated universal family as well as its relations with invariants attached to L-functions. Hence, it allows some freedom in the choice of certain normalizing factors, for instance to ensure non-vanishing, see Chapter 4. This feature is not present in deepest problems concerning L-functions, typically subconvexity questions where the choice of the archimedean analytic conductor might be critical. There is no need to settle this debate here, and the methods borrowed from [17] still hold for any of the choices mentioned above, and for a more general class of size functions.

Conductor as a notion of depth

A more geometric interpretation of this notion of conductor, where the structure of the underlying group appears, would give a better computational grasp on the conductor. This is provided by the so-called theory of local newforms. This section is devoted to B × more than to G, for it lightens notations. This local convention makes no harm, for a representation π of G(A) is viewed as a representation of B × (A) with trivial central character. By Flath's theorem, an irreducible admissible representation of B × (A) decomposes in a unique way as a restricted tensor product π = ⊗ π of irreducible smooth representations where almost every component π is unrami ed. It is hence natural to de ne rst the conductor for the local components π . Setting c(π p ) = 1 for the nite unrami ed components guarantees well-de niteness of the product over all places and is required to get the consistency with the conductors de ned above. The aim of the present section is to de ne the notion of conductor for the remaining local components.

Split local components

Proposition 6. Let π ∈ A(G) and a split place. The local component π is in nitedimensional.

Proof. Since the universal family excludes global characters, a representation π in it is generic. The Jacquet-Langlands correspondence preserves genericity, hence as shown on the diagram below, the global Jacquet-Langlands correspondence associates to it a generic representation JL(π ) of GL(2), thus also its local components JL(π ) . These local components are also the images by the local Jacquet-Langlands corres-pondence JL(π ) of the local components of π .

π ∈A(B × ) JL / / JL(π )∈A(GL(2)) generic π JL id if R / / JL(π ) generic
At split places, the local Jacquet-Langlands correspondence is the identity, for then B × p GL(2, F p ). Moreover, the correspondence is unique, thus the local components π , at split places, are generic hence in nite-dimensional. This in particular implies that local components being a character can only arise at rami ed places. First of all the focus will be on split places, before turning to the rami ed places by the pullback procedure to the split ones.

Non-archimedian split case

K 0,p (p r ) = ∈ GL 2, O p : ≡ 0 mod p r ⊆ B × p , r 0. (2.19)
This sequence is a ltration, i.e. a decreasing sequence of subgroups. Since the representations considered are smooth, the existence of xed vectors for a small enough subgroups is guaranteed. The conductor of an irreducible admissible in nite-dimensional representation π p of B × p with trivial central character is then de ned by the smallest rank for which it happens.

De nition 9. The additive conductor of π

p ∈ G p is f(π p ) = min r ∈ N : π K 0,p (p r ) p 0 , (2.20)
and the multiplicative and analytic conductor of π p are respectively de ned by c(π p ) = p f(π p ) and c(π p ) = Nc π p .

(2.21)

The existence of the conductor is guaranteed by the work of Casselman [21], who also states that the growth of the dimensions of the xed vector spaces are given by dim π K 0,p( p f(πp)+i )

p = i + 1, i 0. (2.22)

Global analytic conductor

De nition 10. Let π be an automorphic representation of GL(2), and write π = ⊗ π for its tensor product decomposition. Its global analytic conductor is de ned to be

c(π f ) = c (π ) . (2.23)
This gives a well-de ned notion of conductor, for the π are almost everywhere unrami ed. One of the fundamental facts of the theory of local newforms is the consistency with the de nition of the arithmetic conductor, i.e. the depth-conductor notion de ned in (2.21) and the the one coming from ε-factors (2.13) of the associated L-functions are compatible in the following way.

Proposition 7. Let π ∈ A(GL(2)). The notions of arithmetic conductor given in (2.23) is compatible [66] with the one coming from L-functions (2.11), that is to say

p c(π p ) = Q π .
(2.24)

Conductor of characters

Note that for now conductors have been de ned only for generic representations. However, characters can arise as local component at rami ed places as discussed above. Every character of B × p is a composition

B × p -→ F × p -→ C, (2.25)
where the rst application is the reduced norm, and the second one a character of F × p . In other words, every character of B × p is of the form χ 0 •N where χ 0 is a character of F × p . In order to stay consistent, de ne the conductor of a local character at a rami ed place as the conductor of its Jacquet-Langlands embedding in PGL(2). Since the character χ 0 • N is sent on the twisted Steinberg representation St ⊗ χ 0 , it follows

c(χ 0 • N ) = p if χ 0 unrami ed; c(χ 0 ) 2 if χ 0 rami ed.
(2.26)

Analytic conductor for quaternion algebras

The notion of analytic conductor de ned above for automorphic representations of GL(2) extends to a de nition for representations of G, viewed as automorphic representations of B × with trivial central character. Indeed, following the pullpack procedure for heights, de ne for a local representation at a rami ed place its conductor as the one of its Jacquet-Langlands transfer.

De nition 11. Let π ∈ G . Its local conductor is de ned by c(π ) = c(JL(π )).

(2.27)

Since for both split and rami ed places the conductor of a local representation of G is de ned, the global conductor can be introduced.

De nition 12. Let π be an automorphic representation of G, and write π = ⊗ π . Its global analytic conductor is de ned to be

c(π ) = c(π ), (2.28)
which is well-de ned, for almost every component π is unrami ed.

Remarks. The choice of de ning the conductor at rami ed place in this roundabout way seems far from natural and appeals for some comments.

(i) The de nition of the conductor (2.11) coming from the completing factor of associated L-functions is a systematic and intrinsic way to de ne the conductor of a generic representation of a group G. This is provided by the Godement-Jacquet [56] construction of L-functions, making possible the de nition of the εconductor without appealing to an embedding in GL(n). The Jacquet-Langlands correspondence stated in Theorem 10 preserves the notion of L-function and hence also makes the inner notion of ε-conductor for G compatible with the one de ned for GL(2). Since the analytic conductor is by de nition compatible between G and GL(2), this choice makes no harm compared to directly de ning the conductor from the associated L-functions on G.

(ii) There is a candidate introduced by Lansky and Raghuram [82] providing a ltration of subgroups of G that are roughly of the same size as those introduced in (2.19). However, it is not known whether or not the attached notion of depthconductor is the same.

Since the notion of size has been properly de ned and motivated, it becomes possible to precisely state the problem and to consider the truncated universal family

A(Q) = {π ∈ A(G) : c(π ) Q }, Q 1.
(2.29)

Equidistribution

The way a family localizes in a given measurable space is formalized in the notion of equidistribution with respect to a measure. It is a weak convergence against a certain class of functions. The limit measure expresses the probability of an element of the family to appear in a given region. This section is dedicated to introduce some elements of equidistribution and then restate Theorem B as a weak convergence against wider and more handable class of functions.

Elements of equidistribution

The whole equidistribution question consists in deciding whether the proportion of elements of a family lying in a given zone of the space tend to the measure of that zone. In the case of the universal family, for every open relatively compact set U with measure zero boundary in the unitary dual, Theorem B aims at estimating the proportion of automorphic forms lying in U , that is to say

# {π ∈ A(Q) : π ∈ U } N (Q) , Q 1. (2.30)
This statistic question su ers from the lack of individueal knowledge of automorphic forms. The measure representing the distribution of the elements in the truncated universal family is the counting measure

µ Q = 1 N (Q) π ∈A(Q) δ π , Q 1.
(2.31)

The very same idea as the Weyl criterion for equidistribution of sequences modulo 1, or more generally Portmanteau theorems, applies to this setting and leads to restating the equidistribution (2.30) in a functional way. For a measure ν on G = G , recall that

ν (f ) = ∫ Π f (π ) d ν (π ),
(2.32)

for functions such that the expression above has a meaning. So that Theorem B states that µ Q (f ) converges to µ(f ) for every f among the characteristic functions of relatively quasi-compact open sets of Π with thin boundary.

The whole point in choosing a functional formulation is to enlarge the scope of the test functions involved in (2.30) to see them as part of a wider but better-behaved space of functions. Let S be a nite set of places of B. De ne F ( G S ) to be the space of complex Plancherel-measurable and bounded functions on G S supported on a nite number of Bernstein components and whose restriction to the tempered spectrum is continuous outside a measure zero set.

Proposition 8 (Sauvageot). For every relatively compact open set U with thin boundary in G S , the characteristic function 1 U belongs to F ( G S ).

A sequence (ν n ) n of Radon positive measures on Π weakly converges to a measure ν if ν n (f ) converges to ν(f ) for every f ∈ F ( G S ) when n goes to in nity, for every nite set S of places. Since the considered characteristic functions lie in F ( G S ), weak convergence of µ Q to µ implies Theorem B.

From now on, in order to avoid the expression of N (Q), the measure considered is slightly modi ed to be

ν Q = 1 Q 2 π ∈A(Q) δ π , Q 1. (2.33)
This is motivated by the fact, from Theorem A, that N (Q) is of asymptotical order CQ 2 , so that Theorem B is equivalent to: ν Q weakly converges to the measure

ν = C µ µ = 1 2 vol(G(F )\G(A))µ.
(2.34)

Plancherel formulas & Fourier transformation

Technical tools suitable to deal with the action (2.32) of spaces of functions are provided in this section. The action of automorphic representations is extended to functions on G by

π (f ) = ∫ G f ( )π ( ) d , π ∈ A(G), f ∈ C c (G). (2.35)
This de nes a trace class operator, and allows to de ne its Fourier transform

f (π ) = tr π (f ), π ∈ A(G), f ∈ C c (G).
(2.36) Speci c choices of subclasses of functions, on which the automorphic representations are considered acting on, lead to better properties of the corresponding operators, as in the following proposition. Proposition 9. Let K be a compact open subgroup of G and π an automorphic representation of G. For every left-K-invariant f in C c (G) and every x in the representation space V π of π , the image π (f )x is K-invariant.

Proof. For every x ∈ V π and k ∈ K, the left-K-invariance of f yields

π (k) (π (f )x) = ∫ G f ( )π (k )x d = ∫ G f (k -1 )π ( )x d = π (f )x
proving the claim.

Of particular interest are the Hecke algebras. For a nite place , the Hecke alegbra H (G ) is the convolution algebra of complex valued, locally constant and compactly supported functions on G . For an archimedean place , the Hecke alegbra H (G ) is the convolution algebra of complex valued, smooth and compactly supported functions on G . The global Hecke algebra is denoted H (G(A)) or H (G), and is the algebra generated by the restricted products ϕ = ϕ , where ϕ is a function of H (G ) and almost every local component ϕ p equals to 1 K p .

The unitary dual group G is endowed with the Fell topology. The Plancherel measure associated to it is the unique positive Radon measure µ Pl on G such that the Plancherel inversion formula of Harish-Chandra holds, i.e.

∫ G ϕ (π ) d µ Pl (π ) = ϕ (1), ϕ ∈ H (G ).
(2.37)

From now on, every integral on G will be written with the convention that d π stands for d µ Pl (π ), leading to no ambiguity.

Sauvageot density theorem

In spite of the class F ( G S ) of test functions being wider than mere characteristic functions, it has a surprusingly good analytical behavior. Indeed, the Sauvageot density theorem [111] states that any function in F ( G S ) can be approximated by Fourier transforms of functions in the Hecke algebra of G S .

Theorem 11 (Sauvageot). For every f ∈ F ( G S ) and ε > 0, there exist functions ϕ and ψ in the Hecke algebra H (G S ) such that

∀π ∈ G S , f (π ) -ϕ(π ) ψ (π ) µ Pl S ( ψ ) ε
Thus, in order to prove the convergence of ν Q (f ) to ν(f ) for every function f in F ( G S ), it is su cient to prove it for such Fourier transforms. Indeed, let f ∈ F ( G S ). For ε > 0, Sauvageot's theorem guarantees the existence of ϕ and ψ in the Hecke algebra H (G S ) such that ϕ and ψ satisfy the conditions above. Thus

|ν Q (f ) -ν (f )| |ν Q (f ) -ν Q ( ϕ)| + |ν Q ( ϕ ) -ν ( ϕ)| + |ν ( ϕ ) -ν (f )| ν Q ( ψ ) + |ν Q ( ϕ ) -ν ( ϕ )| + ν ( ψ ) |ν Q ( ψ ) -ν ( ψ )| + 2ν( ψ ) + |ν Q ( ϕ ) -ν ( ϕ )|
From the de nition of ν and the domination in the Sauvageot theorem, it follows since conductors are at lmeast one that

ν( ψ ) ζ (1) ζ (1) -1 ∫ G ψ (π ) d π c(π ) 2 ζ (1) -1 ∫ G ψ (π ) d π < µ Pl S ( ψ ) ε Thus, |ν Q (f ) -ν(f )| ε + |ν Q ( ψ ) -ν ( ψ )| + |ν Q ( ϕ ) -ν ( ϕ )|.
(2.38)

In order to prove that ν Q weakly converges to ν, it is then su cient to show that the second and third terms vanish for Q → ∞, i.e. to prove the theorem for such functions ϕ and ψ . A sharper result than what is needed for Theorem B can be proven, with a precise remainder term in the case of Fourier transforms.

Theorem E. For every nite set of places S and ϕ ∈ H ( G S ),

ν Q ϕ = ν ϕ +          O Q -1+ε if F = Q and B totally de nite; O Q -δ F if F Q and B totally de nite; O 1 log Q if B is not totally de nite.
Remark. The underlying constants depend on ϕ. This is not a problem since only convergences matter for Theorem B. It follows that Fourier transforms selects representations of bounded conductor.

Admissible functions

Proposition 10. Let ϕ ∈ H (G S ). There exists c ϕ > 0 such that for every generic π ∈ G S in the support of φ, the conductor of π is less than c ϕ .

Proof. Since S is a nite set of places it is su cient to prove the result for a local component. In the case of a nite place p, let ϕ p be the p-component of ϕ, where p ∈ S. The property (ii) of the Trace Paley-Wiener theorem states that its Fourier transform ϕ p is dominated by a certain open compact subgroup K of G p , that is to say is supported on representations π p having nontrivial xed space π K p . Since K is open and the sequence (K 0,p (p i )) i is a ltration in G p , K contains a certain conjugate of a K 0,p (p r ) and hence ϕ p is nonzero only for representations of conductor dividing p r . Since S contains only a nite number of places, this proves that ϕ selects only representations π S ∈ G S with conductor dividing the product of the corresponding p r . However, bounding the conductor is not enough for the purposes of the trace formula, thus some modi cations on ϕ are necessary. However, it is far from obvious that such modi ed functions are still approachable by Fourier transforms. In order to select automorphic representations in the universal family through trace formula methods, it is necessary to restrict the Fourier transforms considered to the generic spectrum, for otherwise there is no notion of conductor attached to a representation. The following proposition states that it is possible, up to another approximation by density.

Proposition 11. Let φ be the restriction of φ to the generic spectrum, extended by zero elsewhere on the unitary dual. Then φ lies in F ( G S ).

Proof. Recall that the Sauvageot density theorem, stated in Proposition 8 provides a criterion for functions to be approachable by Fourier transforms. All the properties of the Sauvageot class F ( G S ) obviously hold for φ safe possibly the condition on the discontinuity points. The proof is done by the explicit classi cation of the representations on PGL(2) on local elds. Since φ is supported on a nite number of Bernstein component, it is possible to assume it is supported on only one component with no loss of generality.

For archimedean places, the only unitary non-generic representation of PGL(2, R) is the trivial representation, and it is of zero Plancherel measure. The Steinberg representation also lies in the boundary of the generic spectrum. Since the Steinberg representation has positive Plancherel measure, it is not a discontinuity point of φ, otherwise it would already have been a discontinuity point of φ, what is incompatible with the Sauvageot conditions. The cases of PGL(2, C) and PGL(2, F p ) should be treated similarly.

The conductor of a representation in the generic spectrum is well-de ned. It is also necessary to restrict the functions to xed conductors, what has a meaning since Proposition 11 allows to restrict to the generic dual. This is the meaning of the next proposition.

Proposition 12. Let q be a an integer ideal. Let φ be the restriction of φ to representations of xed conductor q, extended by zero elsewhere. Then φ lies in F ( G S ).

Proof. Since the support of q contains only a nite number of places, it is enough to prove the proposition for the restriction to conductors p r . This is also possible by the explicit classi cation of the dual of PGL(2, F ).

Universal family

The truncated universal family has to be explored further. A re ned decomposition of A(Q) using the speci c behavior at the di erent places is possible. The explicit notion of conductor given through the depth notion associated to the ltration (2.19) is suitable to treat the corresponding split components of an automorphic representation. Rami ed and archimedean components need speci c treatment yet arise in a nite number of places. This section supply the necessary toolbox, following the presentation of Finis, Lapid and Müller [43] as well as of Brumley and Milićević [17].

Archimedean Langlands classi cation

The local Langlands classi cation of the archimedean admissible dual [73] of GL(2) provides a recipe for constructing the admissible representations of reductive groups over archimedian local elds in terms of representations of Levi subgroups. Since unitary representations are in particular admissible, it induces a parametrization of the unitary dual of GL(2, F R ∞ ). Let L ∞ the nite set of Levi subgroups of GL(2, F R ∞ ) containing the diagonal torus. For such a Levi M, de ne E 2 (M 1 ) to be the set of isomorphism classes of square integrable representations of M 1 . The only nonempty cases are • E 2 (GL(1, R) 1 ) consisting into the trivial character and the sign character;

• E 2 (GL(1, C) 1 ) composed by the characters z k /|z| k for integers k;

• E 2 (GL(2, R) 1 ) that is the set of discrete series representations of weight k 2. Introduce the set D of G R ∞ -classes of conjugation of pairs δ = (M, δ ) with M ∈ L ∞ and δ ∈ E 2 (M 1
): they constitute the discrete spectral data parametrizing the archimedean sprectum. Write h M,C for the trace-zero hyperplane of the complexi ed dual of the Lie algebra of M, which is a nite-dimensional C-vector space. The spectral data δ ∈ D, consisting of a Levi M ∈ L ∞ and a discrete series representation δ ∈ E 2 (M 1 ), along with ν ∈ h M,C , give rise to an admissible representation of G R ∞ in the following way. The unitary induction Ind G P (δ ⊗e ν ) from the unique parabolic subgroup P containing M is not necessarily irreducible, yet the following holds.

Proposition 13 (Archimedean Langlands classi cation). Let δ ∈ D and ν ∈ h M,C , denote W δ the stabilizer of δ in the Weyl group of h M . There is a unique ν in the class of ν modulo translation by W δ such that the induction Ind G P (δ ⊗ e ν ) admits a unique irreducible quotient, denoted by π δ,ν . Moreover, every admissible irreducible representation of GL(2, F R ∞ ) arises uniquely in this way, up to in nitesimal equivalence.

This construction exhausts the admissible dual of GL(2, F R ∞ ) up to in nitesimal equivalence. This is the archimedean Langlands classi cation [72, Theorem 8.54], that can be reformulated as

G R,1 ∞ M∈L ∞ E 2 (M 1 ) × h M,C /W , (2.39)
where G R,1 ∞ stands for the admissible dual of GL(2, F R ∞ ) up to in nitesimal equivalence. Note that D is a discrete set, leading to refer to δ as the discrete archimedean spectral parameter of π , or the discrete parameter of π R ∞ , while ν in h M,C /W is called the continuous archimedean parameter of π .

Decomposition

In order to address the problem of the weak convergence of ν Q to prove Theorem E, it is necessary to decompose the universal family into smaller sets with xed spectral data, amenable to trace formula methods. Let S be a nite set of places and ϕ ∈ H (G S ). The conductor of π ∈ A(G) splits into local conductors, and in particular it can be written c(π

) = c(π R )c(π R ∞ )c(π R S,f )Nc(π R,S f ).
(2.40)

This decomposition emphasizes the di erent kind of information and behavior each type of place is endowed with, and turns to be a guide for decomposing the counting measure ν Q ( φ) of the truncated universal family. Concerning the split archimedean places, introduce the truncated archimedean split dual

Ω(X ) = π R ∞ ∈ G R ∞ : c(π R ∞ ) X , X > 0. (2.41)
This set of archimedean parameters factorizes further through the precise Langlands classi cation recalled in Section 2.4.1, by xing discrete spectral parameters, so that Ω(X ) = Ω comp (X )

δ ∈D δ =(M,δ ) Ω δ (X ), (2.42)
where

Ω δ (X ) = π R ∞ ∈ G R ∞ : ∃ν ∈ ih M , π R ∞ π δ,ν , c(π R ∞ ) X Ω comp (X ) = π R ∞ ∈ G R ∞ : ∃ν ∈ h M,C \ih M , π R ∞ π ,ν , c(π R ∞ ) X
and the notation π ,ν stands for the existence of a δ ∈ D such that the representation is isomorphic to π δ,ν . The set Ω comp is called the complementary part of the archimedean spectrum, while the remaining part is the tempered part of the spectrum. This denomination is motivated by the fact that the representation π δ,ν is tempered if and only if ν lies in ih M .

Concerning the remaining places, recall that every ideal m is decomposed in the form m = m S m S , where such a decomposition always means that m S is the primeto-S part of m, i.e. is such that m S ∧ S = 1, and m S if the S-part of m, i.e. satis es supp(m S ) ⊆ S. The same decomposition is used without further notice for the other letters. The multiplicative conductor of the nite split places lying out of R is xed to a certain ideal q, and the isomorphism class of the rami ed part is xed to a certain σ R ∈ G R .

Recall from Proposition 12 that the function φ is so that the conductor of the Scomponent to be equal to a certain q S . Thus, the universal family admits the following decomposition according to (2.40) and the choices made above:

A(Q) = A comp (Q) N q Q q∧R=1 σ R ∈ G R c(σ R ) Q/N q δ ∈D δ =(M,δ ) A(q, σ R , δ, Ω), (2.43)
where the sets of xed spectral data are

A(q, σ R , δ, Ω) = π ∈ A(G) : π R σ R , c(π R f ) = q, π R ∞ ∈ Ω δ (Q/N qc(σ R )) A comp (q, σ R , Ω) = π ∈ A(G) : π R σ R , c(π R f ) = q, π R ∞ ∈ Ω comp (Q/N qc(σ R ))
and where Ω stands for Ω(Q/N qc(σ R )), convention used from now on to lighten notations.

The decomposition (2.43) of the universal family is critical for it reduces the study of the whole family to harmonic families, easier to grasp in the context of trace formulas. What is critical is to having got rid of the condition of belonging to A(Q), decomposed in local conditions. It induces a decomposition of the counting measure as

ν Q ( ϕ ) = 1 Q 2 π ∈A(Q) ϕ(π ) = 1 Q 2 π ∈A(G) c(π R )c(π R,S f )c(π R S ,f )c(π R ∞ ) Q ϕ(π ) = 1 Q 2 σ R ∈ G R c(σ R ) Q N q Q/c(σ R ) q∧R=1 δ ∈D δ =(M,δ ) π ∈A(q,σ R ,δ,Ω) ϕ(π ) + 1 Q 2 σ R ∈ G R c(σ R ) Q N q Q/c(σ R ) q∧R=1 π ∈A comp (q,σ R ,Ω) ϕ(π ) (2.44)
where the sum over q is meant to run through ideals of O R . The complementary part corresponds to the second sum appearing in the line above and will be dealt with later and shown to contribute as an error term. Denote A(q, σ R , δ, Ω; ϕ) the innermost part of the splitting in the rst summation above, that is to say

A(q, σ R , δ, Ω; ϕ) = π ∈A(q,σ R ,δ,Ω) ϕ(π ).
(2.45)

Old and new forms

The universal family (2.29) sees no multiplicities, but the trace formula counts them. The spectral multiplicities associated to the decomposition of L 2 (G(F )\G(A)), which are more suitable weights for the forthcoming computations, are given by m (π, q) = dim π K 0 (q) , (2.46)

where ZK 0 (q) = p r ||q Z p K 0,p (p r ) ⊆ B × A R f , (2.47)
and K 0 (q) stands for the image of ZK 0 (q) under the natural projection B × → G. The choice is made so that m(π, q) 0 is equivalent to c(π R f ) | q. The analogous sum to (2.45) additionally weighted by the multiplicities is

B(q, σ R , δ, Ω; ϕ) = π ∈B(q,σ R ,δ,Ω) m π S , q S ϕ(π ), (2.48)
where

B(q, σ R , δ, Ω) = π ∈ A(Q) : π R σ R , c π R f | q, π R ∞ ∈ Ω δ (Q/N qc(σ R )) . (2.49)
The sum de ned by (2.45) counts the newforms while (2.48) counts the old ones at nite split places out of S. The relation between them lies in the following lemma.

Lemma 1. Let q prime to R, σ R irreducible unitary representations of G R , δ ∈ D and ϕ ∈ H (G S ). Let λ 2 = µ µ where µ is the Möbius function. For every Q 1,

A (q, σ R , δ, Ω; ϕ) = d | q λ 2 q d B (d, σ R , δ, Ω; ϕ) . ( 2
= i + 1, i 0.
(2.51)

From this immediately follows, after taking the product over all nite split places, that the global multiplicities are

m (π, q) = τ 2 q c(π R ) , (2.52)
where τ 2 = 1 1 is the divisor function. Since π R K 0 (q) 0 implies c(π R ) | q, the sum de ning B(q, σ R , δ, Ω; ϕ) is eventually reduced to a sum over c(π R ) | q. Thus, by the precise knowledge (2.52) of the multiplicities,

B (q, σ R , δ, Ω; ϕ) = d | q π ∈A(d,σ R ,δ,Ω) τ 2 q c(π R f ) ϕ(π ) = d | q τ 2 q d π ∈A(d,σ R ,δ,Ω) ϕ(π ) = d | q τ 2 q d A (d, σ R , δ, Ω; ϕ)
(2.53) so that B = τ 2 A, with a slight abuse of notation. Hence, by Möbius inversion,

A (q, σ R , δ, Ω; ϕ) = d | q λ 2 q d B (d, σ R , δ, Ω; ϕ) , (2.54)
achieving the proof.

Summing over the spectral data appearing in the decomposition (2.44), the counting measure rewrites as

ν Q ( ϕ ) = 1 Q 2 σ R ∈ G R c(σ R ) Q N q Q/c(σ R )) q∧R=1 δ ∈D δ =(M,δ ) d | q λ 2 q d B (d, σ R , δ, Ω; ϕ) .
(2.55)

Spectral selection

The equidistribution property has been formulated as a convergence of spectral measures. The Selberg trace formula translates it as a purely geometrical quantity. (2.59)

Here π go through the isomorphism classes of unitary irreducible subrepresentations of G(A) in L 2 (G(F )\G(A)), and recall that Φ is the Fourier transform of Φ, see 2.3.2.

Remark. The formulaton of the spectral part (2.59) is Selberg's original one. The weights m(π ) are the multiplicities of the π 's in the discrete part of the spectral decomposition of L 2 (G(F )\G(A)). The multiplicity one theorem ensures these to be less than 1, and the indexation by π actually part of L 2 (G(F )\G(A)) makes them nonzero, hence equal to 1.

The admissible dual can be decomposed into tempered representations and nontempered representation. In view of (2.42) and anticipating that the selecting function at split archimedean places behaves di erently on the tempered spectrum and on the complementary one, it is natural to introduce the tempered and complementary spectral parts as

temp (Φ) = π ⊆L 2 (G(F )\G(A)) π R ∞ π ,ν ν ∈Ω temp m(π ) Φ(π ) comp (Φ) = π ⊆L 2 (G(F )\G(A)) π R ∞ π ,ν ν ∈Ω comp m(π ) Φ(π )
As announced in the outlook of the method, in order to have a problem amenable to the trace formula it is necessary to formulate statistics quantities on the universal family as a spectral side, hence needed to select it by the Fourier transforms of suitable test functions. The aim of the present section is to construct a function Φ ∈ H (G) such that spec (Φ) = B (d, σ R , δ, Ω; ϕ) .

(2.60)

In the case of factorizable test functions Φ = ⊗ Φ , the spectral side of the trace formula factorizes as well, reducing the treatment to local statement on local, and hopefully simpler, quantities. The following sections are dedicated to construct local test functions doing so, aim reached in Lemma 4.

Non-archimedean split places

For an ideal d of O, introduce the congruence subgroup given by the product of the corresponding local congruence subgroups in (2.19), that is to say K 0 (d) = p r ||d K 0,p (p r ).

(2.62)

The following result gives a test function whose Fourier transform selects the nite split conductor.

Lemma 2. For an ideal d of O, let

ε d = vol K 0 (d) -1 1 K 0 (d) .
(2.63)

Its

Rami ed places

For rami ed places, less is known concerning the representations and the choice made in the decomposition (2.55) is to x the corresponding isomorphism class. In the nite dimensional case, knowing matrix coe cients is su cient to determine the underlying matrix. This property still holds for supercuspidal representations in the following sense.

Let σ R be a unitary representation of G R . A matrix coe cient associated to σ R is a function of the form, given and w in the space of σ R ,

ξ ,w σ R : G R -→ C -→ σ ( ) , w
(2.65)

Matrix coe cients are continuous functions on G R , are compactly supported since G R is compact, and are locally constant at nite places and smooth at archimedean places.

Remark. The fact that matrix coe cients is considered only for rami ed places is critical for selecting purposes. The loss of the compactness of the support for matrix coe cients in the split case, where some automorphic representations are not supercuspidal, make them fail to select the corresponding isomorphism class. Such purposes can be achieved by means of existence theorem, yet are less precise, see [75]. This is the reason why the non-totally de nite case or the GL(2) case are analytically harder to deal with, see Section 2.5.4.

As for nite-dimensional matrix coe cients, orthogonality relations can be formulated and are the key to selecting a xed representation σ R . For instance, Knightly and Li [75, Corollary 10.26] provide the following proposition.

Proposition 15. Let σ and π be automorphic representations of G R , and introduce d π the formal degree of π . Then for every unit vectors and w in the representation space of σ , π ξ ,w σ w = 1 π σ w, d π .

(2.66)

Taking for a vector of norm d 1/2 π , it follows that π ξ , σ is the orthogonal projection onto C and in the meanwhile selects the π 's isomorphic to σ . So that, considering its trace, the above into a result concerning Fourier transforms can be restated as follows.

Proposition 16. Let σ and π be automorphic representations of G R . Let be a vector of norm one in the representation space of σ . Then, ξ , σ (π ) = 1 π σ .

(2.67)

From now on, denote ξ σ any choice of matrix coe cient as in Proposition 16.

Remark. From now on every error term also depends on ϕ: this is not such a matter since the Sauvageot theorem will ultimately get rid of every error term to just conclude to a convergence result, and the counting law is obtained with no ϕ added. The sought weighted measure is barely reached by the spectral side with Φ d,π R ,δ,Ω,ρ;ϕ , as stated in the following fundamental lemma.

Φ(σ ) = m(σ R , d) ϕ(σ f )1 σ R π R c(σ R ) | d 1 τ ∈W δ ν ∈Ω ϕ(τ , ν ) + ϵ δ,Ω ρ (τ , ν ) .
(2.74)

Nevertheless, these conditions also stand for characters: in order to not being killed by Φ they have to be trivial on K 0 (d), i.e. they have to be unrami ed since det(K 0 (d)) = O R . Moreover, they have to be isomorphic to π R at rami ed places. The Fourier transform of the chosen test function hence does not vanish on unrami ed characters, unlike awaited. The corresponding extra contribution Ξ is treated separately in Lemma 12, for characters are easier to embrace and it will be shown to contribute as an error term.

  Statistics 1.1 Automorphic world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Universal family of quaternion algebras . . . . . . . . . . . . . . 1.1.2 Analogy with the height on algebraic varieties . . . . . . . . . . 1.1.3 Theorem A: Counting law . . . . . . . . . . . . . . . . . . . . . 1.2 Equidistribution results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Theorem B: Equidistribution . . . . . . . . . . . . . . . . . . . . 1.2.2 Sato-Tate conjectures . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Low-lying zeros of L-functions . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Importance of zeros of L-functions . . . . . . . . . . . . . . . . 1.3.2 Pair correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 One-level densities . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Theorem D: Type of symmetry . . . . . . . . . . . . . . . . . . 1.4 Other ground groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Unitary groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Outline of the story . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Universal family decomposition . . . . . . . . . . . . . . . . . . 1.5.2 Spectral count . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Geometric side . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Further consequences . . . . . . . . . . . . . . . . . . .

  [15], Conrey-Duke-Farmer [30], Sarnak [107] and Serre [116].

[

  11pt, a4paper, twoside, openright]book [utf8]inputenc [T1]fontenc [french, english]babel geometry amsfonts, amssymb, amsmath color graphicx [all, cmtip]xy eurosym amsthm stmaryrd appendix scrextend lipsum [linkcolor=blue, citecolor=green, colorlinks=true]hyperref array xy minitoc makeidx indent rst fncychap newtxtext, newtxmath height=23.36cm,width=15.90cm hmargin=2.5cm,2.5cm vmargin=2cm,3cm Lemma Theorem De nition Proposition Conjecture fancyhdr Theorem Contents Chapter 2

  Haar measure on R is the usual Lebesgue measure d x, and the Haar measure on C is 2 d x ∧ d = | d z ∧ d z|. On non-archimedean elds, Haar measures on F × p are normalized so that its ring of integers O p gets measure one. The ring of adeles is endowed with the product measure. On the group of units F × , choose d × x = d x/|x | as Haar measure.

Proposition 4 .

 4 Let F be an algebraic closure of F . An F -algebra A is isomorphic over F to M 2 (F ) if and only if A is a quaternion algebra.Proof. The algebraA is isomorphic over F to M 2 (F ) ifand only if A is simple over F [125, IX, Coro. 2]. Moreover, the simple algebras over F are the quaternion algebras by de nition. This proves quaternion algebras are forms of M 2 (F ). They are also inner forms by the Skölem-Noether theorem. [115, III.1.4].

For

  nite split places p, by de nition B p M 2, F p so that B × p GL(2, F p ). The notion of local conductor for irreducible smooth in nite-dimensional representations of GL(2) has been introduced by Casselman [21]. Consider the sequence of compact open congruence subgroups

  Let ϕ ∈ H (G S ). The action of ϕ(π ), shortcut notation for ϕ(π S ), can be assumed to have a selecting e ect on the spectral data. Indeed, the trace Paley-Wiener theorem of Bernstein, Deligne and Kazhdan [8] provides the fundamental properties of the Fourier transforms.Theorem 12 (Trace Paley-Wiener). The functions on G p lying in the image by the Fourier transform of the Hecke algebra H (G p ) are the functions ϕ on Gp such that (i) for every standard Levi subgroup M of G p and every irreducible representation σ of M, the function ψ → ϕ(ind M G p (ψ σ )) is a regular function on the complex algebraic variety ψ (M) composed of the unrami ed characters of M;(ii) there exists an open compact subgroup K of G p dominating ϕ, i.e. such that ϕ is nonzero only on representations having non trivial K-xed space π K .

2. 5 . 1

 51 Selberg trace formula for compact casesSince the automorphic quotient of G is compact by Proposition 5, the original formulation of the trace formula, due to Selberg [3] in 1956, can be used and combined with the multiplicity one theorem. If Φ is a function in the Hecke algebra H (G(A)), then geom (Φ) = spec (Φ),(2.56)where the spectral and geometrical parts are as follows. The geometrical part isgeom (Φ) := {γ } vol G γ (F )\G γ (A) ∫ G γ (A)\G(A) Φ x -1 γ x d x .(2.57)The sum runs through conjugacy classes {γ } in G(F ). Since Φ is compactly supported and G(F ) is discrete, the sum is nite. However its length depends on the support of Φ what turns to be a critical di culty for estimations, for this support depends on the spectral parameters. The inner integrals appearing in this geometric side are called the orbital integrals, de ned byO γ (Φ) = ∫ G γ (A)\G(A) Φ x -1 γ x d x .(2.58)The spectral part is spec (Φ) = π ⊆L 2 (G(F )\G(A))m(π ) Φ(π ).

Proposition 14 (

 14 Factorization of the spectral side). If π = ⊗ π and Φ = ⊗ Φ , then Φ(π ) = Φ (π ). (2.61) Hence, in order to achieve the spectral selection (2.60) it is su cient locally select the conditions appearing in the decomposition of the universal family (2.55) through Fourier transforms. The places of F fall into four categories: the split nite part, corresponding to p R ∪ S, where the arithmetic conductor is caught by the means of an explicit ltration, see Section 2.19; the split nite part in the support of the test function ϕ, corresponding to p ∈ S\R, whose conductor is xed by ϕ, see Proposition 12; the rami ed part, corresponding to the nite number of ∈ R, which deliberately remains a blackbox and is handled by xing the representations at those places by means of matrix coe cients; the split archimedean part, parametrized by spectral data that are handled by selecting functions provided by Paley-Wiener theorems.

  Fourier transform selects the multiplicity relative to d. More precisely,ε d (π ) = m(π, d), π ∈ A(G).(2.64)Proof. Let π be an automorphic representation of G. Then π (ε d ) is the projection of the representation space V π on the subspace π d of the xed vectors by K 0 (d) under the action of π . Indeed, every π(ε d ) , for in V π , is K 0 (d)-invariant, for it is an averaging over the action of K 0 (d). For k 0 ∈ K 0 (d) and ∈ V π , π (k 0 )π (ε d ) = vol K 0 (d) -1 π (k 0 ) ∫ K 0 (d) π (k) d k = vol K 0 (d) -1 ∫ K 0 (d) π (k 0 k) d k = vol K 0 (d) -1 ∫ K 0 (d) π (k) d k = π (ε d )so that its image lies in π d . The action of π (ε d ) is also idempotent, more precisely the identity on π d . Indeed, for 0 ∈ π d ,π (ε d ) 0 = vol K 0 (d) π (ε d ) is an idempotent endomorphism of image π d , i.e. a projection on π d . The trace of a projection is its rank, that is to say ε d (π ) is the dimension of the xed vector spaces π d . Those are the sought multiplicities m(π, d), in particular are nonzero if and only if c(π ) | d.

ϕ

  The same arguments used byBrumley and Milićević [17] hold with h δ,Ω ρ replaced by h δ,Ω,ϕ ρ. In particular, a version of the Paley-Wiener proven by Clozel and Delorme [24] provides a function f δ,Ω,ϕ ρ whose Fourier transform is h δ,Ω,ϕ ρ .2.5.5 Spectrum selectionThe weighted counting number B(d, σ R , δ, Ω; ϕ) should be written as a spectral side in the trace formula. Introduce the test functionΦ d,π R ,δ,Ω,ρ;ϕ = Φ , (2.69)which is built with the following local functions: is the local component of ϕ on G ; ξ π is a matrix coe cient for π ;ε d is the function introduced in Lemma 2, ε d, its -component; f δ,Ω,ϕ ρis the function constructed Lemma 5, with Ω = Ω(Q/N qc(π R )).

Lemma 4 .

 4 Let Q c ϕ . Let d ∧ R = 1, π R ∈ G R , δ ∈ E 2 (M 1 ) for an M ∈ L ∞ . Then B (d, π R , δ, Ω; ϕ) = temp Φ d,π R ,δ,Ω,ρ;ϕ + O(Ξ(ϕ, π R )) + O(∂ ρ B(d, π R , δ, Ω)),(2.70)where, introducing the set X ur (G) of unrami ed characters of G(A),Ξ(ϕ, π R ) = χ ∈X ur (G) χ R π R m(χ R , d) ϕ(χ ),(2.71)and∂ ρ B(d, π R , δ, Ω) = ∫ π ∈A(d,π R ,δ ) ν ∈ih M τ 2 d c(π R ) ϵ δ,Ω ρ (τ , ν ) d ν, (2.72)where this last integral means an integration over π ∈ A(G) of xed discrete spectral data d, π R and δ , and with continuous parameters varying in ih M .Proof. Let Φ = Φ d,π R ,δ,Ω,ρ;ϕ . In order to determine the Fourier transform of Φ recall the splitting into local components given by Proposition 14: for every places , w and every a ∈ H (G ,w ), a a w = a a w . Thus, Fourier transforms of the local components of the test function have to be determined. The nite prime-to-S split part ε d is shown to transform into the characteristic function of conductors dividing d in Lemma 2 weighted by the corresponding multiplicities. The rami ed local parts ξ π are known to transform into the characteristic functions of the isomorphism class of π by Lemma 16. The transform of the archimedean split part is shown to approximate the selecting function of bounded conductors in Lemma 3, up to a smoothing error term ϵ δ,Ω ρ . The action of the Fourier transform of Φ on the tempered part follows, and (2.73) yields, for σ ∈ A(G) with archimedean split parameters (τ , ν ),

  . . . . . Odds and ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Number theoretic landscape . . . . . . . . . . . . . . . . . . . . 2.1.2 Automorphic world . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Quaternion algebras . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Analytic conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Opening the ε-factor . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2.3 Conductor as a notion of depth . . . . . . . . . . . . . . . . . . 35 2.3 Equidistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.1 Elements of equidistribution . . . . . . . . . . . . . . . . . . . . 39 2.3.2 Plancherel formulas & Fourier transformation . . . . . . . . . . 40 2.3.3 Sauvageot density theorem . . . . . . . . . . . . . . . . . . . . . 41 2.3.4 Admissible functions . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 Universal family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.4.1 Archimedean Langlands classi cation . . . . . . . . . . . . . . . 44 2.4.2 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.3 Old and new forms . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5 Spectral selection .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Geometric error terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Speci c savings in the case of F = Q . . . . . . . . . . . . . . . 77 3.5 Proof of Corollary C: Sato-Tate conjecture . . . . . . . . . . . . . . . . 80 Low-lying zeros of L-functions . . . . . . . . . . . . . . . . . . . . . . . 84

	xii		Contents
	2 Universal Family for Quaternion Algebras 2.2.2 4 Type of Symmetry Counting law for quaternion algebras Theorem B Plancherel equidistribution for quaternion algebras Corollary C Sato-Tate equidistribution for quaternion algebras Theorem D One-level density for quaternion algebras Conjecture ?? Counting law for unitary groups Conjecture ?? Counting law for symplectic groups Theorem E Restricted equidistribution 4.1 Theorem A 2.1 xi xv	4 6 8 14 ?? ?? 42	83

2.2.1 Conductor arising from functional equations . . . . . . . . . . . 2.5.1 Selberg trace formula for compact cases . . . . . . . . . . . . . 49 2.5.2 Non-archimedean split places . . . . . . . . . . . . . . . . . . . 51 2.5.3 Rami ed places . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5.4 Archimedean parameters . . . . . . . . . . . . . . . . . . . . . . 53 2.5.5 Spectrum selection . . . . . . . . . . . . . . . . . . . . . . . . . 54 3 Counting Law & Equidistribution 57 3.1 Geometric reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Identity contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.1 Test function at 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.2 Identity contribution splitting . . . . . . . . . . . . . . . . . . . 62 3.2.3 Main part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2.4 Geometric reformulation of the constant . . . . . . . . . . . . . 66 3.2.5 Smoothing part . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2.6 Extra error term . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 Spectral error terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3.1 Characters contribution . . . . . . . . . . . . . . . . . . . . . . 69 3.3.2 Complementary spectrum . . . . . . . . . . . . . . . . . . . . . 71 3.4 3.4.1 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4.2 Split orbital integrals . . . . . . . . . . . . . . . . . . . . . . . . 74 3.4.3 Non-split orbital integrals . . . . . . . . . . . . . . . . . . . . . 75 3.4.4 Final estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4.5 4.1.1 One-level density . . . . . . . . . . . . . . . . . . . . . . . . . . 84 List of Results and Conjectures All the original results stated in the present thesis are labeled by letters instead of numbers. They are gathered in the following list for reference. The study of unitary groups and symplectic groups in Chapter 5 is a joint ongoing project with Ian Petrow, in particular the conjectural counting laws stated in conjectures ?? and ??.

(Φ) = vol(G(F )\G(A))Φ(1) + O(∂ ρ B(q, σ R , δ, Ω)), (1.55)

Archimedean parameters

Based on the decomposition of the universal family 2.43, for a general bounded set of continuous parameters Ω in h M,C , the question is to select representations lying in sets of the form

This is a feature of non-compact archimedean groups: their dual is no more discrete and hence admits a continuous parametrization. However, many tools in harmonic analysis on Lie groups involve narrow classes of functions among which characteristic functions of such sets Ω δ are not, requiring a smoothing construction to get admissible functions lying nearby them. This procedure is in essence provided by the fundamental work of Duistermaat, Kolk and Varadarajan [39]. Brumley and Milićević [17] adapt this method to the automorphic setting on GL(n) and construct a function localizing around spectral parameters (δ, ν ), where δ is a xed archimedean discrete spectral datum and where ν is a continuous parameter running through a bounded set of parameters Ω δ . Smoothing procedures behave well on tempered parameters, leading to assume Ω to be a bounded set of tempered parameters of xed discrete part δ , leaving the non-tempered part of Ω to be proven negligible compared to the tempered contribution.

Introduce ϕ which in this section is a function in the Hecke algebra of G R ∞ , and should be denoted ϕ R ∞ in the following ones. The aim is to nd a smooth enough function for trace formula purposes approximating the characteristic function of Ω δ . Brumley and Milićević [17, Section 9] achieved this goal, constructing a function h δ,Ω ρ of Paley-Wiener type with exponential type ρ > 0 as tempered spectral-localizing function. Let h δ,Ω,ϕ ρ

(2.68)

A direct consequence of their result is the following lemma, where the remainder term is fundamental yet willingly hidden in order to ease the exposition. What is of critical importance are the bounds on this undisclosed error term, precisely stated in Lemma 10. Lemma 3. For every discrete spectral data δ ∈ D, there is a function ϵ δ,Ω ρ such that for every (M, τ ) ∈ D and ν ∈ h M,C ,

Proof. This is just encapsulating the results of [17, Lemma 9.2] and multiplying them by ϕ.

After summing over the tempered spectrum, it follows by roughly bounding ϕ in the remainder smoothing term,

that achieves the proof.

Remark. This lemma gets rid of treating precisely the contribution of those characters, for it is more suitable to keep the test function as easy to handle as possible in order to ease the estimations in the geometric side below. The lack of details compared to [17] concerning the treatment of the archimedean parameters may be a break to the understanding. Recall from Lemma 3 that the error term ϵ δ,Ω ρ is better when ν if far from the boundary of Ω, so that it should be considered as a smooth bump function concentrating around the boundary, so that the integral (2.72) is a smoothed version of the counting number B(d, π R , δ, ∂ ρ Ω; ϕ), justifying the notation.