Pietro Beri
email: pietro.beri@math.univ-poitiers.fr

Alberto Cattaneo
email: cattaneo@math.uni-bonn.de

ON BIRATIONAL TRANSFORMATIONS OF HILBERT SCHEMES OF POINTS ON K3 SURFACES

Keywords: 2020 Mathematics Subject Classification: 14J50, 14C05, 14C34, 32G13 Irreducible holomorphic symplectic manifolds, Hilbert schemes of points on surfaces, birational equivalence, automorphisms, cones of divisors

We classify the group of birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one. We study whether these automorphisms are symplectic or non-symplectic and if there exists a hyperkähler birational model on which they become biregular. We also present new geometrical constructions of these automorphisms.

Introduction

Hilbert schemes of points on smooth complex K3 surfaces are examples of irreducible holomorphic symplectic (ihs) manifolds, i.e. compact simply connected Kähler manifolds with a unique, up to scalar, holomorphic two-form, which is everywhere non-degenerate. The second integral cohomology group of ihs manifolds admits a lattice structure, provided by the Beauville-Bogomolov-Fujiki (BBF) quadratic form. The global Torelli theorem for K3 surfaces has been generalized to ihs manifolds, albeit in a weaker form, making possible to investigate automorphisms of these manifolds by studying their action on the BBF lattice.

For an integer t ≥ 1 consider a complex algebraic K3 surface S whose Picard group is generated by an ample line bundle H with H 2 = 2t, i.e. a very general element of the 19-dimensional space of 2t-polarized K3 surfaces. A classification of the group of biregular automorphisms of the Hilbert scheme S [n] := Hilb n (S) has been given by Boissière, An. Cattaneo, Nieper-Wißkirchen and Sarti [START_REF] Boissière | The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface, K3 surfaces and their moduli[END_REF] in the case n = 2, and by the second author for all n ≥ 3 [START_REF] Al | Automorphisms of Hilbert schemes of points on a generic projective K3 surface[END_REF]. In particular, Aut(S [n] ) is either trivial or generated by an involution which is non-symplectic, i.e. it does not preserve the generator of H 2,0 (S [n] ). If t = 1, the involution is not induced by an involution of the surface S (we say that it is non-natural). The results of [START_REF] Boissière | The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface, K3 surfaces and their moduli[END_REF] and [START_REF] Al | Automorphisms of Hilbert schemes of points on a generic projective K3 surface[END_REF] provide explicit numerical conditions on the parameters n and t for the existence of a non-trivial biregular automorphism on S [n] . The first aim of the paper is to give a similar classification for the group Bir(S [n] ) of birational automorphisms of the Hilbert scheme, thus generalizing the results of Debarre and Macrí [START_REF] Debarre | On the period map for polarized hyperkähler fourfolds[END_REF][START_REF] Debarre | Hyperkähler manifolds[END_REF] for the case n = 2.

Section 2 briefly recalls the basic theory of Pells equations, which are the fundamental number theoretic tool in our study. Then, in Section 3 we combine the description of the movable cone of S [n] due to Bayer and Macrí [START_REF] Bayer | MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations[END_REF] with Markman's results on monodromy operators [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF] to obtain numerical necessary and sufficient conditions on n and t for the existence of birational automorphisms on S [n] . For σ ∈ Bir(S [n] ) let σ * ∈ O(H 2 (S [n] , Z)) be its pullback action on the BBF lattice. We set H 2 (S [n] , Z) σ * := v ∈ H 2 (S [n] , Z) | σ * (v) = v and H 2 (S [n] , Z) σ * := (H 2 (S [n] , Z) σ * ) ⊥ . Theorem 1.1. Let S be an algebraic K3 surface with Pic(S) = ZH, H 2 = 2t and n ≥ 2.

If t ≥ 2, there exists a non-trivial birational automorphism σ : S [n] S [n] if and only if t(n -1) is not a square and the minimal solution (X, Y ) = (z, w) of Pell's equation X 2 -t(n -1)Y 2 = 1 with z ≡ ±1 (mod n -1) satisfies w ≡ 0 (mod 2) and (z, z) ≡ (j, k) ∈ Z 2(n-1)Z × Z 2tZ with (j, k) ∈ {(1, 1), (1, -1), (-1, -1)}. If so Bir(S [n] ) = σ ∼ = Z/2Z and • if (j, k) = (1, -1), then σ is non-symplectic with H 2 (S [n] , Z) σ * ∼ = 2 ;

• if (j, k) = (-1, -1), then σ is non-symplectic with H 2 (S [n] , Z) σ * ∼ = 2(n -1) ; • if (j, k) = (1, 1), then σ is symplectic with H 2 (S [n] , Z) σ * ∼ = -2(n -1) . If t = 1, let (X, Y ) = (a, b) be the integer solution of (n -1)X 2 -Y 2 = -1 with smallest a, b > 0. If n -1 is a square or b ≡ ±1 (mod n -1), then Bir(S [n] ) = Aut(S [n] ) ∼ = Z/2Z. Otherwise n ≥ 9, Bir(S [n] ) ∼ = Z/2Z × Z/2Z and Aut(S [n] ) ∼ = Z/2Z.

For more details on the action σ * ∈ O(H 2 (S [n] , Z)), see Proposition 3.1. We remark that, by a classical result of Saint-Donat [START_REF] Saint-Donat | Projective models of K3 surfaces[END_REF], the case t = 1 is the only one where the K3 surface S has a non-trivial automorphism (S is a double covering of P 2 ramified over a sextic curve), hence Aut(S [n] ) ∼ = Z/2Z is generated by the corresponding natural (non-symplectic) involution for all n ≥ 2.

It is interesting to notice that, differently from the biregular case, birational involutions of S [n] can be symplectic, for (infinitely many) suitable choices of n, t. In Proposition 3.5 we give examples of sequences of degrees of polarizations t = t k (n) which realize all cases in Theorem 1. [START_REF] Apostolov | Moduli spaces of polarized irreducible symplectic manifolds are not necessarily connected[END_REF].

By [16, §5.2], the closed movable cone of a projective ihs manifold X has a wall-and-chamber decomposition [START_REF] Apostolov | Moduli spaces of polarized irreducible symplectic manifolds are not necessarily connected[END_REF] Mov(X) = g g * Nef(X ′ )

where the union is taken over all non-isomorphic ihs birational models g : X X ′ . The chambers g * Nef(X ′ ) are permuted by the action of any birational automorphism of X. By Markman's Hodge-theoretic global Torelli theorem [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Theorem 1.3] biregular automorphisms are exactly those which map Nef(X) to itself. If X admits a birational automorphism σ, it is natural to ask whether there exists an ihs birational model g : X X ′ such that g • σ • g -1 is biregular on X ′ . We confront this problem for Hilbert schemes S [n] in Section 4, by using the explicit description of the walls between chambers coming from [START_REF] Bayer | MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations[END_REF]Theorem 12.3]. For a fixed n ≥ 2, we say that a value t ≥ 1 is n-irregular if, for a very general 2t-polarized K3 surface S, the group Bir(S [n] ) contains an involution which is not biregular on any ihs birational model of S [n] . We provide a numerical characterization of n-irregular values in Propositions 4.3 and 4.5. In particular, we verify that symplectic birational involutions remain strictly birational on all birational models. On the other hand, in the case of non-symplectic birational automorphisms there are only finitely many n-irregular t's for a fixed n ≥ 2 (see Corollary 4.7).

Section 5 provides an in-depth analysis of the case n = 3. By studying solutions of two particular Pell's equations we show that, when Bir(S [3] ) is not trivial, the number of chambers in the movable cone of S [3] (i.e. the number of non-isomorphic ihs birational models) is either one, two, three or five. For n = 2 it is known that the number of chambers in Mov(S [2] ) is at most three. As n increases, computing an upper bound for the number of chambers becomes more and more difficult, since walls arise from the solutions of an increasing number of Pell's equations. The new results for n = 3 are summed up in Proposition 5. [START_REF] Boissière | A note on automorphisms and birational transformations of holomorphic symplectic manifolds[END_REF].

In Section 6 we show how Theorem 1.1 can be generalized to study birational maps between Hilbert schemes of points on two distinct K3 surfaces of Picard rank one (which need to be Fourier-Mukai partners). In the literature, isomorphic Hilbert schemes of points on two non-isomorphic K3 surfaces are called (strongly) ambiguous. The main result of the section is Theorem 6.2, an improved version of the criterion [START_REF] Meachan | Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational[END_REF]Theorem 2.2] for the determination of ambiguous pairs (up to isomorphism or birational equivalence).

In Section 7 we go back to the case of biregular involutions. For ihs manifolds of K3 [n] -type (i.e. deformation equivalent to Hilbert schemes of n points on a K3 surface) we compare moduli spaces of polarized manifolds and moduli spaces of manifolds with an involution whose action on the second cohomology has invariant lattice of rank one. Theorem 7.5 gives a modular interpretation of the classification of non-symplectic involutions for manifolds of K3 [n] -type (see for instance [START_REF] Camere | Non-symplectic involutions of manifolds of K3 [n] -type[END_REF]), which for n = 2 has been highlighted in [START_REF] Boissière | On the antisymplectic involutions of the Hilbert square of a K3 surface[END_REF]. For n ≥ 3 this requires some additional care, since we deal with moduli spaces which may be non-connected.

Finally, in Section 8 we give new geometrical examples of birational involutions as in Theorem 1.1, for Hilbert schemes of points on quartic surfaces. Finding similar constructions is an interesting problem, since the Torelli-like results used in the proof of Theorem 1.1 give no insight on the geometry. Moreover, we describe up to deformation birational involutions of S [n] , when S is a general K3 surface of degree 2((n -1)k 2 + 1) for some integer k. These involutions are obtained by deforming a combination of Beauville involutions on the Hilbert scheme of n points of a specific K3 surface of Picard rank two, in such a way that the deformation path goes only through ihs manifolds which are still Hilbert schemes of points on K3's.

Acknowledgements. The authors are indebted to Alessandra Sarti for reading a first draft of the paper and for her precious remarks. This work has greatly benefited from discussions with Samuel Boissière, Chiara Camere and Georg Oberdieck. A. C. is grateful to Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. A. C. is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -GZ 2047/1, projekt-id 390685813. 2) is solvable and r is not a square, the (infinite) set of solutions is divided into equivalence classes: two solutions (X, Y ) and (X ′ , Y ′ ) are said to be equivalent if

Preliminaries on Pell's equations

A generalized Pell's equation is a quadratic diophantine equation (2) X 2 -rY 2 = m in the unknowns X, Y ∈ Z, for r ∈ N and m ∈ Z \ {0}. If m = 1, the equation is called standard. If the equation (
XX ′ -rY Y ′ m ∈ Z, XY ′ -X ′ Y m ∈ Z.
If the equation is standard, then it is solvable and all solutions are equivalent. Inside any equivalence class of solutions, the fundamental solution (X, Y ) is the one with smallest non-negative Y , if there is a single solution with this property in the class. Otherwise, the smallest non-negative value of Y is realized by two conjugate solutions (X, Y ), (-X, Y ): in this case, the fundamental solution of the class will be (X, Y ) with X > 0. If (X, Y ) is a fundamental solution, all other solutions (X ′ , Y ′ ) in the same equivalence class are of the form (3)

X ′ = aX + rbY Y ′ = bX + aY where (a, b) is a solution of the standard Pell's equation a 2 -rb 2 = 1. A solution (X, Y ) of (2) is called positive if X > 0, Y > 0.
The minimal solution of the equation is the positive solution with smallest X. In particular, the minimal solution is one of the fundamental solutions. We also use the expression "minimal solution with a property P " to denote the positive solution with smallest X among those which satisfy the property P .

Let (X, Y ) = (z, w) be the minimal solution of the equation

X 2 -rY 2 = 1. The half-open interval [( √ m, 0), (z √ m, w √ m))
on the hyperbola X 2 -rY 2 = m contains exactly one solution for each equivalence class. The solutions in this interval are all the solutions (X, Y ) of ( 2) such that X > 0 and 0 ≤ Y X < w z . Moreover, if (X, Y ) is a fundamental solution of (2) and m > 0, then:

(4) 0 < |X| ≤ (z + 1)m 2 , 0 ≤ Y ≤ w m 2(z + 1)
.

We will often make use of the following lemma.

Lemma 2.1. For t ≥ 1 and n ≥ 2 such that t(n -1) is not a square, let (z, w) be the minimal solution of

X 2 -t(n -1)Y 2 = 1 with z ≡ ±1 (mod n -1). If w ≡ 0 (mod 2), then (i) z ≡ 1 (mod 2(n -1)) and z ≡ 1 (mod 2t) if and only if (z, w) is not the minimal solution of X 2 -t(n -1)Y 2 = 1. (ii) z ≡ -1 (mod 2(n -1)) and z ≡ -1 (mod 2t) if and only if the equation X 2 -t(n -1)Y 2 = -1 has integer solutions. (iii) if z ≡ 1 (mod 2(n-1)
) and z ≡ -1 (mod 2t) then the equation (n-1)X 2 -tY 2 = -1 has integer solutions; if t ≥ 2 the converse also holds. (iv) if z ≡ -1 (mod 2(n-1)) and z ≡ 1 (mod 2t) then the equation (n-1)X 2 -tY 2 = 1 has integer solutions; if n ≥ 3 the converse also holds.

Proof. Let w = 2m for some m ∈ N. (i) -(ii) Write z = 2(n -1)p ± 1 = 2tq ± 1 for p, q ∈ N. Then p((n -1)p ± 1) = tm 2 and (n -1)p = tq, hence r := p t ∈ N. We have r((n -1)rt ± 1) = m 2 and it follows that there exist s, u ∈ N such that r = s 2 , (n-1)tr±1 = u 2 , m = su, hence u 2 -t(n -1)s 2 = ±1. If the sign is +, notice that u < z, therefore (z, w) is not the minimal solution of X 2 -t(n -1)Y 2 = 1. Conversely, assume first that X 2 -t(n -1)Y 2 = -1 has integer solutions and let (a, b) be the minimal one. Then, by [START_REF] Debarre | Hyperkähler manifolds[END_REF]Lemma A.2] the minimal solution of

X 2 -t(n-1)Y 2 = 1 is (z, w) = (2t(n-1)b 2 -1, 2ab
), which satisfies z ≡ -1 (mod 2(n -1)) and z ≡ -1 (mod 2t). Similarly, if the minimal solution (u, s) of X 2 -t(n -1)Y 2 = 1 does not satisfy u ≡ ±1 (mod n -1), then by (3) we have (z, w) = (2t(n -1)s 2 + 1, 2us), hence z ≡ 1 (mod 2(n -1)) and z ≡ 1 (mod 2t). (iii) -(iv) Write z = 2(n -1)p ± 1 = 2tq ∓ 1 for p, q ∈ N. Then p((n -1)p ± 1) = tm 2 and (n -1)p ± 1 = tq, hence pq = m 2 . Since gcd(p, q) = 1, there exist s, u ∈ N such that p = s 2 , q = u 2 and m = su, thus (n -1)s 2 -tu 2 = ∓1.

Vice versa, let (a, b) be the solution of (n -1)X 2 -tY 2 = ±1 with smallest a > 0. By [10, Lemma A.2], the assumption t ≥ 2 (if the sign is -)

or n ≥ 3 (if the sign is +) implies that the minimal solution of X 2 -t(n -1)Y 2 = 1 is (z, w) = (2(n -1)a 2 ∓ 1, 2ab
), which satisfies z ≡ ∓1 (mod 2(n -1)) and z ≡ ±1 (mod 2t).

Birational automorphisms of S [n]

Let S be an algebraic K3 surface with Pic(S) = ZH, H 2 = 2t, t ≥ 1. For n ≥ 2, let S [n] be the Hilbert scheme of n points on S and {h, -δ} a basis for NS(S [n] ) ⊂ H 2 (S [n] , Z), where h is the class of the nef (not ample) line bundle induced by H on S [n] and 2δ is the class of the exceptional divisor of the Hilbert-Chow morphism S [n] → S (n) . We consider H 2 (S [n] , Z) equipped with the (even) lattice structure given by the Beauville-Bogomolov-Fujiki integral quadratic form. In particular, we have

H 2 (S [n] , Z) ∼ = H 2 (S, Z)⊕Zδ ∼ = U ⊕3 ⊕E 8 (-1) ⊕2 ⊕ -2(n-1)
, where for an integer d = 0 we denote by d the rank one lattice generated by an element of square d.

3.1.

The action on cohomology. By [24, Corollary 5.2] the group Bir(S [n] ) is finite and the homomorphism Bir(S [n] ) → O(H 2 (S [n] , Z)), σ → σ * := (σ -1 ) * is injective by [START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0, Classification of algebraic and analytic manifolds[END_REF]Proposition 10]. Moreover, the kernel of Ψ : Bir(S [n] ) → O(NS(S [n] )), σ → σ * | NS(S [n] ) , is the subgroup of natural automorphisms, which is isomorphic to Aut(S) (see [START_REF] Boissière | The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface, K3 surfaces and their moduli[END_REF]Lemma 2.4] and notice that ker(Ψ) ⊂ Aut(S [n] ) by the global Torelli theorem [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Theorem 1.3]). By [26, §5], this implies that ker(Ψ) = {id} if t ≥ 2, while for t = 1 we have ker(Ψ) = ι [n] , where ι is the involution which generates Aut(S) and ι [n] is the natural involution induced by ι on S [n] .

Let Mov(S [n] ) ⊂ NS(S [n] ) R be the movable cone of S [n] , i.e. the open cone generated by the classes of divisors whose base locus has codimension at least two. It contains the ample cone A S [n] . If there exists σ ∈ Bir(S [n] ) non-natural, then the action σ * on NS(S [n] ) is a non-trivial isometry and by [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Lemma 6.22] it preserves the movable cone; more specifically, σ * exchanges the two extremal rays of Mov(S [n] ). By [2, Proposition 13.1], t(n -1) is not a square, (n -1)X 2 -tY 2 = 1 has no integer solutions (if n = 2) and Mov(S [n] ) = h, zh -twδ R ≥0 , where (z, w) is the minimal solution of z 2 -t(n -1)w 2 = 1 with z ≡ ±1 (mod n -1); moreover, we have z ≡ ±1 (mod 2(n -1)) and w ≡ 0 (mod 2).

By imposing the conditions σ * (h) = zh -twδ, σ * (zh -twδ) = h, one computes that the matrix which describes σ * ∈ O(NS(S [n] )) with respect to the basis {h, -δ} is [START_REF] Boissière | The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface, K3 surfaces and their moduli[END_REF] z -(n -1)w tw -z .

This is an involution of NS(S [n] ) thus from the description of ker(Ψ) that we recalled above we conclude that the group Bir(S [n] ) is either {id}, Z/2Z or Z/2Z × Z/2Z. Assuming that there exists a non-natural birational involution σ ∈ Bir(S [n] ), then if t ≥ 2 we have Bir(S [n] ) = σ , while Bir(S [n] ) = ι [n] , σ if t = 1. Notice that the isometry σ * is the reflection of NS(S [n] ) which fixes the line spanned by (n -1)wh -(z -1)δ. Let ν = bh -aδ ∈ NS(S [n] ) be the primitive generator of this line with a, b > 0, (a, b) = 1.

In the following, for any even lattice L with bilinear form (-, -) we denote by A L := L ∨ /L the discriminant group, which is a finite abelian group equipped with a finite quadratic form q L : A L → Q/2Z induced by the quadratic form on L. If A L is cyclic of order m, we write A L ∼ = Z mZ (α) if q L takes value α ∈ Q/2Z on a generator of A L . For g ∈ O(L) we denote by g the isometry induced by g on A L . In the case of the lattice H 2 (S [n] , Z) we have

A H 2 (S [n] ,Z) ∼ = Z 2(n-1)Z -1 2(n-1)
. For any x ∈ H 2 (S [n] , Z) let div(x) be the divisibility of x in H 2 (S [n] , Z), i.e. the positive generator of the ideal (x, H 2 (S [n] , Z)) ⊂ Z. The transcendental lattice of S [n] is Tr(S

[n] ) = NS(S [n] ) ⊥ ⊂ H 2 (S [n] , Z). Proposition 3.1. Let S be an algebraic K3 surface with Pic(S) = ZH, H 2 = 2t, t ≥ 1. For n ≥ 2, let σ ∈ Bir(S [n]
) be a non-natural automorphism and ν = bh -aδ ∈ NS(S [n] ) be the primitive generator of the line fixed by σ * with a, b > 0. Then one of the following holds:

• σ * | Tr(S [n] ) = -id and • either σ * = -id and ν 2 = 2, i.e. (a, b) is an integer solution of (n - 1)X 2 -tY 2 = -1; • or σ * = id, ν 2 = 2(n-1) and div(ν) = n-1, i.e. (a, b n-1 ) is an integer solution of X 2 -t(n -1)Y 2 = -1. • n ≥ 9, σ * | Tr(S [n] ) = id, σ * = -id and ν 2 = 2t, i.e. the minimal solution of X 2 -t(n -1)Y 2 = 1 is (b, a t
) and b ≡ ±1 (mod n -1). Proof. Let (z, w) be the minimal solution of X 2 -t(n -1)Y 2 = 1 with z ≡ ±1 (mod n-1); then, as we recalled before, z ≡ ±1 (mod 2(n-1)) and w ≡ 0 (mod 2). By [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Theorem 1.3], σ * is a monodromy operator (see [16,

if z ≡ 1 (mod 2t) and σ * | Tr(S [n] ) = -id if and only if z ≡ -1 (mod 2t). On the other hand σ * = id if and only if z ≡ -1 (mod 2(n -1)) and σ * = -id if and only if z ≡ 1 (mod 2(n -1)), for all t ≥ 1 (see [8, Remark 5.2]).
If σ * acts as -id on Tr(S [n] ) and t ≥ 2, then the statement follows as in the proof of [START_REF] Al | Automorphisms of Hilbert schemes of points on a generic projective K3 surface[END_REF]Proposition 5.1] and by [START_REF] Al | Automorphisms of Hilbert schemes of points on a generic projective K3 surface[END_REF]Lemma 6.3].

Assume that z ≡ 1 (mod 2t) (hence, σ * | Tr(S [n] ) = id or t = 1). By Lemma 2.1 and by the hypothesis that (n -1)X 2 -tY 2 = 1 has no integer solutions we have z ≡ 1 (mod 2(n -1)), i.e. σ * acts as -id on the discriminant group of H 2 (S [n] , Z). Moreover, (z, w) = (2t(n -1)s 2 + 1, 2us), where (u, s) is the minimal solution of

X 2 -t(n -1)Y 2 = 1. The axis of the reflection σ * | NS(S [n] ) is spanned by (n -1)wh -(z -1)δ = 2s(n -1) (uh -tsδ) .
Since gcd(u, ts) = 1, we conclude that the primitive generator of the axis of the reflection is ν = uh -tsδ, whose square is 2t. By looking at quadratic residues modulo n -1, we see that the minimal solution (u, s) of

X 2 -t(n -1)Y 2 = 1 can have u ≡ ±1 (mod n -1) only if n ≥ 9.
Remark 3.2. Let ω be a generator of H 2,0 (S [n] ). Then NS(S [n] ) = ω ⊥ ∩H 2 (S [n] , Z), therefore an automorphism σ ∈ Bir(S [n] ) is symplectic (i.e. σ * ω = ω) if and only if σ * | Tr(S [n] ) = id. As a consequence of [18, Lemma 3.5, Corollary 5.1] (see also [START_REF] Al | Automorphisms of Hilbert schemes of points on a generic projective K3 surface[END_REF]Proposition 4.1]) any symplectic non-trivial birational automorphism of S [n] is not biregular. We will return to this point in Section 4.

Classification of birational automorphisms.

Let L be an even lattice and l ∈ L such that (l, l) = 0. We define the reflection

R l : L R → L R , m → m -2 (m, l) (l, l) l.
This map restricts to a well-defined isometry

R l ∈ O(L) if and only if the divisibility of l is either |(l, l)| or |(l, l)| /2. Recall that the real spinor norm of L is the group homomorphism sn L R : O(L) → R * / (R * ) 2 ∼ = {±1} defined as sn L R (g) = - (v 1 , v 1 ) 2 . . . - (v r , v r ) 2 (mod (R * ) 2 )
where

g R = R v1 • . . . • R vr is the factorization of g R ∈ O(L R )
with respect to reflections defined by elements v i ∈ L. In particular, if the signature of L is (l + , l -), after diagonalizing the bilinear form of L over R it is immediate to check that sn L R (-id) = (-1) l+ . Proof of Theorem 1.1 for t ≥ 2. By Proposition 3.1, Lemma 2.1 and the previous discussion, the numerical conditions in the statement of the theorem are necessary for the existence of a birational automorphism of S [n] with non-trivial action on NS(S [n] ) (i.e. a non-trivial automorphism, since t = 1). Assume now that these conditions hold. If n = 2 the equation (n -1)X 2 -tY 2 = 1 has no integer solutions, since (j, k) = (-1, 1) (Lemma 2.1). We therefore have Mov(S [n] ) = h, zh-twδ R ≥0 by [2, Proposition 13.1]. Using again Lemma 2.1, we give the following definitions depending on (j, k).

(i) If (j, k) = (1, -1), then (n -1)X 2 -tY 2 = -1 has integer solutions. Let (a, b) be the solution with smallest a > 0 and set ν = bh -aδ. By Lemma 2.1 we have (z, w) = (2(n -1)a 2 + 1, 2ab), since we are assuming t ≥ 2.

As

ν 2 = 2, we can define φ = -R ν ∈ O(H 2 (S [n] , Z)). Then φ = -id ∈ O(A H 2 (S [n] ,Z) ) by [12, Proposition 3.1]. (ii) If (j, k) = (-1, -1), then X 2 -t(n -1)Y 2 = -1 has integer solutions.
Let (a, b) be the minimal solution and set ν = (n -1)bh -aδ. Then (z, w) = (2t(n-1)b 2 -1, 2ab). The class ν has square 2(n-1) and divisibility n -1 (see [START_REF] Al | Automorphisms of Hilbert schemes of points on a generic projective K3 surface[END_REF]Lemma 6.3]), so we can define φ

= -R ν ∈ O(H 2 (S [n] , Z)) and φ = id by generalizing [12, Corollary 3.4]. (iii) If (j, k) = (1, 1), let (b, a) be the minimal solution of X 2 -t(n -1)Y 2 = 1 and set ν = bh -taδ. Then (z, w) = (2t(n -1)a 2 + 1, 2ab). The primitive element γ = (n -1)ah -bδ satisfies γ 2 = -2(n -1), while the divisibility of γ in H 2 (S [n] , Z) is either n -1 or 2(n -1). This implies that φ = R γ ∈ O(H 2 (S [n] , Z)
) and moreover φ = -id by [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Proposition 9.12].

Notice that there is no ambiguity in the definitions of ν and φ when n = 2. By construction, φ| Tr(S [n] ) = -id in cases (i), (ii) and φ| Tr(S [n] ) = id in case (iii). Thus φ extends to a Hodge isometry of H 2 (S [n] , C). Moreover, φ is orientationpreserving. Indeed, let sn R := sn

H 2 (S [n] ,Z) R ; then if φ = -R ν we have sn R (φ) = sn R (-id)sn R (R ν ) = -sn R (R ν ) = sgn(ν 2 ) = +1
where we used the fact that H 2 (S [n] , Z) has signature [START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0, Classification of algebraic and analytic manifolds[END_REF][START_REF] Nagell | Introduction to Number Theory[END_REF]. If φ = R γ , then sn R (φ) = -sgn(γ 2 ) = +1. We conclude, by [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Lemma 9.2], that φ ∈ Mon 2 Hdg (S [n] ) (in the case φ = R γ , also see [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Proposition 9.12]).

From the relations between (a, b) and (z, w) that we remarked for each of the three pairs (j, k), it is immediate to check that ν ∈ Mov(S [n] ) and that φ| NS(S [n] ) is the isometry (5) fixing the line spanned by ν, hence φ(Mov(S [n] )) = Mov(S [n] ). Then, by [16, Lemma 6.22 and Lemma 6.23] φ ∈ Mon 2 Bir (S [n] ), i.e. there exists a birational automorphism σ ∈ Bir(S [n] ) such that σ * = φ.

We are left to discuss the case t = 1. For all n ≥ 2 the group Aut(S [n] ) contains the natural involution induced by the generator of Aut(S). Here, the equation (n -1)X 2 -tY 2 = -1 has always integer solutions, while X 2 -t(n -1)Y 2 = -1 has no integer solutions if we assume that (n -1)X 2 -tY 2 = 1 has none.

Proof of Theorem 1.1 for t = 1. Let (z, w) be the minimal solution of z 2 -(n -1)w 2 = 1 with z ≡ ±1 (mod n-1). If there exists a non-natural automorphism then w ≡ 0 (mod 2), which implies z ≡ 1 (mod 2t). Thus, by the proof of Proposition 3.1, (z, w) is not the minimal solution of the equation, i.e. b ≡ ±1 (mod n -1). On the other hand, if we assume b ≡ ±1 (mod n -1) then (n -1)X 2 -Y 2 = 1 has no integer solutions by [START_REF] Debarre | Hyperkähler manifolds[END_REF]Lemma A.2]. The existence of a symplectic automorphism follows as in the proof for t ≥ 2, case (iii). By [8, Proposition 1.1], this automorphism is not biregular.

If S is a 2-polarized K3 surface of Picard rank one, the smallest n such that Bir(S [n] ) ∼ = Z/2Z × Z/2Z is n = 9. Remark 3.3. Notice that t = 1 is the only value of t such that the isometry (5) of NS(S [n] ) can be glued to both + id, -id ∈ O(Tr(S [n] )). This follows from [21, Corollary 1.5.2], since the discriminant group of Tr(S [n] ) is

Z 2tZ . If t = 1, let ν = bh -aδ ∈ NS(S [n] ) be as in the proof of Theorem 1.1. Then (-R ν ) ⊕ (-id) ∈ O(NS(S [n] ) ⊕ Tr(S [n] )) extends to -R ν ∈ O(H 2 (S [n] , Z)), while (-R ν ) ⊕ id extends to R γ , with γ = (n -1)ah -bδ.
If n -1 is not a square and b ≡ ±1 (mod n -1), both isometries R γ and -R ν of the cohomology lattice are Hodge monodromies which lift to (distinct) non-natural birational automorphisms of S [n] . The lift of R γ is symplectic while the lift of -R ν is non-symplectic. Each automorphism can be recovered by composing the other with ι [n] , where ι is the covering involution which generates Aut(S). Indeed, (ι

[n] ) * | NS(S [n] ) = id and (ι [n] ) * | Tr(S [n] ) = -id (since ι [n] is a natural involution).
Remark 3.4. If for n ≥ 2, t ≥ 2 there exists a non-trivial birational automorphism on S [n] , it is biregular if and only if n, t satisfy condition (iii) of [START_REF] Al | Automorphisms of Hilbert schemes of points on a generic projective K3 surface[END_REF]Theorem 6.4], which guarantees that A S [n] = Mov(S [n] ). If t ≤ 2n -3 the automorphism is not biregular by [8, Proposition 1.1].

The following proposition provides examples of polarization degrees for the K3 surface S so that S [n] has a non-natural birational automorphism, for each of the three different actions on cohomology of Proposition 3.1. Proposition 3.5. Fix n ≥ 2. For t ≥ 1, we denote by S an algebraic K3 surface with Pic(S) = ZH, H 2 = 2t.

(i) There exist infinitely many t's such that Bir(S [n] ) contains a non-natural, non-symplectic automorphism σ with

H 2 (S [n] , Z) σ * ∼ = 2 , e.g. t = (n - 1)k 2 + 1 for k ≥ 1. (ii)
There exists t such that Bir(S [n] ) contains a non-natural, non-symplectic automorphism σ with H 2 (S [n] , Z)

σ * ∼ = 2(n -1) if and only if -1 is a quadratic residue modulo n -1.
If so, this happens for infinitely many t's, e.g. t = (n -1)k 2 + 2qk + q 2 +1 n-1 for k, q ≥ 1 and q 2 ≡ -1 (mod n -1). (iii) There exists t such that Bir(S [n] ) contains a non-natural, symplectic automorphism σ if and only if n -1 = q 2 -1 h for some q ≥ 3, h ≡ 0 (mod q ± 1). If so, H 2 (S [n] , Z) σ * ∼ = -2(n -1) and this happens for infinitely many t's, e.g.

t = (n -1)k 2 + 2qk + h for k ≥ 1.
Proof. As before, we denote by (z, w) the minimal solution of

X 2 -t(n -1)Y 2 = 1 with z ≡ ±1 (mod n -1). (i) If t = (n -1)k 2 + 1 and k ≥ 1, then t(n -1) is not a square and (z, w) = (2t -1, 2k) = (2(n -1)k 2 + 1, 2k) by [10, Lemma A.2], since (k, 1
) is the solution of (n -1)X 2 -tY 2 = -1 with smallest X > 0. We conclude with Theorem 1.1. (ii) If there exists t such that Bir(S [n] ) contains a non-natural, non-symplectic automorphism σ with H 2 (S [n] , Z) σ * ∼ = 2(n -1) , then by Theorem 1.1 and Lemma 2.1 the equation X 2 -t(n -1)Y 2 = -1 is solvable, hence -1 is a quadratic residue modulo n -1. On the other hand, let t = (n -1)k 2 + 2qk + q 2 +1 n-1 for k, q ≥ 1 and q 2 ≡ -1 (mod n -1). The minimal solution of X 2 -t(n -1)Y 2 = -1 is (q + (n -1)k, 1), therefore t(n -1) is not a square and (z, w) = (-1 + 2t(n -1), 2q + 2(n -1)k), again by [START_REF] Debarre | Hyperkähler manifolds[END_REF]Lemma A.2]. The existence of the automorphism and its action on cohomology follow then from Theorem 1.1. (iii) If Bir(S [n] ) contains a symplectic automorphism σ, then by Theorem 1.1 and Lemma 2.

1 the minimal solution (u, s) of X 2 -t(n -1)Y 2 = 1 has u ≡ ±1 (mod n -1). Hence n -1 = u 2 -1 ts 2 ≥ 8 and u ± 1 ∤ ts 2 . Let now t = (n -1)k 2 + 2qk + h for k, q, h as in the statement. The minimal solution of X 2 -t(n -1)Y 2 = 1 is (u, s) = (q + (n -1)k, 1
). Since h ≡ 0 (mod q ± 1), we have q ≡ ±1 (mod n-1), therefore (z, w) = (1+2t(n-1), 2q+2(n-1)k), which allows us to conclude by using Theorem 1.1.

Decomposition of the movable cone and automorphisms of birational models

We consider the wall-and-chamber decomposition (1) of the closed movable cone Mov(S [n] ) ⊂ NS(S [n] ) R = Rh ⊕ Rδ. As shown in [8, Lemma 2.5] by using [2, Theorem 12.1], the walls are spanned by the classes of the form Xh -2tY δ which lie in the movable cone, for (X, Y ) a positive solution of one of Pell's equations

X 2 -4t(n -1)Y 2 = α 2 -4ρ(n -1) with X ≡ ±α (mod 2(n -1))
, where the possible values of ρ and α are:

     ρ = -1 and 1 ≤ α ≤ n -1; ρ = 0 and 3 ≤ α ≤ n -1; 1 ≤ ρ < n-1 4 and 4ρ + 1 ≤ α ≤ n -1. (6)
Definition 4.1. For n ≥ 2, let S be an algebraic K3 surface with Pic(S) = ZH,

H 2 = 2t, t ≥ 1.
The value t is said to be n-irregular if there exists a birational automorphism σ ∈ Bir(S [n] ) such that for all ihs birational models g :

S [n] X the involution g • σ • g -1 ∈ Bir(X) is not biregular. Remark 4.2.
If the decomposition of Mov(S [n] ) has an odd number of chambers, and there exists a non-natural automorphism σ ∈ Bir(S [n] ), then the generator ν of the axis of the reflection σ * | NS(S [n] ) is in the interior of one of the chambers. This follows from the fact that the isometry σ * acts on the set of chambers by [16, Lemma 5.12], hence if their number is odd, one of them is preserved by σ * , and therefore it contains ν. As a consequence, there exists an ihs birational model (corresponding to the preserved chamber) on which the involution becomes biregular, by [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Theorem 1.3]. Notice that Mov(S [n] ) has only one chamber if and only if σ ∈ Aut(S [n] ). On the other hand, if there is an even number of chambers in the decomposition of Mov(S [n] ), then ν lies on one of the walls in the interior of the cone, and therefore σ is not biregular on any of the ihs birational models of S [n] .

This implies that t is n-irregular if and only if Bir(S [n] ) contains a non-natural birational automorphism and the wall-and-chamber decomposition of Mov(S [n] ) has an even number of chambers.

] ) = h, zh-twδ R ≥0 with (z, w) = (2t(n -1)a 2 + 1, 2ab). Since (b + 1)(b -1) = t(n -1)a 2 , we have c := max{gcd(n -1, b -1), gcd(n -1, b + 1)} ≥ 2. We define α := max{4, 2(n-1) c }, which is an even integer such that 4 ≤ α ≤ n -1. We consider Pell's equation X 2 -4t(n -1)Y 2 = α 2 . The pair (X, Y ) = (bα, a α
2 ) is a solution with the property 2Y X = a b . We also have X = bα ≡ ±α (mod 2(n -1)) by construction, hence ν lies on the wall (in the interior of the movable cone) spanned by the class Xh-2tY δ.

By the previous proposition, a non-natural birational automorphism of S [n] which acts on Tr(S [n] ) as id (or as ± id if t = 1) is not biregular on any ihs birational model of S [n] , which is stronger than what we stated in Remark 3.2. By Proposition 3.5, for n fixed the number of n-irregular values t for which this happens is either zero or infinite.

Example 4.4. For any odd k ≥ 5 with k ≡ ±1 (mod 8), define t = k 2 -1 8 ∈ N. We can readily check that S [9] admits a symplectic non-natural involution by Theorem 1.1, hence t is 9-irregular by Proposition 4.3.

ON BIRATIONAL TRANSFORMATIONS OF HILBERT SCHEMES OF POINTS ON K3'S 11

For non-symplectic automorphisms when t = 1, the behaviour is different. We now show that, for a fixed n ≥ 2, there is only a finite number of n-irregular values t for which the non-natural birational automorphism of S [n] acts as -id on Tr(S [n] ), and we provide an algorithm to compute them. Proposition 4.5. Let t ≥ 2 and n ≥ 2, such that t(n -1) is not a square and (n -1)X 2 -tY 2 = 1 has no integer solutions (if n = 2). Assume that one between (n -1)X 2 -tY 2 = -1 and X 2 -t(n -1)Y 2 = -1 admits integer solutions; define ℓ = 1 if the first equation is solvable or ℓ = n -1 if the second one is, and let (a, b) be the solution with smallest a > 0. Then t is n-irregular if and only if there exists a pair (α, ρ) as in [START_REF] Boissière | A note on automorphisms and birational transformations of holomorphic symplectic manifolds[END_REF] and a positive solution (X, Y ) of Pell's equation

X 2 -4t(n -1)Y 2 = α 2 -4ρ(n -1) with X ≡ ±α (mod 2(n -1)) such that 4tℓY 2 = (α 2 -4ρ(n -1))a 2 .
Proof. Let ν = ℓbh -aδ ∈ Mov(S [n] ) be the class of square 2ℓ as in the proof of Theorem 1.1. Then ν lies on one of the walls in the interior of Mov(S [n] ) if and only if a ℓb = 2t Y X , where (X, Y ) is a positive solution of one of the equations X 2 -4t(n -1)Y 2 = α 2 -4ρ(n -1) with X ≡ ±α (mod 2(n -1)) and (α, ρ) as in [START_REF] Boissière | A note on automorphisms and birational transformations of holomorphic symplectic manifolds[END_REF]. We write the condition on the slopes as:

Y X • ℓb 2(n -1)a = 1 4t(n -1)
and we observe that (2(n-1)a) 2 -4t(n-1)(ℓb) 2 = -4ℓ(n-1). Hence this becomes:

1 4t(n -1) - α 2 -4ρ(n -1) 4t(n -1)X 2 1 4t(n -1) + 4ℓ(n -1) 4t(n -1)(2(n -1)a) 2 = 1 4t(n -1)
.

If we rearrange the equation, we obtain:

(7) ℓX 2 -(n -1)(α 2 -4ρ(n -1))a 2 = ℓ(α 2 -4ρ(n -1)) i.e. 4tℓY 2 = (α 2 -4ρ(n -1))a 2 .
Remark 4.6.

• Notice that, by ( 7) and (n -1)a 2 -t(ℓb) 2 = -ℓ, we have X 2 = (α 2 -4ρ(n-1))tℓb 2 . However, tℓ has to divide α 2 -4ρ(n-1) (because tℓ is coprime with a), hence α 2 -4ρ(n-1) = tℓr 2 for some r ∈ N. This gives an easy way to compute a (finite) list of candidates for the n-irregular values t, among the divisors of α 2 -4ρ(n -1) for (α, ρ) as in [START_REF] Boissière | A note on automorphisms and birational transformations of holomorphic symplectic manifolds[END_REF]. In particular, t ≤ (n -1)(n + 3).

• In order to check if a value t ≥ 2 is n-irregular in Proposition 4.5, it is enough to consider the pairs (α, ρ) such that α 2 -4ρ(n -1) > 0 and the positive solutions (X, Y ) with smallest X in each equivalence class of solutions of X 2 -4t(n -1)Y 2 = α 2 -4ρ(n -1), otherwise the wall spanned by Xh -2tY δ is not in the interior of the movable cone.

Corollary 4.7. For a fixed n ≥ 2, the number of n-irregular values t for which Bir(S [n] ) contains a non-symplectic birational automorphism is finite. In particular, for n ≤ 8 there is only a finite number of n-irregular values t.

Proof. By Proposition 3.1, assuming t ≥ 2 a non-natural automorphism σ ∈ Bir(S [n] ) satisfies σ * | Tr(S [n] ) = -id if and only if the minimal solution of X 2t(n -1)Y 2 = 1 has X ≡ ±1 (mod n -1). In particular, this always holds for n ≤ 8.

Then, by Theorem 1.1, either (n -1)X 2 -tY 2 = 1 or X 2 -t(n -1)Y 2 = 1 is solvable. The statement follows from Proposition 4.5 and Remark 4.6.

In Table 1, for n ≤ 14 we list all n-irregular values t as in Corollary 4.7. We separate the values of t for which the middle wall of the movable cone is spanned by a primitive class ν of square 2 (i.e. ℓ = 1), from those where the generator has square 2(n -1) and divisibility n -1 (i.e. ℓ = n -1). For n = 9, 12, the values t in the table are all the n-irregular values, 9 and 12 being the only n ≤ 14 which satisfy Proposition 3.5 (iii). We observe that, for n ≤ 5, all n-irregular values are of the form t = n or t = 4n -3. Proof. Notice that t = n and t = 4n -3 are values of the form t = (n -1)k 2 + 1, for k = 1, 2 respectively. By Proposition 3.5 the Hilbert scheme S [n] has a non-natural birational automorphism which acts as -id on Tr(S [n] ). The minimal solution of (n -1)

X 2 -tY 2 = -1 is (a, b) = (k, 1). • If t = n and n ≥ 3, consider the equation X 2 -4t(n -1)Y 2 = α 2 -4ρ(n -1) for ρ = -1 and α = 2. Its minimal solution is (X, Y ) = (2n, 1), which satisfies X ≡ α (mod 2(n -1)). • If t = 4n -3 and n ≥ 2, consider the equation X 2 -4t(n -1)Y 2 = α 2 -
4ρ(n-1) for ρ = -1 and α = 1. Its minimal solution is (X, Y ) = (4n-3, 1), which satisfies X ≡ α (mod 2(n -1)). In both cases the relation 4tY 2 = (α 2 -4ρ(n -1))a 2 holds, hence the statement follows from Proposition 4.5.

In the case t = n, it is known that the automorphism of S [n] which acts as the reflection in the (only) wall contained in the interior of Mov(S [n] ) is Beauville's involution [3, §6], which is biregular if and only if n = 2.

In Proposition 3.5 we showed that for t = (n -1)k 2 + 1, k ≥ 1 the Hilbert scheme S [n] has a non-symplectic birational involution. We observed in Lemma 4.8 that the involution is not biregular for k = 1, 2, while by [8, Proposition 6.7] it is biregular (i.e. A S [n] = Mov(S [n] )) whenever k ≥ n+3 2 . It seems however that this lower bound on k is far from optimal. Conjecture 4.9. Let n ≥ 2 and t = (n -1)k 2 + 1, for k ∈ N. Let S be an algebraic K3 surface with Pic(S) = ZH, H 2 = 2t. If k ≥ 3, then the non-natural non-symplectic involution which generates Bir(S [n] ) is biregular.

We checked computationally that the conjecture holds for all n ≤ 14. The results of [8, Proposition 6.7] can be adapted also to the other family of polarizations in Proposition 3.5 giving rise to non-symplectic birational involutions. Lemma 4.10. Let n ≥ 3 and q ∈ N be such that q 2 ≡ -1

(mod n -1). Let t = (n -1)k 2 + 2qk + q 2 +1
n-1 for k ≥ 1 and S an algebraic K3 surface with

Pic(S) = ZH, H 2 = 2t. If k ≥ n+3
2 , then the non-natural non-symplectic involution which generates Bir(S [n] ) is biregular.

Proof. By Proposition 3.5 we just need to prove that A S

[n] = Mov(S [n] ). De- fine Q = (n -1)k + q. For k ≥ n+3 2 we have 4t(n -1) > (n + 3) 2 (n -1) 2 ≥ α 2 -4ρ(n -1)
2 for all (α, ρ) as in ( 6), hence the solutions of X 2 -4t(n -

1)Y 2 = α 2 -4ρ(n -1) are encoded in the convergents of the continued fraction of 4t(n -1) = 2 Q 2 + 1 = [2Q; Q, 4Q] (for details see [9, Chapter XXXIII, §16]
). The proof then follows as for [8, Proposition 6.7].

Kähler-type chambers and automorphisms for n = 3

It is known that for a K3 surface S of Picard rank one the number of Kähler-type chambers in the decomposition of Mov(S [2] ) is d ∈ {1, 2, 3} (see for instance [START_REF] Debarre | Hyperkähler manifolds[END_REF]Example 3.18]). We now detail the computation of the number of chambers for n = 3 in the cases where Mov(S [n] ) = h, zh -twδ R ≥0 , with (z, w) the minimal solution of X 2 -t(n -1)Y 2 = 1 with z ≡ ±1 (mod n -1). In particular, this holds whenever Bir(S [3] ) = {id}.

For an integer t ≥ 2 such that 2t is not a square, consider the following two generalized Pell's equations:

P 8t (9) : X 2 -8tY 2 = 9 P 8t (12) : X 2 -8tY 2 = 12.
Remark 5.1.

• If we write the two equations as X 2 -8tY 2 = 8 + α 2 , with α ∈ {1, 2}, then it is clear that all solutions (X, Y ) satisfy X ≡ ±α (mod 4). 12 9 is not rational. • Assume that P 8t (12) is solvable and let (X, Y ) be a solution. Then X = 2Z for some [START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF], if it is solvable, has one class of solutions when t ≡ 0 (mod 3), two classes otherwise. We also remarked that solutions (X, Y ) of P 8t [START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF] are of the form (2Z, Y ) for (Z, Y ) solution of P 2t (3), and that P 8t [START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF] is not solvable if t ≡ 0 (mod 2).

• If (X, Y ) is a solution of P 8t (9) and (X ′ , Y ′ ) is a solution of P 8t (12), then Y X = Y ′ X ′ since
Z ∈ Z such that (Z, Y ) is a solution of P 2t (3) : Z 2 -2tY 2 = 3. Notice that Z ≡ 1 (mod 2), Y ≡ 1 (
If t ≡ 1 (mod 3), write t = 3q + 1 for q ∈ N 0 and assume that (Z, Y ) is a solution of P 2t (3). Then Z 2 -(6q + 2)Y 2 = 3, i.e. Z 2 ≡ 2Y 2 (mod 3). This implies that Z ≡ Y ≡ 0 (mod 3), which gives a contradiction. Hence, P 8t [START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF] has no solutions.

If t ≡ 0 (mod 9), write t = 9q for q ∈ N. If (X, Y ) is a solution of P 8t [START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF], then X = 6X ′ for some X ′ ∈ N and 3(X ′ ) 2 -6qY 2 = 1, which is impossible. Hence, P 8t [START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF] has no solutions.

If t ≡ 6 (mod 9), write t = 9q + 6 for q ∈ N 0 . If (X, Y ) is a solution of P 8t (12), then X = 3X ′ for some X ′ ∈ N and 3(X ′ ) 2 -8(3q + 2)Y 2 = 4. This implies Y 2 ≡ -1 (mod 3), which cannot be. Hence, P 8t [START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF] has no solutions. Lemma 5.3. If t ≡ 1 (mod 3) or t ≡ 3, 6 (mod 9), then all solutions of P 8t [START_REF]Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, Part II[END_REF] are equivalent to (X, Y ) = (3, 0). If t ≡ 0 (mod 9), then P 8t (9) has either one, two or three classes of solutions. If t ≡ 2 (mod 3), then P 8t (9) has either one or three classes of solutions.

Proof. If t = 3q + 1, for q ∈ N 0 , then X 2 ≡ 2Y 2 (mod 3), hence all solutions (X, Y ) are of the form (3X ′ , 3Y ′ ), with (X ′ ) 2 -8t(Y ′ ) 2 = 1. This is now a standard Pell's equation, which has only one class of solutions, thus the same holds for P 8t [START_REF]Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, Part II[END_REF].

If t = 3q for q ∈ N and (q, 3) = 1, then X = 3X ′ for some X ′ ∈ Z such that 3(X ′ ) 2 -8qY 2 = 3. Since 3 ∤ q we need (X, Y ) = (3X ′ , 3Y ′ ) with (X ′ ) 2 -24q(Y ′ ) 2 = 1. There exists only one class of solutions (X ′ , Y ′ ) for this standard Pell's equation, thus also the solutions of P 8t (9) form a single class.

Assume now that t = 9q for q ∈ N. Then a solution (X, Y ) of P 8t (9) is of the form (X, Y ) = (3X ′ , Y ) with (X ′ ) 2 -8qY 2 = 1. Let (z, w) be the minimal solution of P 8t (1) : z 2 -8tw 2 = 1. Notice that the solutions of P 8t (1) are the pairs (X ′ , Y

3 ) for all solutions (X ′ , Y ) of (X ′ ) 2 -8qY 2 = 1 such that Y ≡ 0 (mod 3). Let (a, b) be the minimal solution of (X ′ ) 2 -8qY 2 = 1. Since this is a standard Pell's equation, by (3) its next two solutions (for increasing values of X ′ ) are (a 2 + 8qb 2 , 2ab) and (a 3 + 24qab 2 , 8qb 3 + 3a 2 b). We observe that one among these first three positive solutions (X ′ , Y ) has Y ≡ 0 (mod 3), since either a 2 (hence, a) or 8qb 2 = a 2 -1 is divisible by 3. The positive solution (X ′ , Y ) with this property and smallest X ′ is therefore equal to (z, 3w), thus the corresponding solution (X, Y ) = (3X ′ , Y ) of P 8t (9) satisfies Y X = w z , i.e. it is the first positive solution in the same equivalence class of (3, 0). We conclude that P 8t (9) has either one, two or three classes of solutions.

Finally, assume that t ≡ 2 (mod 3). Let (z, w) be the minimal solution of P 8t (1) and let (u 1 , v 1 ), (u 2 , v 2 ) be two positive solutions of P 8t (9) such that, for i = 1, 2:

(8) 0 < u i ≤ 3 z + 1 2 , 0 < v i ≤ 3w 2(z + 1)
.

By [START_REF] Boissière | On the antisymplectic involutions of the Hilbert square of a K3 surface[END_REF], this is equivalent to asking that either (u i , v i ) or (-u i , v i ) is a fundamental solution of P 8t (9), different from (3, 0). Thus u 1 , v 1 , u 2 , v 2 are not divisible by three. From u 2 1 -8tv 2 1 = 9 and u 2 2 -8tv 2 2 = 9 we get ( 9)

u 1 v 2 ≡ ±u 2 v 1 (mod 9), u 1 u 2 ≡ ±8tv 1 v 2 (mod 9)
where the signs in the two congruences coincide. If we now multiply u 2 1 -8tv 2 1 = 9 and u 2 2 -8tv 2 2 = 9 member by member, we obtain

(10) u 1 u 2 ∓ 8tv 1 v 2 9 2 -8t u 1 v 2 ∓ u 2 v 1 9 2 = 1
where by ( 9) the two squares in the LHS term are integers. If we assume that (u 1 , v 1 ) and (u 2 , v 2 ) are distinct, then u 1 v 2 ∓ u 2 v 1 = 0. Since (z, w) is the minimal solution of P 8t (1), from [START_REF] Debarre | Hyperkähler manifolds[END_REF] we have

|u 1 v 2 ∓ u 2 v 1 | ≥ 9w. However, from (8) we compute |u 1 v 2 ∓ u 2 v 1 | < 9w, which is a contradiction. We conclude u 1 = u 2 , v 1 = v 2 .
Thus, there are at most three classes of solutions for P 8t (9): the class of (3, 0) and possibly the classes of (u 1 , v 1 ) and (-u 1 , v 1 ). Notice that the latter two classes are always distinct: in order for them to coincide we would need

u 2 1 + 8tv 2 9 ∈ Z, 2u 1 v 1 9 ∈ Z which does not happen, since (u 1 , 3) = (v 1 , 3) = 1.
Hence, the equation has either one or three classes of solutions.

We remark that all cases in the statements of Lemma 5.2 and Lemma 5.3 occur, for suitable values of t.

Let S be an algebraic K3 surface such that Pic(S) = ZH, H 2 = 2t, t ≥ 2. As explained in Section 4, if Bir(S [3] ) = {id} then 2t is not a square, 2X 2 -tY 2 = 1 has no integer solutions and the minimal solution of X 2 -2tY 2 = 1 has Y ≡ 0 (mod 2). As a consequence, Mov(S [3] ) = h, zh -2twδ R ≥0 , where (z, w) is the minimal solution of P 8t (1) : z 2 -8tw 2 = 1. The walls in the interior of the movable cone are the rays through Xh-2tY δ, for (X, Y ) positive solution of X 2 -8tY 2 = 8+α 2 such that α ∈ {1, 2}, X ≡ ±α (mod 4) and 0 < Y X < w z . By Remark 5.1, this implies that the number of chambers in the interior of the cone coincides with the number of combined equivalence classes of solutions for P 8t (9) and P 8t [START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF] (the class of the solution (3, 0) of P 8t (9) determines the two extremal rays of the movable cone). Hence, Lemma 5.2 and Lemma 5.3 give the following result. Proposition 5.4. Let t ≥ 2 such that 2t is not a square, 2X 2 -tY 2 = 1 has no integer solutions and the minimal solution of X 2 -2tY 2 = 1 has Y ≡ 0 (mod 2).

The following table provides the numbers of classes of solutions for the equations P 8t (9) and P 8t [START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF] and the number of chambers in the movable cone of S [3] , for S an algebraic K3 surface such that Pic(S) = ZH, H 2 = 2t.

t mod 18 0 1 2 3 4 5 6 7 8 # eq. classes P 8t (9) 1, 2, 3 1 1, 3 1 1 1, 3 1 1 1, 3 # eq. classes P 8t (12) 0 0 0 0, 1 0 0, 2 0 0 0 # chambers 1, 2, 3 1 1, 3 1, 2 1 1, 3,
(9) 1, 2, 3 1 1, 3 1 1 1, 3 1 1 1, 3 # eq. classes P 8t (12) 0 0 0, 2 0 0 0 0 0 0, 2 # chambers 1, 2, 3 1 1, 3, 5 1 1 1, 3 1 1 1, 3, 5
In all cases where Mov(S [3] ) has no walls in its interior (i.e. there is only one chamber), then any birational automorphism is biregular by the global Torelli theorem for ihs manifolds.

Corollary 5.5. Let S be an algebraic K3 surface such that Pic(S) = ZH, H 2 = 2t, t ≥ 1. If t ≡ 1, 4, 6, 7, 10, 12, 13, 15, 16 (mod 18), then Bir(S [3] ) = Aut(S [3] ).

From Theorem 1.1, Corollary 4.7 and the results of this section we conclude the following.

Proposition 5.6. Let S be an algebraic K3 surface such that Pic(S) = ZH, H 2 = 2t. If t = 1, then Bir(S [3] ) = Aut(S [3] ) ∼ = Z/2Z. If t ≥ 2, then Bir(S [3] ) = {id} if and only if:

• 2t is not a square; • 2X 2 -tY 2 = 1 has no integer solutions; • either 2X 2 -tY 2 = -1 or X 2 -2tY 2 = -1 has integer solutions.
If Bir(S [3] ) = {id}, let d be the number of chambers in the decomposition of Mov(S [3] ). Then d ∈ {1, 2, 3, 5} and one of the following holds:

• d = 1 and Bir(S [3] ) = Aut(S [3] ) ∼ = Z/2Z; • d = 2, t = 3 or t = 9, Aut(S [3] ) = {id} and Bir(S [3] ) ∼ = Z/2Z; • d = 3, 5, Aut(S [3] ) = {id} and Bir(S [3] ) ∼ = Z/2Z If t = 3, 9 and σ ∈ Bir(S [3] ), there exists an ihs sixfold X and a birational map g : S [3] X such that g • σ • g -1 ∈ Aut(X).

Table 2 lists the number d of chambers in the decomposition of Mov(S [3] ) and the structure of the groups Aut(S [3] ), Bir(S [3] ), for an algebraic K3 surface S with Pic(S) = ZH, H 2 = 2t, 1 ≤ t ≤ 30 when Bir(S [3] ) = {id}. Table 2. Chambers and automorphisms for n = 3. t d Aut(S [3] ) Bir(S [3] )

1 1 Z/2Z Z/2Z 3 2 {id} Z/2Z 5 3 {id} Z/2Z 9 2 {id} Z/2Z 11 5 {id} Z/2Z 13 1 Z/2Z Z/2Z 19 1 Z/2Z Z/2Z 25 1 Z/2Z Z/2Z 27 3 {id} Z/2Z 29 3 {id} Z/2Z

Ambiguous Hilbert schemes and birational models

Having classified the group of birational automorphisms of S [n] , for a K3 surface S with Picard rank one, we explain in this section how the same approach can be used to study the more general problem of whether there exists a K3 surface Σ, again of Picard rank one, and a birational map φ : S [n] Σ [n] which do not come from an isomorphism S → Σ. This is related to the notion of (strong) ambiguity for Hilbert schemes of points on K3 surfaces, which for n = 2 (and φ biregular) has been investigated in [START_REF] Debarre | On the period map for polarized hyperkähler fourfolds[END_REF] and [START_REF] Zuffetti | Strongly ambiguous Hilbert squares of projective K3 surfaces with Picard number one[END_REF]. Some of the results of this section overlap (even though they are proved differently) with those of [START_REF] Meachan | Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational[END_REF], where birationality of derived equivalent Hilbert schemes of K3 surfaces is studied.

It is known that, if S [n] and Σ [n] are birational, then S and Σ are Fourier-Mukai partners and S [n] and Σ [n] are derived equivalent (see [START_REF] Ploog | Equivariant autoequivalences for finite group actions[END_REF]Proposition 10]). If Pic(S) = ZH with H 2 = 2t, t ≥ 1, then by [START_REF] Oguiso | K3 surfaces via almost-primes[END_REF]Proposition 1.10] the number of non-isomorphic FM partners of S is 2 ρ(t)-1 , where ρ(t) denotes the number of prime divisors of t (and ρ(1) = 1).

We can classify the Fourier-Mukai partners Σ of S as follows (for details see for instance [START_REF] Oguiso | K3 surfaces via almost-primes[END_REF]Section 4]). The overlattice L = H 2 (Σ, Z) of Tr(S) ⊕ NS(S) (with integral Hodge structure defined by setting L 2,0 = Tr(S) 2,0 ) corresponds to an isotropic subgroup

I L ⊂ A Tr(S) ⊕ A NS(S) = Z 2tZ (-1 2t ) ⊕ Z 2tZ ( 1 2t
) (as in [21, Section 1.4]). Since NS(S) and Tr(S) are primitive in L, the group I L is of the form I L = I a := ǫ + aη for some a ∈ (Z/2tZ) × , a 2 ≡ 1 (mod 4t), where ǫ (respectively, η) is a generator of A Tr(S) (respectively, A NS(S) ) on which the finite quadratic form takes value -1 2t (respectively, + 1 2t ). For each a ∈ (Z/2tZ) × , a 2 ≡ 1 (mod 4t), there exists a K3 surface Σ a (unique up to isomorphism) such that H 2 (Σ a , Z) is Hodge-isometric to the overlattice L a of Tr(S) ⊕ NS(S) defined by I a . Moreover, Σ a ∼ = Σ b if and only if b ≡ ±a (mod 2t). Indeed, 2 ρ(t)-1 (which is the number of non-isomorphic FM partners of S) is the cardinality of a ∈ (Z/2tZ) × , a 2 ≡ 1 (mod 4t) / ± id.

Remark 6.1. If there exists a birational map φ : S [n] Σ [n] as above, then φ * (Mov(Σ [n] )) = Mov(S [n] ) ( [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Corollary 5.7]). Let {h S , δ S } and {h Σ , δ Σ } be the canonical bases for NS(S [n] ) and NS(Σ [n] ) respectively, as in Section 3.1. Then by [START_REF] Boissière | A note on automorphisms and birational transformations of holomorphic symplectic manifolds[END_REF]Theorem 1] the map φ is induced by an isomorphism of the underlying K3 surfaces if and only if φ * (δ Σ ) = δ S . As a consequence, if φ does not come from an isomorphism S → Σ then φ * (h Σ ) is the primitive generator of the extremal ray of Mov(S [n] ) not spanned by h S . This implies that Σ is uniquely defined, up to isomorphism: in fact, if we also have φ ′ : S [n] (Σ ′ ) [n] which does not come from an isomorphism S → Σ ′ , then φ ′ • φ -1 : Σ [n] (Σ ′ ) [n] is induced by an isomorphism Σ → Σ ′ , since its pullback maps h Σ ′ to h Σ (hence, also δ Σ ′ to δ Σ ). Theorem 6.2. Let S be an algebraic K3 surface such that Pic(S) = ZH, with H 2 = 2t, t ≥ 1. For n ≥ 2, let (z, w) be the minimal solution of X 2 -t(n-1)Y 2 = 1 with z ≡ ±1 (mod n -1). There exists a K3 surface Σ and a birational map φ : S [n] Σ [n] which is not induced by an isomorphism S → Σ if and only if:

• t(n -1) is not a square; • if n = 2, (n -1)X 2 -tY 2 =

has no integer solutions;

• z ≡ ±1 (mod 2(n -1)) and w ≡ 0 (mod 2). If so, the K3 surfaces S and Σ are isomorphic if and only if z ≡ ±1 (mod 2t). Moreover, φ is biregular if and only if, for all integers ρ, α as follows:

• ρ = -1 and 1 ≤ α ≤ n -1, or • ρ = 0 and 3 ≤ α ≤ n -1, or • 1 ≤ ρ < n-1 4 and max 4ρ + 1, 2 ρ(n -1) ≤ α ≤ n -1 if Pell's equation X 2 -4t(n -1)Y 2 = α 2 -4ρ(n -1) is solvable, the minimal solution (X, Y ) with X ≡ ±α (mod 2(n -1)) satisfies Y X ≥ w 2z . Proof.
If there exists a birational map φ : S [n] Σ [n] which does not come from an isomorphism S → Σ then both extremal rays of Mov(S [n] ) correspond to Hilbert-Chow contractions (Remark 6.1). By [2, Theorem 5.7], t(n -1) is not a square, (n -1)X 2 -tY 2 = 1 has no integer solutions (if n = 2) and z ≡ ±1 (mod 2(n -1)), w ≡ 0 (mod 2). On the other hand, assume that these conditions are satisfied. Let a ∈ (Z/2tZ) × , a 2 ≡ 1 (mod 4t), such that S ∼ = Σ a . Since z 2 -t(n -1)w 2 = 1 and w is even, we can consider the FM partner Σ za of S. As Tr(S [n] ) ∼ = Tr(S) and NS(S [n] ) ∼ = NS(S) ⊕ -2(n -1) , the groups I a and I za (defined at the beginning of the section) can also be seen as isotropic subgroups of A Tr(S [n] ) ⊕ A NS(S [n] ) (in particular, we assume 2tη = h ∈ NS(S [n] )). It is then immediate to check that the overlattices of Tr(S [n] ) ⊕ NS(S [n] ) defined by these two subgroups are H 2 (Σ

[n] a , Z) and H 2 (Σ [n]
za , Z) respectively. Let µ ∈ O(NS(S [n] )) be the isometry [START_REF] Boissière | The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface, K3 surfaces and their moduli[END_REF]. We have that id ⊕µ ∈ O(Tr(S [n] ) ⊕ NS(S [n] )) extends to a Hodge isometry ψ :

H 2 (Σ [n] a , Z) → H 2 (Σ [n] za , Z), because µ(aη) = zaη ∈ A NS(S [n]
) (here we use the fact that w is even). Notice that the discriminant group of H 2 (S [n] , Z) is generated by the class of δ 2(n-1) and µ δ 2(n-1) = -(n-1)wh 2(n-1) -zδ 2(n-1) = ± δ 2(n-1) . By [START_REF] Nikulin | Integer symmetric bilinear forms and some of their geometric applications[END_REF]Corollary 1.5.2], ψ extends to a Hodge isometry H * (Σ a , Z) → H * (Σ za , Z) between the Mukai lattices of the two K3 surfaces Σ a , Σ za . We conclude that Σ

[n] a and Σ

[n] za are birationally equivalent, by [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Corollary 9.9]. As stated before, Σ a and Σ za are isomorphic if and only if a ≡ ±za (mod 2t), i.e. z ≡ ±1 (mod 2t). The isomorphism φ : S [n] Σ [n] is biregular if and only if the Hodge isometry φ * : H 2 (Σ [n] , Z) → H 2 (S [n] , Z) is effective, that is if it maps ample classes to ample classes. Since φ * permutes the two extremal rays of the movable cones, this is equivalent to asking that the movable cone of S [n] coincides with the ample cone, i.e. that there is only one chamber in the decomposition of Mov(S [n] ). The last part of the statement follows then as in the proof of [START_REF] Al | Automorphisms of Hilbert schemes of points on a generic projective K3 surface[END_REF]Theorem 6.4]. Remark 6.3. If S ∼ = Σ the existence of a birational map S [n] Σ [n] which does not come from an isomorphism of the K3 surfaces is equivalent to the existence of a non-natural birational automorphism in Bir(S [n] ). Indeed, in this case Theorem 6.2 gives the same conditions of Theorem 1.1. Remark 6.4. It can be readily checked that the conditions in the first part of Theorem 6.2 are equivalent to those of [START_REF] Meachan | Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational[END_REF]Theorem 2.2]. Write z = 2(n -1)k + ǫ and w = 2h for k, h ∈ N and ǫ ∈ {±1}. Then, z 2 -t(n -1)w 2 = 1 implies k(k(n -1) + ǫ) = th 2 . Since the two factors in the LHS term are coprime, there exist p, q, r, s ∈ N such that k = sp 2 , k(n-1)+ǫ = rq 2 , h = pq, t = rs. In particular, (n -1)sp 2 -rq 2 = ±1, which is what is requested in [START_REF] Meachan | Derived equivalent Hilbert schemes of points on K3 surfaces which are not birational[END_REF]Theorem 2.2]. The FM partner Σ of S such that there exists a birational map S [n] Σ [n] not coming of square 2d and divisibility γ.

Then M ρ is isomorphic to a connected component of M [k] ⊂ M n 2d,γ . Proof. Let (X, η) be (ρ, 2d )-polarized, with i ∈ Aut(X) the non-symplectic in- volution such that ρ = η • i * • η -1 . If D is the primitive ample divisor which generates H 2 (X, Z) i * = η -1 ((Λ n ) ρ ), then (X, D) ∈ M n 2d,γ . The morphism α : M ρ → M n 2d,γ , (X, i) → (X, D
) is clearly well-defined at the level of moduli spaces and its image is contained in a connected component M ⊂ M n 2d,γ . Indeed, if (X 1 , η 1 , i 1 ), (X 2 , η 2 , i 2 ) are (ρ, 2d )-polarized then g = η -1 2 • η 1 is a parallel transport operator (because (X 1 , η 1 ) and (X 2 , η 2 ) lie in the same connected component of M Λn ) such that g(D 1 ) = D 2 . The connected component M is determined by the explicit choice of the isometry ρ. By construction, M ⊂ M [k] with (Λ n ) ρ = k . Let now (X, D) be a polarized manifold in M. By [8, Proposition 5.3] there is a (unique) non-symplectic involution i ∈ Aut(X) such that i * = -R D . Since (X, D) is in the same connected component of any α(X ′ , i ′ ) and ρ = -R k there exists a marking η :

H 2 (X, Z) → Λ n such that (X, η) ∈ M Λn and ρ = η • i * • η -1 , hence we have a well-defined map M → M ρ , (X, D) → (X, i) which is the inverse of α. Remark 7.6. Let (d, γ) ∈ {(1, 1), (1, 2), (n -1, n -1)} and fix a connected com- ponent M ⊂ M n 2d,γ . For (X, D) ∈ M let η : H 2 (X, Z) → Λ n be a marking such that (X, η) ∈ M 0 Λn and define ρ = -R η(D) ∈ O(Λ n ).
Then it is clear from the proof of the previous proposition that M is the connected component of M n 2d,γ which is isomorphic to M ρ via the map α.

Geometrical constructions

Let S be a 2t-polarized K3 surface of Picard rank one and n ≥ 2. Theorem 1.1 allows us to determine the values n, t for which Bir(S [n] ) = {id}, however it does not provide any indication on how to construct these automorphisms geometrically. If t = n, then Bir(S [n] ) is generated by Beauville's (non-symplectic) involution [3, §6]. For 2 ≤ n ≤ 10, 2 ≤ t ≤ 7, t = n, the pairs (n, t) such that Bir(S [n] ) = {id} are the following:

(n, t) = (2, 5), [START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0, Classification of algebraic and analytic manifolds[END_REF][START_REF] Boissière | The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface, K3 surfaces and their moduli[END_REF], (4, 7), (6, 2), (8, 2), [START_REF] Al | Automorphisms of Hilbert schemes of points on a generic projective K3 surface[END_REF][START_REF] Boissière | On the antisymplectic involutions of the Hilbert square of a K3 surface[END_REF], [START_REF]Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, Part II[END_REF][START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0, Classification of algebraic and analytic manifolds[END_REF], [START_REF]Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, Part II[END_REF][START_REF] Boissière | The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface, K3 surfaces and their moduli[END_REF].

The involution which generates Bir(S [n] ) is symplectic for (n, t) = (9, 3), [START_REF]Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, Part II[END_REF][START_REF] Boissière | The automorphism group of the Hilbert scheme of two points on a generic projective K3 surface, K3 surfaces and their moduli[END_REF], non-symplectic in the other cases.

The general K3 surface S of degree 2t = 10 is a transverse intersection Gr(2, 5) ∩ Γ ∩ Q ⊂ P 9 , where Γ ∼ = P 6 and Q is a quadric. The birational involution of S [2] was constructed by O'Grady in [22, Section 4.3], while a geometric description for the involution of S [3] has been provided in [START_REF] Debarre | Hyperkähler manifolds[END_REF]Example 4.12].

We give new constructions of the non-symplectic involutions for (n, t) = (6, 2), (8, 2). Let K 4 be the moduli space of 4-polarized K3 surfaces. We denote by D x,y ⊂ K 4 the Noether-Lefschetz divisor corresponding to polarized K3 surfaces (S, H) for which there exists B ∈ Pic(S) with (H, B) = x, B 2 = y and such that the sublattice of Pic(S) generated by B, H is primitive. A general K3 surface S of degree 4 can be embedded as a smooth quartic surface in P 3 and we consider the polarization H ∈ Pic(S) given by a hyperplane section.

Example 8.1 (n = 6, t = 2). Let S ⊂ P 3 be a smooth quartic surface which does not contain any twisted cubic curve, e.g. (S, H) / ∈ D 3,-2 . If p 1 , . . . , p 6 ∈ S are in general linear position there exists a single rational normal curve (i.e. a twisted cubic) passing through them, which we denote by C 3 . Since C 3 does not lie on S, the intersection S ∩ C 3 consists of twelve points. By associating the p i 's with the six residual points of intersection, we obtain a birational involution of S [6] .

For n = 6, t = 2 the equation X 2 -t(n -1)Y 2 = -1 is solvable, with minimal solution (a, b) = (3, 1). Thus for (S, H) ∈ K 4 with Pic(S) = ZH, the birational nonsymplectic involution σ which generates Bir(S [6] ) satisfies H 2 (S [6] , Z) σ * = Zν ∼ = 10 with ν = 5h -3δ.

Example 8.2 (n = 8, t = 2). Let S ⊂ P 3 be a smooth quartic surface such that, if a curve C ⊂ P 3 of degree four is contained in two different quadric surfaces, then C ⊂ S. This holds for instance outside the divisors D 4,0 , D 4,2 (notice that we do not have to exclude D 4,-2 , since a rational quartic curve in P 3 is contained in a unique quadric surface). Consider p 1 , . . . , p 8 ∈ S in general linear position. The linear system of quadric surfaces in P 3 passing through p 1 , . . . , p 8 is a pencil. Its base locus is a quartic curve C 4 , given by the intersection of two different divisors in the pencil. By the initial assumption C 4 is not contained in S, hence C 4 intersects S along sixteen points, eight of which are the p ′ i s. This gives rise to a birational involution of S [8] .

For n = 8, t = 2 the equation (n -1)X 2 -tY 2 = -1 is solvable, with minimal solution (a, b) = (1, 2). Thus for (S, H) ∈ K 4 with Pic(S) = ZH, the birational nonsymplectic involution σ which generates Bir(S [8] ) satisfies H 2 (S [8] , Z) σ * = Zν ∼ = 2 with ν = 2h -δ.

Take n, t which satisfy ( * ). If (n -1)X 2 -tY 2 = -1 admits solutions let (a, b) be the minimal one, otherwise we take (a, b) such that (a, b n-1 ) is the minimal solution of X 2 -t(n -1)Y 2 = -1. Consider the rational map n φ 2t : K 2t M(n, t) and U ⊂ K 2t as in Proposition 7.3. In the following, for a K3 surface S and L ∈ Pic(S) we denote by L ∈ Pic(S [n] ) the line bundle induced by L on S [n] (and, with a small abuse of notation, its class in NS(S [n] )). Proposition 8.3. For n ≥ 2, t ≥ 2 which satisfy ( * ) and (Ω, Θ) ∈ K 2t , assume that there is j ∈ Bir(Ω [n] ) which acts on H 2 (Ω [n] , Z) as -R D , for D = b Θ -aδ. Then for every (S, H) ∈ U there exists a deformation family of ihs manifolds π : X → B over an analytic connected base B, a line bundle L on X and points 0, p ∈ B such that • (π -1 (0), L | π -1 (0) ) ∼ = (Ω [n] , b Θ -aδ) and (π -1 (p), L | π -1 (p) ) ∼ = (S [n] , b H -aδ);

• for every b ∈ B \ {0} the pair (π -1 (b),

L | π -1 (b)
) is of the form (Σ [n] , b L-aδ) for some (Σ, L) ∈ U .

Proof. Since U is path-connected, for any (S 1 , H 1 ), (S 2 , H 2 ) ∈ U there exists a polarized deformation of K3 surfaces π ′ : (X ′ , H) → B ′ between (S 1 , H 1 ) and (S 2 , H 2 ) with B ′ ⊂ U Passing to the Hilbert schemes, this gives a polarized deformation π : (X , F ) → B := n φ 2t (B ′ ) between (S 2 , b H 2aδ), whose fibers are all of the form (Σ [n] , b L -aδ) for (Σ, L) ∈ U .

Consider now (Ω, Θ) as in the statement. If D is ample then (Ω, Θ) ∈ U , hence the previous argument allows us to conclude. On the other hand, if j is not biregular then D is movable. However, since U is open, we can still find a family of ihs manifolds with a line bundle π : (X , L) → B such that the fiber over a point 0 ∈ B is (Ω [n] , D), while for all b ∈ B, b = 0 the fiber π -1 (b) is of the form (Σ [n] , b L -aδ) for (Σ, L) ∈ U .

Combined with the results of Section 7, the proposition gives a description of a deformation path of ihs manifolds with an involution, from (Ω [n] , j) to (S [n] , σ), where σ is the non-symplectic involution of the Hilbert scheme of n points on a very general 2t-polarized K3 surface S. This deformation path has the advantage to be explicit, in the sense that it is induced by a deformation of K3 surfaces from Ω to S. Hence it lies in a 19-dimensional subspace of the 20-dimensional moduli space of ( 2d , ρ)-polarized ihs manifolds of K3 [n] -type with the prescribed action on cohomology. This allows us to keep track of the action of the involution on length n subschemes of K3 surfaces, along the deformation path.

As an application of Proposition 8.3, we give a geometrical description (up to deformation) of the biregular involutions of Hilbert schemes of very general K3 surfaces of degree 2t = 2((n -1)k 2 + 1). This extends the results of [4, §6] for n = 2.

For n, k ≥ 2 we consider the hyperbolic rank two lattice M n,k whose Gram matrix, with respect to a suitable basis, is

2n 2n + k -1 2n + k -1 2n
.

By [19, Corollary 2.9] there exists a K3 surface Σ whose Picard group is isomorphic to M n,k . Let )d 1 + yd 2 , with y ∈ Z and ∆ = y 2 (k -1)(4n + k -1) -4n. A direct computation shows that (N, d 2 ) • (N, L) > 0, hence d 2 and L are in the interior of the same chamber. By convexity, the same holds for d 1 = d2+L k+1 . By [14, Corollary 8.2.9] there exists an isometry ̺ ∈ H 2 (Σ, Z) which maps d 1 , d 2 , L to the ample cone. We denote D i = ̺(d i ) and H = ̺(L).

For i = 1, 2, we observe that D i is very-ample by [26, §8], since there is no B ∈ Pic(Σ) such that (B, B) = 0 and (B, D i ) ∈ {1, 2}. As a consequence, we have two distinct embeddings ϕ |Di| : Σ ֒→ P n+1 whose image is a surface of degree (D i , D i ) = 2n. Each of these embeddings gives rise to a Beauville involution ι i ∈ Bir(Σ [n] ), which acts on H 2 (Σ [n] , Z) as the reflection fixing the line spanned by D i -δ. Now we take j = ι 1 ι 2 ι 1 ∈ Bir(Σ [n] ): its invariant lattice is generated by ι * 1 ( D 2 -δ) = H -kδ.

Recall that for t = (n -1)k 2 + 1 the minimal solution of (n -1)X 2 -tY 2 = -1 is (k, 1). Thus, from the lemma and Proposition 8.3 we obtain the following geometric description.

Proposition 8.5. For n ≥ 2 and k ≥ n+3 2 , the non-natural, non-symplectic involution of the Hilbert scheme of n points on a K3 surface of Picard rank one in K 2((n-1)k 2 +1) can be described, up to a deformation as in Proposition 8.3,as the

  Definition 1.1]) hence it acts on A H 2 (S [n] ,Z) as ± id by [16, Lemma 9.2]. Moreover, since Tr(S [n] ) ∼ = Tr(S) has odd rank, σ * | Tr(S [n] ) = ± id (because σ * is a Hodge isometry; see [14, Corollary 3.3.5]). If t = 1, by imposing that σ * | Tr(S [n] ) glues with the isometry σ * | NS(S [n] ) of the form (5), we conclude (by [21, Corollary 1.5.2]) that σ * | Tr(S [n] ) = id if and only

Proposition 4 . 3 .

 43 For n ≥ 2 and t ≥ 1, assume that there exists a non-natural birational automorphism σ ∈ Bir(S[n] ) and either t = 1 or σ * | Tr(S [n] ) = id. Then t is n-irregular. Proof. By the proof of Proposition 3.1, if t = 1 or σ * | Tr(S [n] ) = id the axis of the reflection σ * | NS(S [n] ) is spanned by the class ν = bh-taδ, where (b, a) is the minimal solution of X 2 -t(n-1)Y 2 = 1. Moreover, n ≥ 9 and Mov(S [n

d 1 , d 2 ∈Lemma 8 . 4 .

 1284 Pic(Σ) be primitive generators such that (d1 , d 1 ) = (d 2 , d 2 ) = 2n, (d 1 , d 2 ) = 2n + k -1. For n ≥ 2 and k ≥ 3, there exists H ∈ Pic(Σ) ample with H 2 = 2((n -1)k 2 + 1). The Hilbert scheme Σ [n] has a birational involution whose action on H 2 (Σ [n] , Z) is -R ( H-kδ) .Proof. Let L = (k + 1)d 1 -d 2 ∈ Pic(Σ), which satisfies L 2 = 2((n -1)k 2 + 1). Consider the chamber decomposition of {x ∈ Pic(Σ) R : (x, x) > 0} where the walls are orthogonal to (-2)-classes. We show that d 1 , d 2 , L are in the interior of the same chamber. Any (-2)-class in Pic(Σ) is of the form N = ( -(2n+k-1)y± √ ∆ 2n

Table 1 .

 1 n-irregular values t for n ≤ 14 as in Corollary 4.7.Lemma

	n 2	n-irregular t's s.t. ν 2 = 2	5	n-irregular t's s.t. ν 2 = 2(n -1)
	3	3, 9		/
	4	4, 13		/
	5	5, 17		/
	6	6, 9, 21		/
	7	7, 25, 49		/
	8	2, 4, 8, 11, 16, 29, 37		/
	9	1, 9, 33, 57		/
	10	10, 13, 37, 61, 85		/
	11	11, 19, 41, 49, 121		/
	12	3, 4, 5, 12, 15, 25, 27, 45, 125		/
	13	1, 13, 49		/
	14 14, 17, 22, 38, 49, 53, 77, 121, 133		5

4.8

. For all n ≥ 3, the value t = n is n-irregular. For all n ≥ 2, the value t = 4n -3 is n-irregular.

  The next two lemmas give bounds for the number of equivalence classes of solutions of P 8t[START_REF]Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, Part II[END_REF] and P 8t[START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF], depending on t. Lemma 5.2. If t ≡ 3 (mod 18), then P 8t (12) is either not solvable or it has one class of solutions. If t ≡ 5, 11, 17 (mod 18), then P 8t (12) is either not solvable or it has two classes of solutions. In all other cases, P 8t (12) is not solvable. Proof. By Remark 5.1 the equation P 8t

	mod 2) and t ≡ 1 (mod 2), since 3 is not a quadratic residue modulo 4. By [20, Theorem 110], if P

2t

[START_REF] Beauville | Some remarks on Kähler manifolds with c 1 = 0, Classification of algebraic and analytic manifolds[END_REF]

is solvable it has one equivalence class of solutions if t ≡ 0 (mod 3), otherwise it has two classes of solutions, which are conjugate. Then one can easily see that the same holds for P 8t

[START_REF] Gritsenko | The Kodaira dimension of the moduli of K3 surfaces[END_REF]

.

from an isomorphism S → Σ is the moduli space M S (p 2 s, pqH, q 2 r) ∼ = M S (s, H, r). Notice that z ≡ ±1 (mod 2t) (i.e. Σ ∼ = S) if and only if {r, s} = {1, t}. Corollary 6.5. Let S be an algebraic surface with Pic(S) = ZH, H 2 = 2t and n ≥ 2 an integer such that S [n] admits an involution which is not biregular. Assume that t is not n-irregular, hence there exists an ihs birational model X of S [n] with a biregular involution. Then X is not isomorphic to the Hilbert scheme of n points on a K3 surface.

Proof. Let (z, w) be the minimal solution of X 2 -t(n -1)Y 2 = 1 with z ≡ ±1 (mod n -1). By contradiction, assume that X ∼ = Σ [n] for some K3 surface Σ. The groups Aut(Σ [n] ) and Aut(S [n] ) are different so S is not isomorphic to Σ, while Σ [n] and S [n] are birational. So z ≡ ±1 (mod 2t) by Theorem 6.2, which is in contrast with Theorem 1.1.

By [8, Proposition 1.1], Corollary 6.5 is applicable for the values t ≤ 2n-3 which satisfy Theorem 1.1 and which are not n-irregular (see Table 1), e.g. (n, t) = (6, 2).

Moduli spaces of polarized manifolds of K3 [n] -type

We say that an ihs manifold is of K3 [n] -type if it is deformation equivalent to the Hilbert scheme of n points on a K3 surface. Let Λ n = U ⊕3 ⊕E 8 (-1) ⊕2 ⊕ -2(n-1) , which is isometric to the second cohomology lattice of any ihs manifold of K3 [n] type (with the BBF quadratic form). A polarization type is the choice of a O(Λ n )orbit [k], with k ∈ Λ n primitive and (k, k) > 0. We denote by M [k] the moduli space of polarized manifolds (X, D), with X of K3 [n] -type and D a primitive ample class on X such that there exists an isomorphism η : H 2 (X, Z) → Λ n (called a marking) such that η(D) = k. More generally, we denote by M n 2d,γ the moduli space which parametrizes manifolds of K3 [n] -type with a polarization of square 2d and divisibility γ, for d, γ ∈ N (for more details on these moduli spaces, we refer to [START_REF]Moduli spaces of irreducible symplectic manifolds[END_REF] and [START_REF] Apostolov | Moduli spaces of polarized irreducible symplectic manifolds are not necessarily connected[END_REF]).

From [16, §7.1], two polarized manifolds (X 1 , D 1 ), (X 2 , D 2 ) ∈ M n 2d,γ are in the same connected component of the moduli space if and only if there exists a parallel transport operator g : 1) ⊕2 be in the canonical O(Λ)-orbit of primitive embeddings in the Mukai lattice (see [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Corollary 9.5]). We denote by T (X i , D i ) the rank-two positive definite lattice which arises as saturation, inside Λ, of the sublattice generated by ι Xi (D i ) and ι Xi (H 2 (X i , Z)) ⊥ . Then there exists a polarized parallel transport operator between (X 1 , D 1 ) and (X 2 , D 2 ) if and only

Let X be a manifold of K3 [n] -type with a non-symplectic biregular involution i ∈ Aut(X) such that H 2 (X, Z) i * has rank one. By [START_REF] Camere | Non-symplectic involutions of manifolds of K3 [n] -type[END_REF]Proposition 4.3] there exists an i * -invariant primitive class D ∈ A X such that (X, D) belongs to one of the following moduli spaces: • The moduli space M n 2,1 is connected; the polarization type is unique.

• If n ≡ 0 (mod 4), then M n 2,2 is connected; the polarization type is unique.

,n-1 has 2 ρ(n-1)-1 connected components; the polarization type is unique.

Given n, t ≥ 2 we define the property ( * ) : for an algebraic K3 surface S with Pic(S) = ZH, H 2 = 2t, the Hilbert scheme S [n] is equipped with a biregular non-natural automorphism.

Equivalently, this means that n, t satisfy the numerical conditions given in [8, Theorem 6.4].

Take n, t which satisfy ( * ). Let (a, b) (resp. (a, b n-1 )) be the integer solution of (n -1)X 2 -tY 2 = -1 (resp. X 2 -t(n -1)Y 2 = -1) with minimal X > 0, depending on which of the two equations is solvable (see Proposition 3.1). We consider the moduli space K 2t of 2t-polarized K3 surfaces and U ⊂ K 2t the subset of elements (Σ, H) such that D := bh -aδ ∈ NS(Σ [n] ) is ample, where as usual h ∈ NS(Σ [n] ) is the class of the line bundle induced by H and 2δ is the class of the exceptional divisor of the Hilbert-Chow contraction Σ [n] → Σ (n) . In particular, by Proposition 3.1 U contains all 2t-polarized K3 surfaces of Picard rank one. For (Σ, H) ∈ U , the polarized manifold (Σ

If D 2 = 2(n -1), we are interested in determining the connected component of M n 2(n-1),n-1 which contains (Σ [n] , D). The canonical embedding ι Σ [n] in the Mukai lattice satisfies (see [START_REF] Markman | A survey of Torelli and monodromy results for holomorphic-symplectic varieties[END_REF]Example 9.6])

where we identify Λ with

With respect to the basis {v, w}, the polarization is ι Σ [n] (D) = av -(n -1)w and the bilinear form of the lattice is given by the matrix 2(n -1) 2a 2a 2tβ 2 .

In the following we assume that n, t satisfy ( * ) and we denote by M(n, t) the connected component (of one the moduli spaces M n 2,1 , M n 2,2 , M n 2(n-1),n-1 ) which contains the polarized manifolds (Σ [n] , bh -aδ) as above, for (Σ, H) ∈ U . Lemma 7.2. For n ≥ 2, let t 1 , t 2 ≥ 2 be such that X 2 -t i (n -1)Y 2 = -1 is solvable, and let (a i , β i ) be the minimal solution. Assuming that both t 1 , t 2 satisfy ( * ), the connected components M(n, t 1 ), M(n, t 2 ) of M n 2(n-1),n-1 coincide if and only if a 1 ≡ ±a 2 (mod n -1).

Proof. By the previous discussion, the connected components coincide if and only if there exists an isometry λ :

In particular, λ(w 1 ) = ±w 2 and λ((n -1)w 1 ) = (n -1)w 2 -(a 2 ± a 1 )v 2 . Such isometry exists if and only if a 1 ≡ ±a 2 (mod n -1).

We now consider the map n φ 2t : U → M(n, t), (Σ, H) → (Σ [n] , bh -aδ). Clearly the image of n φ 2t is contained in the Noether-Lefschetz locus of M(n, t), that is the subset of polarized manifolds whose Picard group has rank at least two. Let C 2t ⊂ M(n, t) be the locus which parametrizes polarized manifolds (X, D) whose Picard group contains a primitive hyperbolic rank-two sublattice K ∋ D and such that K ⊥ ⊂ H 2 (X, Z) has discriminant -2t. The proof of [START_REF] Debarre | On the period map for polarized hyperkähler fourfolds[END_REF]Proposition 7.1] can be generalized to obtain the following result. Proposition 7.3. Given n, t which satisfy ( * ), the subset U ⊂ K 2t is open and the rational map n φ 2t :

For n ≥ 2, let M Λn be the moduli space of marked ihs manifolds of K3 [n] -type and fix a connected component

Following [START_REF] Joumaah | Non-symplectic involutions of irreducible symplectic manifolds of K3 [n] -type[END_REF] and [START_REF] Camere | Non-symplectic involutions of manifolds of K3 [n] -type[END_REF], we can consider the moduli space M n 2d ,ρ ⊂ M 0 Λn of (ρ, 2d )polarized manifolds of K3 [n] -type. An element of this moduli space is a marked manifold (X, η) which admits a non-symplectic involution i ∈ Aut(X) such that ρ = η•i * •η -1 . Notice that i is uniquely determined, by the injectivity of Aut(X) → O(H 2 (X, Z)), i → i * . By [START_REF] Camere | Non-symplectic involutions of manifolds of K3 [n] -type[END_REF]Theorem 3.3], the moduli space M n 2d ,ρ is non-empty if and only if the induced isometry ρ ∈ O(A Λn ) is ± id AΛ n .

Let (X, η), (X ′ , η ′ ) be two (ρ, 2d )-polarized manifolds, with involutions i ∈ Aut(X) and i

We say that the pairs (X, i), (X ′ , i ′ ) are isomorphic if there exists an isomorphism f :

Proof. This follows from [15, Theorem 10.5]: since (Λ n ) ρ has rank one, M ρ is irreducible by [START_REF] Joumaah | Non-symplectic involutions of irreducible symplectic manifolds of K3 [n] -type[END_REF]Theorem 9.11], i.e. all (ρ, 2d )-polarized manifolds of K3 [n] type deform in the same family. For the same reason every pair (X, i) ∈ M ρ is simple in the sense of [ • The proposition holds for all k ≥ 3 such that the birational involution which generates Bir(S [n] ), for S a 2((n -1)k 2 + 1)-polarized K3 surface of Picard rank one, is biregular (cfr. Conjecture 4.9).

• For k = 2 (i.e. t = 4n -3) the wall orthogonal to the (-2)-class d 1 -d 2 separates d 1 and d 2 , therefore the construction of Lemma 8.4 cannot be performed. We already know that this value of t is n-irregular (Lemma 4.8), hence Proposition 8.3 is not applicable.

For n = 2 both Beauville involutions in Proposition 8.5 are biregular, as one can show that there are no lines on the quartic surfaces ϕ |Di| (Σ) ⊂ P 3 . For k odd, this construction is [4, Theorem 6.1].


P Béjot

J Kasparian

Energy conservation in self-phase modulation

Keywords: numbers: 52.30.Cv, 52.35.Py, 05.45.-a, 52.55.Tn

Spectral broadening of ultrashort laser pulses is simultaneously described by either self-phase modulation (SPM) or four-wave mixing (FWM). The latter implies the instantaneous conservation of both the photon number and energy, while the former describes a time-dependent frequency shift, implying a violation of the energy conservation if the number of photons is to be conserved in each time slice. We resolve this paradox by considering the transient energy storage in the propagation medium, that can be calculated in the SPM formalism via the dephasing between the incident pulse and the medium polarization leading to an effective imaginary part in the third-order susceptibility. In parallel, considering the temporal variation of the incident intensity in FWM offsets the instantaneous frequency.

INTRODUCTION

Non-linear optics has been first investigated in condensed matter. The spectacular phenomenon of supercontinuum generation was observed as early as 1970 by Alfano and Shapiro [START_REF] Alfano | Emission in the region 4000 to 7000 Å via four-photon coupling in glass[END_REF][START_REF] Alfano | Observation of Self-Phase Modulation and Small-Scale Filaments in Crystals and Glasses[END_REF], who identified four-wave mixing (FWM) [START_REF] Alfano | Emission in the region 4000 to 7000 Å via four-photon coupling in glass[END_REF] and self-phase modulation (SPM) [START_REF] Alfano | Observation of Self-Phase Modulation and Small-Scale Filaments in Crystals and Glasses[END_REF] as the origin of this wide spectral broadening [START_REF] Zheltikov | Ultrashort light pulses in hollow waveguides[END_REF][START_REF] Dudley | Supercontinuum generation in photonic crystal fiber[END_REF][START_REF] Knight | Nonlinear waveguide optics and photonic crystal fibers[END_REF][START_REF] Alfano | The Supercontinuum laser source, Fundamentals with updated references[END_REF][START_REF] Boyd | Non-linear Optics[END_REF][START_REF] He | Physics of nonlinear optics[END_REF][START_REF] Loudon | Quantum theory of light[END_REF][START_REF] Govind | Nonlinear Fiber Optics[END_REF][START_REF] Shen | The Principles of Nonlinear Optics[END_REF]. The advent of chirped pulse amplification [START_REF] Strickland | Compression of amplified chirped optical pulses[END_REF] in 1985 allowed even more efficient spectral broadening, as well as its observation in gases including atmospheric pressure air.

FWM describes spectral broadening as the interaction of plane monochromatic waves. Two photons at frequencies ω 1 and ω 2 interact through the χ (3) susceptibility to generate two photons at ω 3 and ω 4 . The energy conservation imposes ω 1 + ω 2 = ω 3 + ω 4 , so that the spectrum should remain symmetrical after broadening via FWM, and the number of photons is conserved at any time.

The same spectral broadening is alternatively described by SPM as a deformation of the pulse. After propagating over a distance z, the carrier wave experiences a time-dependent frequency shift ∆ω(t) = -k 0 zn 2 I ′ (t)z, [START_REF] Alfano | Emission in the region 4000 to 7000 Å via four-photon coupling in glass[END_REF] where the prime denotes the temporal derivation, k 0 = n 0 ω/c, n 0 and n 2 = 3χ (3) /4n 0 ǫ 0 c are the linear and non-linear refractive indices, respectively, ω the angular frequency of the incident pulse, I its intensity, and c the velocity of light. SPM intrinsically induces an asymmetric time-dependent frequency shift. In most common media like air or glass n 2 > 0, so that the leading edge of the pulse is red-shifted while its trailing edge is blue-shifted. Such time-dependent frequency shift is incompatible with the simultaneous conservation of the pulse energy and of the number of photons.

In this Letter, we address this paradox. By taking into account the dephasing induced by SPM between the incident electric field and the polarization of the propagation medium, we show that the latter transiently stores energy, which restores energy conservation. Furthermore, considering the fast variation of the pulse intensity in the FWM formalism yields the asymmetric transient frequency shifts that are usually described in the SPM formalism.

DISCUSSION

Energy conservation in SPM.

We consider a pulse described within the slowly-varying envelope approximation (SVEA), so that the electric field expresses as E(t) = E 0 (t)(e -iωt + c.c.). It induces a third-order polarization

P (3) (t) = ǫ 0 χ (3) E(t) 3 (2)
ǫ 0 being the permittivity of vacuum. If we neglect absorption and dispersion, the evolution of this third-order polarization in a medium with an eigenfrequency ω e can be written in a perturbative approach [START_REF] Ettoumi | Generalized Miller formulae[END_REF]:

d 2 P (3) dt 2 + ω e 2 P (3) = N eQ (3) P (1) 3 (t) (3)
where P (1) = χ (1) E is the elastic polarizability, χ (1) the first-order susceptibility, E the electric field, N the local density of electrons, -e their charge, m their mass, and

Q (3) = N 3 e 4 mǫ 3 0 χ (3) χ (1) 4 (4)
Expanding the first-order polarization P (1) (t) = P

(1) 0 (t) e i(ωt-kz-φ (1) ) + c.c. in the right term of Equation (3), we get

d 2 P (3)
dt 2 + ω e 2 P (3) = N eQ (3) P

(1) 0

3 × e 3i(ωt-kz-φ (1) ) + 3e i(ωt-kz-φ (1) ) + c.c.

We decompose the polarization P (3) into its components P SPM and P TH , respectively oscillating at the fundamental and third harmonic (TH) frequencies ω and 3ω:

P (3) (t) = P TH 0 (t) e i(3(ωt-kz)-φ TH ) + c.c. + P SPM 0 (t) e i(ωt-kz-φ SPM ) + c.c. (6)
Here, φ TH and φ SPM are the dephasings of the polarization relative to the electric field for each spectral component. They are chosen such that P TH 0 and P SPM 0 are real. We focus on the terms oscillating at the fundamental frequency ω, consider the SVEA approximation, identify the real and imaginary parts of P SPM , and calculate φ (1) from the linear polarization equation ( 14) [START_REF] Ettoumi | Generalized Miller formulae[END_REF].

The SVEA implies dφ SPM /dt ≪ ω and d 2 P SPM 0 /dt 2 ≪ ω 2 P SPM 0 (t), so that the terms at frequency ω in Equation ( 5) rewrite:

ω 2 e -ω - dφ SPM dt 2 P SPM 0 (t) + 2i ω - dφ SPM dt dP SPM 0 dt + d 2 P SPM 0 dt 2 = 3N eQ (3) P (1) 0 3 (t) e i(φ SPM -φ (1) ) + c.c. = 3N eQ (3) ǫ 0 χ (1) E 0 3 e i(φ SPM -φ (1) ) + c.c. (7)
Identifying the components parallel and orthogonal to P SPM 0 in the complex plane yields

ω 2 e -ω 2 P SPM 0 (t) = 6N eQ (3) ǫ 0 χ (1) E 0 3 × cos φ SPM -φ (1) (8)
2ω dP SPM 0 dt = 6N eQ (3) ǫ 0 χ (1) E 0 3 × sin φ SPM -φ (1) (9)
which combine into

φ SPM -φ (1) ≈ tan φ SPM -φ (1) (10) = 2ω ω 2 e -ω 2 dP SPM 0 P SPM 0 (t)dt (11) = 6ω ω 2 e -ω 2 dE 0 E 0 (t)dt (12) = 3ω ω 2 e -ω 2 dI I(t)dt (13)
φ (1) can be estimated with a similar derivation starting from the linear propagation equation [START_REF] Ettoumi | Generalized Miller formulae[END_REF]:

d 2 P (1) dt 2 + ω 2 P (1) = N e 2 m E(t) (14)
resulting in

φ (1) = ω ω 2 e -ω 2 dI I(t)dt (15)
which finally yields

φ SPM = 4φ (1) = 4ω ω 2 e -ω 2 dI I(t)dt (16)
Far from resonance (ω ≪ ω e ), φ SPM ≈ 4ω/ω 2 e T ≪ 1, T being the pulse duration. For example, in air ω e ≈ 80 nm [START_REF] Zhang | Precision refractive index measurements of air, N2, O2, Ar, and CO2 with a frequency comb[END_REF] so that for typical experiments with a 100 fs pulse centered at 800 nm, φ SPM ≈ 10 -4 . Therefore, the supplementary self-phase modulation induced by this dephasing is fully negligible.

However, the dephasing φ SPM between the driving field E and the resulting polarization P SPM implies an energy transfer between the incident electric field and the propagation medium. The instantaneous power per unit volume transferred to the propagation medium amounts to

P SPM (t) = E(t) dP SPM dt ( 17
)
where the value of dP SPM 0 /dt is defined in Eq. ( 9), so that

P SPM (t) = 2E 0 (t) cos (ωt -kz) -2ωP SPM 0 (t) sin ωt -kz -φ SPM + 2 dP SPM 0 dt cos ωt -kz -φ SPM (18)
Averaging over one or a few optical cycles yields

P SPM (t) = -4ωE 0 P SPM 0 ω 2π 2π ω 0 cos(ωt) sin ωt -φ SPM dt + 4E 0 dP SPM 0 dt ω 2π 2π ω 0 cos(ωt) cos ωt -φ SPM dt (19) = 4ωE 0 P SPM 0 sin φ SPM 2 + 4E 0 dP SPM 0 dt cos φ SPM 2 (20)
Plugging the values of P SPM 0 and dP SPM 0 dt from Equations ( 8) and ( 9), we obtain

P SPM (t) ≈ 6ǫ 0 E 4 0 ωχ (3) cw sin φ SPM + χ (3) cw ω 2 e -ω 2 2ω sin φ SPM -φ (1) (21)
≈ 6ǫ 0 E 4 0 χ (3) cw φ (1) 4ω + 3 ω 2 e -ω 2 2ω (22) = 3ǫ 0 χ (3) cw E 4 0 φ (1) 3ω 2 e + 5ω 2 ω ( 23
)
Introducing the relations I = 2ǫ 0 cn 0 E 2 0 and χ

cw = 4n 0 ǫ 0 cn 2 /3, this equation revrites:

P SPM (t) = n 2 I n 0 c 3ω 2 e + 5ω 2 ω 2 e -ω 2 dI dt (24)
In typical conditions (e.g., at a wavelength of 800 nm), ω e ≫ ω, resulting in

P SPM (t) ≈ 3n 2 I n 0 c dI dt , (25)
On the leading edge of the pulse, dI/dt > 0 so that P SPM > 0: The field transfers energy to the medium and initiates the dipole oscillation, while on the trailing edge the dipoles return to rest and release their energy into the electromagnetic field. The net energy loss by the pulse is ∞ -∞ P SPM dt = 0 since I = 0 at both t = ±∞. The energy storage in the propagation medium is therefore transient and results in a net energy transfer from the red-shifted leading edge of the pulse towards its blue-shifted trail. It therefore reconciles red-and blue-shifts with the simultaneous conservation of energy and of the photon number: A temporal slice of the pulse cannot be considered as an isolated system. Rather, the completion of the system requires either to consider the propagation medium together with the pulse. For typical ultrashort laser filaments in air [START_REF] Kasparian | Physics and applications of atmospheric non-linear optics and filamentation[END_REF][START_REF] Bergé | Ultrashort filaments of light in weakly ionized, optically transparent media[END_REF][START_REF] Couairon | Femtosecond filamentation in transparent media[END_REF][START_REF] Chin | The propagation of powerful femtosecond laser pulses in optical media: physics, applications, and new challenges[END_REF][START_REF] Kasparian | White-Light Filaments for Atmospheric Analysis[END_REF] the relative energy transfer can reach 1 %/cm, enabling the strong reshaping occurring during their propagation.

An alternative way to understand the transient energy storage in the medium consists in grouping the dephasing φ SPM with the third-order susceptibility χ (3) and the associated non-linear refractive index n 2 , resulting in the effective values

χ (SPM) eff = χ (3) e iφ SPM (26) n SPM 2,eff = n 2 e iφ SPM ( 27
)
Their imaginary components χ (3) sin φ SPM ≈ χ (3) φ SPM and n 2 sin φ SPM ≈ n 2 φ SPM are intrinsically associated with gain or loss.

Link with self-steepening.

The above-derived transient energy storage has to translate into a depletion of the pulse intensity in the front, and a growth in its trail. Indeed, let us consider the self-steepening term affecting the envelope [START_REF] Bergé | Ultrashort filaments of light in weakly ionized, optically transparent media[END_REF]:

dE dz = -n 2 n 0 c d dt |E| 2 E ( 28
)
where |E| 2 = I. The relative variation of E is therefore:

dE Edz = -n 2 |E| 2 n 0 c d|E| 2 |E| 2 dt + dE Edt ( 29
)
which can convert into the relative variation of the intensity:

dI 2Idz = -n 2 I n 0 c dI Idt + dI 2Idt (30)
The local intensity variations due to self-steepening therefore amount to

dI dz = -3n 2 I n 0 c dI dt ( 31
)
which identifies with the instantaneous power gained from the medium, -P SPM (Eq. ( 25)), providing an interpretation of the self-steepening in terms of the deformation of the envelope due to the conservation of the photon number density in spite of their energy drift due to the SPM-induced frequency change. This interpretation is fully compatible with the usual one in terms of the stretching of the temporal pulse slices due to gradients in the group velocity. The latter focuses on the point of view of the wave deformation, while the former translates this deformation and the associated frequency shifts into photon energy and considers its implications for the simultaneous conservation of energy and the photon number.

Time-dependent frequency shift in FWM.

We shall now highlight how the FWM formalism accounts for a time-dependent frequency shift in spite of the energy conservation. We define the instantaneous spectrum at time t 0 as the Fourier transform of the pulse convolved by a temporal gate centered at t 0 and of width τ such that 1/ω ≪ τ ≪ T , where T is the pulse duration. To allow analytical derivations, we will consider a Gaussian gate in the following. However, other gate shapes may be considered without loss of generality.

Four-wave mixing is typically described by assuming an instantaneous response function of the third-order nonlinear susceptibility (Equation ( 2)) and focusing on the nonlinear polarization component in EE * E, oscillating at ω:

P SPM (t) = 4ǫ 2 0 cn 0 n 2 E 0 (t) 3 e -iωt + c.c. ( 32
)
Under the paraxial approximation, this non-linear polarization contributes to the field evolution:

2ik 0 dE(t) dz NL,ω = - ω 2 ǫ 0 c 2 P SPM (t) (33)
k 0 being the wavevector in vacuum. After plugging (32) into (33), the field evolves as

dE(t) dz = iC NL E(t) 3 , (34)
where C NL = 2k 0 ǫ 0 cn 0 n 2 . In order to evaluate the frequency shift of the instantaneous spectrum after a short propagation distance dz, we develop the electrical field at the first order in dz as

E(t, dz) = E(t, 0) + iC NL E(t, 0) 3 dz + O(dz). (35)
Now let us locally Fourier-transform Eq. ( 35) in the vicinity of t 0 after Taylor-expanding the slowly-varying field envelope as E 0 (t) ≈ E 0 (t 0 ) + (t -t 0 )E ′ 0 (t 0 ). Introducing ∆ω = ω -ω, and neglecting the influence of dephasing on the field amplitude, we get the intensity spectrum at first order in dz

I(ω ∝ Ê(ω), t 0 , dz) 2 ≈ e -τ 2 2 ∆ω 2 τ 2 E 2 0 (t 0 ) × 1 2 -3C NL E 0 (t 0 )E ′ 0 (t 0 )∆ωdz , (36)
where all terms involving E ′ 0 2 have been discarded based on the SVEA approximation. Deriving with respect to ∆ω and Taylor-expanding again at order 1 in the vicinity of ω yields the spectral peak of the spectrum at time t 0 :

∆ω * (t 0 ) = -6C NL E 0 E ′ 0 dz (37) = -k 0 n 2 I ′ 0 (t 0 )dz (38)
where we have introduced the intensity I 0 = 2n 0 ǫ 0 cE 2 0 . Similarly, the mean frequency is, at first order in dz:

ω (t 0 ) ≡ ω| Ê| 2 dω | Ê| 2 dω = ω - 24C NL E 3 0 (t 0 )E ′ 0 (t 0 )dz 4E 2 0 (t 0 ) + E ′ 0 2 τ 2 (39) ≈ ω -6C NL E 0 E ′ 0 dz (40) = ω -k 0 n 2 I ′ (t 0 )dz (41)
where E ′ 0 2 τ 2 ≪ E 2 0 due to the SVEA. Therefore, both the peak (Eq. ( 38)) and mean (Eq. ( 41)) transient frequency offset calculated with the FWM formalism are equal to those predicted by SPM (Eq. ( 1)), provided the temporal intensity variation of the incident pulse is taken into account. Note that the frequency offset does not depend on the width τ of the gate, which confirms that it is only a computation intermediate.

CONCLUSION

As a conclusion, we reconciled the time-dependent frequency shift described by self-phase modulation, with the energy and photon number conservations implied by the four-wave mixing formalism. An energy transfer occurs from the red-shifted leading edge of the pulse to the blue-shifted trailing edge, mediated by a transient energy storage in the propagation medium. The latter can be evidenced by considering the dephasing between the driving pulse and the medium polarization, induced by the Kerr effect. This energy transfer is also the origin of self-steepening, which translates as a depletion (resp. replenishment) of the pulse envelope on its front (resp. trailing edge). Furthermore, considering the time-dependent intensity in the FWM formalism reproduces this frequency shift, i.e., this transient energy storage in the medium. Our results straightforwardly generalize to non-degenerate FWM and XPM, although the calculation is slightly more tedious. We also note that we did not consider group velocity dispersion (GVD). Taking it into account would affect FWM via the phase matching conditions, and equivalently SPM via the envelope deformation, without impact on our argument.

Acknowledgements. This work was supported by the Swiss NSF (contracts 200021 155970 and 200020 175697). We gratefully acknowledge fruitful discussions with Jean-Pierre Wolf, Nicolas Berti and Wahb Ettoumi.