Vincent Bosser

Federico Pellarin

Hankel-type determinants and Drinfeld quasi-modular forms *

Keywords: Drinfeld quasi-modular forms, Hankel determinants, function fields of positive characteristic, AMS Classification 11F52, 14G25, 14L05

In this paper we introduce a class of determinants "of Hankel type". We use them to compute certain remarkable families of Drinfeld quasi-modular forms.

Introduction

To motivate this paper, we will first review some problem in the classical theory of quasi-modular forms. Let M Z be the Z-algebra generated by classical modular forms (for SL 2 (Z)) whose qexpansion has coefficients in Z. It is well known that M Z is the polynomial algebra Z[E 4 , E 6 , ∆] where E 4 , E 6 are the normalised (1 ) Eisenstein series of weights 4, 6 respectively, and where ∆ = (E 3 4 -E 2 6 )/1728 is the unique normalised cusp form of weight 12, so that, in particular, M Z is finitely generated.

Let now M Q be the Q-algebra of classical quasi-modular forms, as defined by Kaneko and Zagier in [START_REF] Kaneko | A generalized Jacobi theta function and quasimodular forms[END_REF], with the additional condition that their q-expansions have coefficients in Q. It is easy to show that

M Q = Q[E 2 , E 4 , E 6 ]
, where E 2 is the (non-modular) normalised Eisenstein series of weight 2, so this again is a finitely generated algebra, but over Q.

We may then formulate the following:

Problem 1. Compute a minimal set of generators for M Z = M Q ∩Z[[q]], the Z-algebra generated by quasi-modular forms of M Q whose q-expansions have coefficients in Z.

This problem is likely to be a difficult one. The examination of the q-expansions of the quasi-modular forms DE w with E w normalised Eisenstein series of weight w, D = qd/dq and Clausen-von Staudt Theorem, indicate that the algebra M Z is more likely not finitely generated, in contrast with the structure of M Z . What does a minimal set of generators of M Z look like?

In [START_REF] Kaneko | On extremal quasimodular forms[END_REF], Kaneko and Koike introduced a notion of extremal quasi-modular form (2 ). An extremal quasi-modular form of weight w and depth ≤ l is a non-vanishing polynomial in E 2 , E 4 , E 6 which is isobaric of weight w, whose degree in E 2 is not bigger than l, and such that the order of vanishing at q = 0 of its q-expansion is maximal. If w ≥ 0 is even and l ≥ 0, such a form exists and is proportional to a unique normalised form in M Q denoted by f l,w .

Kaneko and Koike, in [7, Conjecture 2], made a prediction on the size of the denominators of the coefficients of such forms which resembles in some way to a generalisation of Clausen-von Staudt Theorem. Indeed, if Conjecture 2 of loc. cit. holds, then f l,w ∈ Z p [[q]] for every prime number p such that p ≥ w, provided that l ≤ 4. In addition, the following question can be addressed.

Question. Let l be a non-negative integer, and denote by E l the set of w's such that f l,w exists, and belongs to Z [[q]]. For which l's is E l infinite?

Although very few of the f l,w 's are known to have q-expansion defined over Z (3 ), the feeling that we have, after extensive numerical computations, is that E l is infinite for 0 ≤ l ≤ 4 and finite for l > 4. In these circumstances, we would suggest to use these forms f l,w with w in E l to construct a set of generators for M Z but we refrain from making any kind of written prediction in this direction because this hypothesis is, so far, largely conjectural.

In this paper, we want to discuss similar problems, arising in the theory of Drinfeld quasimodular forms, where we have a slightly better understanding of what is going on. Let q = p e be a power of a prime number p with e > 0 an integer, let F q be the finite field with q elements. Let us consider, for an indeterminate θ, the polynomial ring A = F q [θ] and its fraction field

K = F q (θ).
Let K ∞ be the completion of K for the θ -1 -adic valuation and let us embed an algebraic closure of K ∞ in its completion C ∞ for the unique extension of that valuation. Following Gekeler in [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF], we denote by Ω the set C ∞ \ K ∞ , which has a structure of a rigid analytic space over which the group Γ = GL 2 (A) acts discontinuously by homographies, and with the usual local parameter at infinity u (denoted by t in [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF] and [START_REF] Bosser | Differential properties of Drinfeld quasi-modular forms[END_REF]). These facts lead quite naturally to the notion of Drinfeld quasi-modular forms, rather parallel to that of classical quasi-modular forms for SL 2 (Z), which are studied in [START_REF] Bosser | Differential properties of Drinfeld quasi-modular forms[END_REF], and to which we refer for the required background.

Following [START_REF] Bosser | Differential properties of Drinfeld quasi-modular forms[END_REF], we have three remarkable formal series E, g, h ∈ A[[u]] algebraically independent over K(u), representing respectively: the u-expansion of a Drinfeld quasi-modular form of weight 2, type 1 and depth 1 (the false Eisenstein series of weight 2 of Gekeler [START_REF] Gekeler | On the coefficients of Drinfeld modular forms[END_REF]), the u-expansion of an Eisenstein series of weight q -1 and type 0, and the u-expansion of a Poincaré series of weight q + 1 and type 1. The first terms of these formal series are as follows, where [i] = θ q i -θ for i > 0 integer (see [START_REF] Bosser | Differential properties of Drinfeld quasi-modular forms[END_REF]Lemma 4.2]):

E = u + u q 2 -2q+2 + • • • ∈ uA[[u q-1 ]] g = 1 -[1]u q-1 -[1]u q 3 -2q 2 +2q-1 + • • • ∈ A[[u q-1 ]] h = -u -u q 2 -2q+2 + • • • ∈ uA[[u q-1 ]].
Let M w,m be the K-vector space of Drinfeld modular forms of weight w, type m, whose uexpansions are defined over K, which also is the space of isobaric polynomials (for weights and types) in g and h with coefficients in K (4 ). The K-vector space of Drinfeld quasi-modular forms of weight w, type m and depth ≤ l, defined over K is the space

M ≤l w,m = M w,m ⊕ M w-2,m-1 E ⊕ • • • ⊕ M w-2l,m-l E l .
All these spaces are finite dimensional subspaces of K[[u]] and we may form the K-algebra of Drinfeld quasi-modular forms

M K = K[E, g, h] = w,m l M ≤l w,m .
In analogy with the Problem 1, we have:

Problem 2.
Compute a minimal set of generators for M A , the A-algebra generated by quasimodular forms of M K whose u-expansions have coefficients in A.

We say that an element f of M ≤l w,m \{0} is an extremal Drinfeld quasi-modular form if ord u=0 f is maximal among the orders at u = 0 of non-zero elements of that vector space. If there exists an extremal Drinfeld quasi-modular form of M ≤l w,m (5 ), we denote by f l,w,m the unique normalised such form.

To present our main result, we need to define a certain double sequence of quasi-modular forms (E j,k ) j∈Z,k≥1 .

Let us denote as usual by ∆ = -h q-1 ∈ A[[u]] the opposite of the unique normalised cusp form of weight q 2 -1 for GL 2 (A), and let us extend the notation [j] to non-positive integers by simply writing [j] = θ q j -θ for j ∈ Z, so that [0] = 0 and [-1] = θ 1/q -θ. The sub-sequence (E j,1 ) j∈Z is defined inductively in the following way. We set

E 0,1 = E, E 1,1 = -Eg+h [1]
and then, for j ≥ 0, by

E j+2,1 = - 1 [j + 2]
(∆ q j E j,1 + g q j+1 E j+1,1 ), and for j ≤ 1, by E j-2,1 = -1 ∆ q j-2 ([j]E j,1 + g q j-1 E j-1,1 ). For example, we have the following particular cases:

E q -1,1 = -h, E q 2 -2,1 = -hg q , E q 3 -3,1 = -h(g q+1 -[1] q h q-1 ) q ,
and in general, for all j ≤ -1, it is possible to check that E q j -j,1 is a Drinfeld cusp form of weight q j + 1 and type 1.

Let us write

B k (t) := 0≤i<j<k (t q j -t q i ) ∈ F q [t].
For k ≥ 2 and j ∈ Z, we then define E j,k with the following determinant of Hankel type:

E j,k = 1 B k (θ) E j,1 E j+1,1 • • • E j+k-1,1 E q j-1,1 E q j,1 • • • E q j+k-2,1 E q 2 j-2,1 E q 2 j-1,1 • • • E q 2 j+k-3,1 . . . . . . . . . E q k-1 j-k+1,1 E q k-1 j-k+2,1 • • • E q k-1 j,1
.

We shall show:

Theorem 1 The following properties hold, for j ≥ 0 and k ≥ 1.

1. There exists a constant C(q, k) and a sequence of integers (l k ) k≥1 such that for all j ≥ C(q, k), E j,k ∈ M ≤(q k -1)/(q-1)

(q k -1)(q j +1)/(q-1),k \ M ≤l k (q k -1)(q j +1)/(q-1),k with l k → ∞ for k → ∞.
2. For all j, k with j ≥ 0, we have ord u=0 E j,k = q j (q 2k -1)/(q 2 -1).

3. For all j, k with j ≥ 0, we have

E j,k ∈ A[[u]
] and E j,k is normalised.

4. For k = 1 and for k = 2 if q ≥ 3, we have E j,k = f (q k -1)/(q-1),(q k -1)(q j +1)/(q-1),k for all j ≥ 0.

In particular, for all j, k, E j,k is non-zero, property which does not seem to follow directly from the definition above. The interest of the theorem is that it provides in an explicit way a family of normalised Drinfeld quasi-modular forms parametrised by Z ≥0 × Z >0 , with unbounded depths and weights, with high order of vanishing at u = 0, and with u-expansions defined over A. The theorem gives a partial answer to the analogue of the Question above. Indeed, denoting by E l,m the set whose elements are the weights w such that f l,w,m is defined over A, we have the following obvious consequence of Theorem 1.

Corollary 2 If l = 1 and for any value of q, or if q ≥ 3 and l = q + 1, we have that f l,l(q j +1),l ∈ A[[u]] for all j ≥ 0. Therefore, for the selected values of q, l, m, the set E l,m has infinitely many elements.

It can be shown that for k > 2, the degree of E j,k in E is not equal to (q k -1)/(q -1), that is, it is not maximal (it is maximal only for k = 1, 2), which may mean that for such values, E j,k is not extremal. However, the fact that l k → ∞ suggests that no natural threshold for the depth (as l = 4 in the classical case, as suggested by [START_REF] Kaneko | On extremal quasimodular forms[END_REF]Conjecture 2]) exists in the Drinfeldian framework. Moreover, the presence of infinitely many f l,w,m 's defined over A detected by Theorem 1 suggests that the A-algebra M A generated by the Drinfeld quasi-modular forms with u-expansions defined over A could have, as a minimal set of generators, the f l,w,m 's with w ∈ E l,m for all l, m's.

Remark. With the help of a formula appearing in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF], it is possible to explicitly compute the u-expansions of E q j -j,1 for j ≥ 0: we have E q j -j,1 = a∈A + a q j u a with the notations of loc. cit. These forms, which are Hecke eigenforms, are also object of investigations by A. Petrov (private communication).

Determinants of Hankel's type

An inversive difference field (K, τ ) is the datum of a field K together with an automorphism τ that will be supposed of infinite order. The τ -constant subfield K τ is by definition the subfield of K of all the elements x ∈ K such that τ x = x. Every inversive difference field can be embedded in an existentially closed field K ex , that is a field endowed with an extension of τ such that K τ = (K ex ) τ , in which every polynomial τ -difference equation has at least a non-trivial solution.

We need now to choose a field K with two distinguished automorphisms to serve our purposes. Consider two indeterminates t, u and the field of formal series

R = K((t))((u)).
The Frobenius F q -linear endomorphism F of R splits as a product

F = χτ = τ χ,
where χ, τ : R → R are respectively K((u))-and F q ((t))-linear, uniquely determined by χ(t) = t q , τ (u) = u q and τ θ = θ q . The perfection

K = R perf = i≥0 F q (θ 1/q i )((t 1/q i ))((u 1/q i ))
of R is then endowed with extensions of τ and χ such that both the difference fields (K, τ ) and (K, χ) are inversive. Also, K τ is equal to the perfect closure F q ((t)

) perf of F q ((t)) in K and K χ is equal to the perfect closure K((u)) perf of K((u)) in K.
Let x 1 , . . . , x s be elements of K. Their τ -wronskian is the determinant:

W τ (x 1 , . . . , x s ) = det      x 1 τ x 1 • • • τ s-1 x 1 x 2 τ x 2 • • • τ s-1 x 2 . . . . . . . . . x s τ x s • • • τ s-1 x s      .
We recall from [10] that x 1 , . . . , x s are K τ -linearly independent if and only if W τ (x 1 , . . . , x s ) = 0. Similarly, the χ-wronskian W χ (x 1 , . . . , x s ) of x 1 , . . . , x s can be introduced, and x 1 , . . . , x s are K χ -linearly independent if and only if W χ (x 1 , . . . , x s ) = 0. For f ∈ K, we introduce the following sequence of determinants of Hankel type:

H k (f ) = f τ f τ 2 f • • • τ k-1 f χf χτ f χτ 2 f • • • χτ k-1 f . . . . . . . . . . . . χ k-1 f χ k-1 τ f χ k-1 τ 2 f • • • χ k-1 τ k-1 f
.

The proposition below will be used later.

Proposition 3

The following conditions are equivalent.

(i) H k (f ) = 0 for some k ≥ 1.
(ii) There exist s ≥ 1, elements λ 1 , . . . , λ s in F q ((t)) perf and elements b 1 , . . . , b s in some algebraic closure K((u)) alg of K((u)) such that, in some existentially closed extension of (K, τ

) containing K((u)) alg , f = λ 1 b 1 + • • • + λ s b s .
(iii) For some s ≥ 1, there exist elements µ 1 , . . . , µ s in K((u)) perf , and elements b ′ 1 , . . . , b ′ s in F q ((t)) alg , an algebraic closure of F q ((t)), such that

f = µ 1 b ′ 1 + • • • + µ s b ′ s , in some existentially closed extension of (K, χ) containing F q ((t)) alg .
Proof. It is easy to show that each of the second and the third conditions separately implies the first. Let us show that the first condition implies the second. Assuming that H k (f ) = 0 for some k ≥ 1 is equivalent to say that W χ (f , τ f , . . . , τ k-1 f ) = 0. Hence, there exist a 0 , . . . , a s ∈ K χ = K((u)) perf with a 0 a s = 0, such that

a 0 f + a 1 τ f + • • • + a s τ s f = 0.
On the other hand, the algebraic equation

a 0 X + a 1 X q + • • • + a s X q s = 0 has s solutions b 1 , . . . , b s in an algebraic closure K((u)) alg of K((u)), which are linearly indepen- dent over the field (K τ ) F = F q . In particular, W F (b 1 , . . . , b s ) = 0.
Let us consider the compositum F of K and K((u)) alg in some existentially closed extension of the difference field (K, τ ) (so we embed K((u)) alg in the existentially closed difference field (K ex , τ )). The restriction τ | K((u)) alg of τ is equal to the restriction of the Frobenius

F | K((u)) alg . Moreover, obviously, W F (b 1 , . . . , b s ) = W τ (b 1 , . . . , b s ) so that b 1 , . . . , b s are also F τ -linearly in- dependent, F τ being equal to F q ((t)) perf . Since b 1 , . . . , b s span the F τ -vector space of solutions of the equation a 0 X + a 1 τ X + • • • + a s τ s X = 0,
we obtain the second property.

The proof that the first property implies the third is similar and left to the reader, who will notice that it suffices to transpose the matrix used to define H k (f ).

Remark. It is easy to show, writing H s,k at the place of τ s H k (f ) for a better display, that the following formula holds:

H q+1 s,k -H q s,k-1 H s,k+1 = H q s-1,k H s+1,k , (s ∈ Z, k ≥ 2). (1)
Formula (1) plays a role for (τ, χ)-difference fields similar to that of Sylvester's formula expressing determinants

∂ i+j f ∂z i 1 ∂z j 2 0≤i,j≤k-1
as in [START_REF] Amice | Les nombres p-adiques. PUF. Collection SUP[END_REF].

The elements f = i,j c i,j t i u j that we choose are either d, either E = -hτ d, where d is the unique solution (cf. [START_REF] Pellarin | Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms[END_REF]) in

F q [t, θ][[u]] ⊂ A[[t]][[u]] of the linear τ -difference equation (t -θ q )∆(τ 2 X) + g(τ X) -X = 0, (2)
with c 0,0 = 1 and c i,0 = 0 for i > 0. We point out that in [START_REF] Pellarin | Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms[END_REF] we have computed some coefficients of the u-expansion of d. See also the remark after Lemma 9 below. The relationship between H k (d) and H k (E) is simple. Since χh = h, we have

χ i-1 τ j-1 (E) = -h q j-1 τ χ i-1 τ j-1 (d) (1 ≤ i, j ≤ k), hence H k (E) = (-1) k h 1+•••+q k-1 τ (H k (d)) = (-1) k h q k -1 q-1 τ (H k (d)). (3)
Lemma 4 We have, for j ∈ Z and k ≥ 1:

E j,k = τ j H k (E) B k t=θ .
Proof. For all k, H k (f ) can be rewritten, thanks to the identity χ = F τ -1 , as

H k (f ) = f τ f τ 2 f • • • τ k-1 f (τ -1 f ) q f q (τ f ) q • • • (τ k-2 f ) q . . . . . . . . . . . . (τ 1-k f ) q k-1 (τ 2-k f ) q k-1 (τ 3-k f ) q k-1 • • • f q k-1 . ( 4
)
It is proved in [START_REF] Pellarin | Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms[END_REF] that E| t=θ = E = E 0,1 . Moreover, by Lemma 22 of [START_REF] Pellarin | Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms[END_REF] we have

τ E = 1 t-θ q (gE + h), hence (τ E)| t=θ = E 1,1
. Now, as one sees from Equation (2) above, or by Proposition 9 of [START_REF] Pellarin | Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms[END_REF], the function E satisfies the linear τ -difference equation

(t -θ q 2 )(τ 2 E) = g q (τ E) + ∆E.
It easily follows from this, by induction, that (τ j E)| t=θ is well defined for all j ∈ Z, and is equal to E j,1 . Comparing the definition of E j,k with (4) we immediately recover that the forms E j,k of Theorem 1 are, for k ≥ 1, precisely the formal series of K[[u]] obtained by substituting t with θ in τ j H k (E)/B k , a licit operation.

Properties of the determinants H k (d)

Let k ≥ 1 be an integer. Either H k (d) = 0, or there exists ν k ∈ Z ≥0 such that

H k (d) = s≥ν k κ k,s u s
with κ k,s ∈ F q [t, θ] and κ k,ν k = 0. We will prove the Theorem below, from which we will deduce Theorem 1.

Theorem 5

We have H k (d) = 0 for all k ≥ 1, and the following properties hold.

1. ν k = (q k -1)(q k-1 -1)

q 2 -1 , 2. κ k,ν k = B k (t), 3. H k (d)/κ k,ν k lies in F q [t, θ][[u q-1
]] and is normalised.

Corollary 6

The function d can be expressed neither as a finite linear combination λ 1 b 1 + • • • + λ s b s with λ 1 , . . . , λ s ∈ F q ((t)) perf and b 1 , . . . , b s ∈ K((u)) alg , nor as a finite linear combination

µ 1 b ′ 1 + • • • + µ s b ′ s with µ 1 , . . . , µ s ∈ K((u)) perf and b ′ 1 , . . . , b ′ s ∈ F q ((t)
) alg . Proof. By Theorem 5, H k (d) = 0 for all k. Therefore, we can apply Proposition 3.

The rest of this section is devoted to the proof of Theorem 5. Since the u-expansions of many forms involved (like g, ∆, d . . .), are actually expansions in powers of u q-1 , it will be convenient to set v := u q-1 .

In Section 3.1, we first prove a general divisibility property for the coefficients of the u-expansion

of H k (f ) for formal series f ∈ F q [t, θ][[v]].
Then, in Section 3.2, we carefully study the growth of the degree in t of the coefficients of d. Finally, we complete the proof of Theorem 5 in Section 3.3.

Computation of normalisation factors

Proposition 7 Let f be a formal series in F q [t, θ][[v]], so that we have a formal series expansion H k (f ) = s≥0 κ s v s with κ s ∈ F q [t, θ] for all s. Then, the polynomial B k (t) divides κ s for all s ≥ 0.

Proof. We observe that if for 1 ≤ i, j ≤ k we have formal expressions f i,j = s∈I c i,j,s , then, by multilinearity:

f 1,1 • • • f 1,k . . . . . . f k,1 • • • f k,k = s1,...,s k ∈I c 1,1,s1 • • • c 1,k,s k . . . . . . c k,1,s1 • • • c k,k,s k . ( 5
)
Let us write f = s≥0 c s v s with c s ∈ F q [t, θ]. We set

f i,j = χ i-1 τ j-1 (f ) = s≥0 χ i-1 τ j-1 (c s v s ) = s≥0 c s (t q i-1
, θ q j-1 )u q j-1 s so that c i,j,s = c s (t q i-1 , θ q j-1 )v sq j-1 . By ( 5), we obtain that

H k (f ) = s1,s2,...,s k v s1+s2q+•••+s k q k-1 d s1,s2,...,s k , (6)
where

d s1,s2,...,s k = c s1 (t, θ) c s2 (t, θ q ) • • • c s k (t, θ q k-1 ) c s1 (t q , θ) c s2 (t q , θ q ) • • • c s k (t q , θ q k-1 ) . . . . . . . . . c s1 (t q k-1 , θ) c s2 (t q k-1 , θ q ) • • • c s k (t q k-1 , θ q k-1 ) . (7)
We use the fact that c s = µ κ µ,s θ µ ∈ F q [t, θ], with κ µ,s ∈ F q [t]. Let us apply (5) again, this time with f i,j = χ i-1 τ j-1 c sj = µ (χ i-1 κ µ,sj )θ µq j-1 and c i,j,µ = (χ i-1 κ µ,sj )θ µq j-1 . We obtain that d s1,...,s k = µ1,...,µ k θ µ1+µ2q+•••+µ k-1 q k-1 e µ1,...,µ k , where e µ1,...,µ k =

η 1 • • • η k . . . . . . χ k-1 η 1 • • • χ k-1 η k ,
with η j = κ µj ,sj . Now, by multilinearity, e µ1,...,µ k is a sum of Moore's determinants:

M (ν 1 , . . . , ν k ) = t ν1 • • • t ν k . . . . . . t ν1q k-1 • • • t ν k q k-1
.

We then apply the following lemma, which completes the proof of Proposition 7.

Lemma 8

The formula

M (0, 1, . . . , k -1) = B k (t) holds. Moreover, for any choice of ν 1 , . . . , ν k , B k (t) divides M (ν 1 , . . . , ν k ).
Proof. The explicit formula is a well known application, either of Moore's determinants, or Vandermonde's determinants. As for the divisibility property, this follows from an old and well known result of Mitchell, [START_REF] Mitchell | Note on Determinants of Powers[END_REF], as M (ν 1 , . . . , ν k ) can be viewed as a generalised Vandermonde's determinant.

The degrees of the coefficients of d

To prove Theorem 5, we will need a precise estimate of the growth of the degree in t of the coefficients of d. Recall that the function d lies in F q [t, θ][[v]], where v = u q-1 . We will write in what follows

d = s≥0 c s v s , (8)
where c s ∈ A[t]. The aim of this section is to prove the following lemma.

Lemma 9 Let s ≥ 0 and l ≥ 0 be integers satisfying

s < 1 + q 2 + • • • + q 2l . Then deg t c s ≤ l.
Moreover, for all l ≥ 0 we have

c 1+q 2 +•••+q 2(l-1) (t) = (-1) l t l + • • • , ( 9
)
where the dots stand for terms of degree < l.

Remark. We have used here the convention that the empty sum is zero, so we have 1 +

q 2 + • • • + q 2(l-1) = 0 when l = 0. Proof. Write g = 1 -[1]v + • • • = s≥0 γ s v s ∈ A[[v]],
and

∆ = -v(1 -v q-1 + • • • ) = s≥0 δ s v s ∈ vA[[v]].
As in [START_REF] Pellarin | Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms[END_REF], we will use the following recursion formula for the coefficients c s , which easily follows from the τ -difference equation (2) (see [9, Formula (30)]) :

c s = i+jq=s γ i (τ c j ) + (t -θ q ) i+jq 2 =s δ i (τ 2 c j ). ( 10
)
We first prove by induction on s ≥ 0 that deg t c s ≤ l for all l satisfying 1 + q 2 + • • • + q 2l > s. This statement is clearly true for s = 0 and s = 1, since c 0 = 1 and c 1 = -(t -θ). Let now s ≥ 2 and l ≥ 0 be such that s < 1 + q 2 + • • • + q 2l , and consider the formula [START_REF] Pellarin | τ -recurrent sequences and modular forms[END_REF]. If j is an index occurring in the first sum, then we have j ≤ s/q < s, hence

deg t τ c j = deg t c j ≤ l (11)
by induction hypothesis. Let now (i, j) be a pair of indices occurring in the second sum. If i = 0, then δ i = 0 and δ i (τ

2 c j ) = 0. If i ≥ 1, then j ≤ (s -1)/q 2 < 1 + • • • + q 2(l-1) , so deg t τ 2 c j = deg t c j ≤ l -1 ( 12
)
by induction hypothesis applied to j and l -1 (note that j < s). Since the coefficients γ i and δ i do not depend on t, it follows from ( 11), ( 12) and ( 10) that deg t c s ≤ l as required.

Let us now prove the second part of the lemma. We argue by induction on l. For l = 0 and l = 1 the assertion is true. Let now l ≥ 2 be an integer, and suppose that the formula (9) holds for l -1. Put s := 1 + • • • + q 2(l-1) , and consider again the recursion formula [START_REF] Pellarin | τ -recurrent sequences and modular forms[END_REF]. If j is any index appearing in the first sum, then, as before, j < s = 1 + • • • + q 2(l-1) . Hence deg t τ c j = deg t c j ≤ l -1 by the first part of the lemma. Let us now consider a pair (i, j) appearing in the second sum of [START_REF] Pellarin | τ -recurrent sequences and modular forms[END_REF]. The smallest possible value for i is i = 1 (since s ≡ 1 (mod q 2 )), for which we have j = 1 + • • • + q 2(l-2) . In this case, the induction hypothesis yields (since 2) , hence deg t (τ 2 c j ) = deg t c j ≤ l -2 by the first part of the lemma. It follows from these considerations that

δ 1 = -1) δ i (τ 2 c j ) = (-1) l t l-1 + • • • If now i > 1, then j < 1 + • • • + q 2(l-
(t -θ q ) i+jq 2 =s δ i (τ 2 c j ) = (t -θ q )((-1) l t l-1 + • • • ) = (-1) l t l + • • • Summing up, we have proved that c 1+q 2 +•••+q 2(l-1) (t) = (-1) l t l + • • • Remark.
The following explicit formula can be deduced from [START_REF] Pellarin | τ -recurrent sequences and modular forms[END_REF], see [START_REF] Pellarin | Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms[END_REF].

d =1 -(t -θ)v -(t -θ)v q 2 -q+1 + (t -θ)v q 2 + (t -θ)(t -2θ q + θ)v q 2 +1 -(t -θ)(t -θ q )v q 2 +q + (θ q -θ)(t -θ)(t -θ q )v q 2 +q+1 + • • • (13)

Proof of Theorem 5

We can now begin the proof of Theorem 5. We write as before

d = s≥0 c s v s ,
where c s ∈ A[t]. It will be convenient to introduce the following notation. If s = (s 1 , . . . , s k ) ∈ N k (where N = {0, 1, . . .}), we define

||s|| := k i=1 s i q i-1
and, as in Section 3.1, Equation [START_REF] Kaneko | On extremal quasimodular forms[END_REF],

d s := det(χ i-1 τ j-1 c sj ) 1≤i,j≤k = det(C s1 , C s2 , . . . , C s k ),
where C sj is the column vector defined by

C sj =      τ j-1 c sj χ(τ j-1 c sj ) . . . χ k-1 (τ j-1 c sj )      .
With this notation, the formula (6) writes

H k (d) = s d s v ||s|| , (14)
where s runs over all k-tuples of N k . To prove Theorem 5, we will show that the first non zero coefficient in the v-expansion (14) is obtained for only one multi-index s, namely for

s 0 := (1 + q 2 + • • • + q 2(k-2) , . . . , 1 + q 2 , 1, 0).
This will easily yield the theorem. We will need for this three lemmas.

Lemma 10 Set s 0 := (1 + q 2 + • • • + q 2(k-2) , . . . , 1, 0) ∈ N k . Then we have ||s 0 || = (q k -1)(q k-1 -1) (q 2 -1)(q -1)
and

d s 0 = B k (t).
Proof. The first part of the lemma amounts to compute the double sum

k i=1 k-1-i j=0 q 2j q i-1 ,
which is an exercise left to the reader.

To prove the second part, we use Lemma 9, Equality (9) :

d s0 = (-1) k-1 t k-1 + • • • • • • -t + θ q k-2 1 (-1) k-1 t (k-1)q + • • • • • • -t q + θ q k-2 1 . . . . . . . . . (-1) k-1 t (k-1)q k-1 + • • • • • • -t q k-1 + θ q k-2 1
.

Let us denote by C 1 , . . . , C k the columns of this matrix. If we substract θ q k-2 C k to C k-1 , then we eliminate the constant terms in C k-1 , that is, we get the new penultimate column C ′ k-1 = t (-t, -t q , . . . , -t q k-1 ). By substracting now to the column C k-2 a suitable linear combination (with coefficients in F q [θ]) of the last two columns, we get the new column C ′′ k-2 = t (t 2 , t 2q , . . . , t 2q k-1 ). Repeating this process for the columns C , j = k -3, . . . , 1, we see by induction that

d s 0 = (-1) k-1 t k-1 • • • -t 1 (-1) k-1 t (k-1)q • • • -t q 1 . . . . . . . . . (-1) k-1 t (k-1)q k-1 • • • -t q k-1 1 . Now, this determinant is equal to 1 t • • • t k-1 1 t q • • • t (k-1)q . . . . . . . . . 1 t q k-1 • • • t (k-1)q k-1
, which is equal to B k (t) (Vandermonde determinant; see also Lemma 8).

The next lemma roughly says that if a coefficient d s is not zero in (14), and if we reorder the coefficients c si such that the sequence (deg t c si ) i is non decreasing, then the degrees deg t c si grow at least linearly in i.

Lemma 11 Let d = (s 1 , . . . , s k ) ∈ N k such that d s = 0. Let (i 1 , . . . , i k ) be a permutation of the set {1, . . . , k} such that

deg t c si 1 ≤ • • • ≤ deg t c si k .
Then, for all l, we have

deg t c si l ≥ l -1.
Proof. Let us write d s = det(C s1 , . . . , C s k ). Suppose that there exists an l such that deg t c si l ≤ l -2. Then, since the operator τ does not change the degree in t, the family (c si 1 , . . . , τ i l -1 c si l ) consists of l polynomials in K[t] of degree ≤ l -2, so they are linearly dependent over K. Hence there exist elements λ j ∈ K, not all zero, such that l j=1 λ j τ ij -1 c si j = 0.

If we now apply the operator χ i-1 (1 ≤ i ≤ k), we find :

l j=1 λ j χ i-1 τ ij -1 c si j = 0 (1 ≤ i ≤ k).
In other words, we get l j=1 λ j C ij = 0, that is, a non trivial linear combination of the columns (i 1 , . . . , i l ) in d s . Hence d s = 0, which is a contradiction.

We introduce a further notation. If σ ∈ S {1,...,k} is a permutation of the set {1, . . . , k} and if s = (s 1 , . . . , s k ) is an element of N k , we define s σ := (s σ(1) , . . . , s σ(k) ). We recall that s 0 was defined in Lemma 10.

Lemma 12 Let σ be a permutation of the set {1, . . . , k} such that σ = Id. Then

||s σ 0 || > ||s 0 ||.
Proof. We argue by induction on k. For k = 1 there is nothing to prove. Let now k ≥ 2 be an integer and let σ be a permutation as in the lemma. For l ≥ 1, define t l by t l := 1 + • • • + q 2(l-2) . We will also use the notation s t l q k-l .

Let further τ denote the permutation of {1, . . . , k} such that s (k) 0 σ = (t τ (k) , . . . , t τ (1) ).

First, suppose that τ (k) = k. Then τ induces a non trivial permutation of the set {1, . . . , k -1}, and

|| s (k) 0 σ || -||s (k) 0 || = k-1 l=1 (t τ (l) -t l )q k-l = q || s (k-1) 0 σ ′ || -||s (k-1) 0 ||, where σ ′ is the (non trivial) permutation of {1, . . . , k-1} such that s (k-1) 0 σ ′ = (t τ (k-1) , . . . , t τ (1) ). By induction hypothesis, it immediately follows that || s (k) 0 σ || -||s (k) 0 || > 0. Suppose now that τ (k) = k. Then || s (k) 0 σ || ≥ t k q k-τ -1 (k) ≥ qt k = q(q 2(k-1) -1) q 2 -1 > (q k -1)(q k-1 -1) (q 2 -1)(q -1) , hence || s (k) 0 σ > ||s (k) 0 || by Lemma 10.
Proof of Theorem 5. We define s 0 = (s 0,1 , . . . , s 0,k ) as in Lemma 10. Thus we have

s 0,l = 1 + • • • + q 2(k-1-l) (1 ≤ l ≤ k).
Let now s = (s 1 , . . . , s k ) ∈ N k be such that d s = 0. Choose a permutation σ of {1, . . . , k} such that deg t c s

σ(k) ≤ • • • ≤ deg t c s σ(1)
. By Lemma 11 (note the different order that we have chosen here), we have deg t c s σ(l) ≥ k -l for all l. Hence, by Lemma 9,

s σ(l) ≥ 1 + • • • + q 2(k-1-l) = s 0,l , or s l ≥ s 0,σ -1 (l) . (15)
: H k (d) = d s 0 v ||s0|| + higher terms
The points 1 and 2 of Theorem 5 follow at once from this and Lemma 10 (recall that v = u q-1 , so ν k = (q -1)||s 0 ||). The point 3 is then a consequence of Proposition 7.

Proof of Theorem 1

In order to prove Theorem 1, we need to introduce a few notation. For any integer l ≥ 0 and any triple (µ, ν, m) ∈ Z × Z × Z/(q -1)Z, we denote by M ≤l µ,ν,m the K((t))-module of almost A-quasimodular forms of weight (µ, ν), type m and depth ≤ l (see [START_REF] Pellarin | Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms[END_REF], Section 4.2), and we set

M µ,ν,m = l≥0 M ≤l µ,ν,m .
We will also write l(f ) for the depth of the form f . As in [9, § 5.1], we further set h = hd, and we finally define

M ♯ µ,ν,m = K((t))[g, h, ∆ -1 , E, h] ∩ M µ,ν,m . We have : Lemma 13 1. If f ∈ M ♯ µ,ν,m , then τ f ∈ M ♯ qµ,ν,m and χf ∈ M ♯ µ,qν,m .
2. For all j ≥ 0 we have

l(τ j E) ≤ 1, l(τ j h) ≤ 1, l(χ j E) ≤ q j , l(χ j h) ≤ q j .
Proof. We first prove that the following equalities hold :

τ h = ∆E, τ E = 1 t -θ q (gE + h), χh = (t -θ) q E q - g ∆ h q , χE = h q ∆ . (16)
The first equality follows at once from the definitions of h and E and the second is Lemma 22 of [START_REF] Pellarin | Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms[END_REF]. The last one then follows from the first :

χE = χ τ h ∆ = h q ∆ .
Finally, to prove the third equality, we use the following one, which follows for instance from [9, Proposition 9] or [4, Proposition 2.7] :

h = t -θ q ∆ q (τ 2 h) - g ∆ τ h.
Applying χ to both sides of this equality, and using the formula τ h = ∆E, we get χh = t q -θ q ∆ q τ (h q ) -g ∆ h q = (t -θ) q E qg ∆ h q .

The first part of the lemma follows at once from the relations (16) (we recall that E ∈ M ≤1

1,1,1

and h ∈ M ≤0 q,1,1 ). The second part is a simple induction, noticing that the depth of a form f in M ♯ µ,ν,m is nothing else than the degree deg E f , when f is seen as an element of the polynomial ring K((t))(g, h, ∆ -1 )[E, h].

We now have all the elements to prove Theorem 1.

Proof of Theorem 1. For all i, j ∈ {1, . . . , k} we have, by Lemma 13: χ i-1 τ j-1 E ∈ M ≤q i-1 q j-1 ,q i-1 ,1 .

It follows, by a straightforward computation, that H k (E) ∈ M ≤(q k -1)/(q-1)

(q k -1)/(q-1),(q k -1)/(q-1),k .

Replacing t by θ, we then obtain the value of the weight and the type of E j,k = (τ j H k (E)/B k )| t=θ . We prove the last part of the first property of the Theorem asserting that the degree in E of E j,k is not smaller than some integer l k with l k → ∞ as k → ∞.

By the main theorem of [START_REF] Pellarin | Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms[END_REF], if f ∈ M ≤l w,m is non-zero and if w ≥ 4l(2q(q + 2)(3 + 2q)l + 3(q 2 + 1)) 3/2 , (18) then ord u=0 f ≤ 16q 3 (3 + 2q) 2 lw.

We can choose C(q, k) big enough so that if j ≥ C(q, k), then (18) holds with f = E j,k , w = (q k -1)(q j + 1)/(q -1) and l = deg E (E j,k ). Then, we get l ≥ q j-3 1 + q j 1 16(1 + q)(3 + 2q) 2 (1 + q k ) so that, enlarging C(q, k) if necessary, we get, for j ≥ C(q, k), l ≥ 1 32(1 + q)(3 + 2q) 2 (1 + q k ) which gives the required property of growth of the sequence (l k ) k .

Using now Theorem 5 and (3), we find, for all j ≥ 0 :

τ j H k (E) κ k,ν k = (-1) k h q j q k -1 q-1 κ k,ν k τ j+1 κ k,ν k u ν k + • • • = (-1) k h q j q k -1 q-1 (u q j+1 ν k + • • • ) ∈ A[[t, u]].
Substituting t = θ in this equality yields E j,k = (-1) k h q j q k -1 q-1 (u

q j+1 ν k + • • • ) ∈ A[[u]].
The properties 2, 3 of Theorem 1 follow at once from this and from (17). It remains to show the property 4. We consider first the case k = 1. By definition, H 1 (E) = E and E j,1 = (τ j E)| t=θ with ord u=0 E j,1 = q j . By [3, Theorem 1.2, Proposition 2.3], E j,1 is proportional to the function x j defined there, and hence extremal. Moreover, it is normalised, so that E j,1 = f 1,q j +1,1 for j ≥ 0.

Let us assume now that k = 2, q ≥ 3. In this case, by [3, Theorem 1.3, Proposition 2.13], we see that E j,2 is proportional to the form ξ j defined there, and hence extremal. Since it is normalised and defined over A, the proof of Theorem 1 is complete.

0

  instead of s 0 to indicate the dependence on k. Thus we have s 0 = s (k) 0 = (t k , . . . , t 1 ) and ||s

  and the equality ||s|| = ||s 0 || holds only if σ = Id. In that case, the inequality (15) shows that ||s|| = ||s 0 || only if s = s 0 . Thus, we have shown that in the v-expansion (14), the first non zero coefficient is d s0

	It follows, by Lemma 12, that we have
	||s|| ≥ ||s σ -1 0	|| ≥ ||s 0 ||,

A formal power series i≥i 0 c i q i (or i≥i 0 c i u i ) is said to be normalised if c i 0 = 1. A modular form is normalised, by definition, if its q-expansion is normalised. A similar definition will be used for quasi-modular forms and for Drinfeld quasi-modular forms.

Notice that in fact, the definition of extremality of Kaneko and Koike slightly differs from ours.

For example, it is not known whether f 1,14 ∈ Z[[q]] but this looks true from numerical evidence.

Properly speaking, to call these spaces "spaces of Drinfeld modular forms" is an abuse of language; these spaces are just generated by the u-expansions associated to such forms, but since we will work here with formal series in u only, it looked advantageous to make the identification between forms and formal series. We will do the same for Drinfeld quasi-modular forms; see[START_REF] Bosser | Differential properties of Drinfeld quasi-modular forms[END_REF] for further explanations.

This occurs if and only if M ≤l w,m = (0), that is, if and only if w ≡ 2m (mod q -1) with w, l ≥ 0), it is unique up to multiplication by an element of K × := K \ {0}.

contract ANR "HAMOT", BLAN-0115-01.


Prof. Dr Johan P Mackenbach
email: j.mackenbach@erasmusmc.nl

Can we reduce health inequalities? An analysis of the English strategy (1997-2010)

Background

England has been the first European country to pursue a systematic policy to reduce socioeconomic inequalities in health. This paper assesses whether this strategy has worked, and what lessons can be learned.

Methods

Review of documents. Analysis of entry-points chosen, specific policies chosen, implementation of these policies, changes in intermediate outcomes, and changes in final health outcomes.

Findings

Despite some partial successes, the strategy failed to reach its own targets, i.e. a 10% reduction of inequalities in life expectancy and infant mortality. This is due to the fact that it did not address the most relevant entry-points, did not use effective policies, and was not delivered at a large enough scale for achieving population-wide impacts.

Interpretation

Health inequalities can only be reduced substantially if governments have a democratic mandate to make the necessary policy changes, if demonstrably effective policies can be developed, and if these policies are implemented on the scale needed to reach the over-all targets.

Introduction

Over the past three decades, research into socioeconomic inequalities in health has gradually moved from description to explanation to intervention [START_REF] Mackenbach | Tackling socioeconomic inequalities in health: an analysis of recent European experiences[END_REF]. As a result, some European countries have been able to start systematic government action to reduce health inequalities. England has been the first to enter this stage, after a failed attempt in the early 1980s. The Black report, commissioned by a Labour government to investigate the causes and possible remedies for Britain's undiminished social class inequalities in mortality, had proposed a radical change in social and health care policies but had disappeared in a desk drawer after the Conservatives won the 1979 elections [START_REF]Inequalities in Health: Report of a research working group[END_REF]. However, research continued and when Labour came back into power almost 20 years later, it again commissioned an expert report to bring together all the evidence -the Independent Inquiry into Inequalities in Health [START_REF]Independent Inquiry into Inequalities in Health (the Acheson report)[END_REF] -and developed a comprehensive program to tackle health inequalities.

England has been followed by a small number of other countries which have all developed and (partly) implemented national strategies to reduce health inequalities, including the Netherlands [START_REF] Mackenbach | A strategy for reducing health inequalities in the Netherlands[END_REF], Sweden [START_REF]The 2005[END_REF], Norway [START_REF]National strategy to reduce social inequalities in health[END_REF] and Finland [START_REF]National Action Plan to Reduce Health Inequalities 2008-2011[END_REF]. While all these strategies are similar at the surface, the English strategy stands out as having been more systematically developed, better resourced, more stringently implemented, and more extensively monitored. Furthermore, the Labour government has stayed in power for a long time, and was in a position to pursue its goal to reduce health inequalities for a unique 13 years. The strategy has come to an end with the 2010 parliamentary elections and the formation of a Conservative/Liberal Democrat coalition government. It is therefore extremely important to know whether the English strategy has worked, and what are the lessons to be learned. Can we reduce health inequalities?

The English strategy to reduce health inequalities The English strategy to reduce health inequalities was shaped in two steps (table 1). The first was in 1999, when the Department of Health issued "Reducing health inequalities: an action report" [START_REF]Reducing health inequalities: an action report[END_REF]. This was the government's response to the "Independent Inquiry into Inequalities in Health", which had been published in 1998 and had made 39 recommendations with 123 subclauses, ranging from higher living standards of households in receipt of social security benefits to improvements in school nutrition, and from improved housing insulation to more equitable allocation of resources within the National Health Service [START_REF]Independent Inquiry into Inequalities in Health (the Acheson report)[END_REF]. One of the reasons for the delay between the 1997 elections and the 1999 Action report, was that the Labour party had promised not to exceed the previous government's spending commitments in the first two years after their take-over.

The 1999 "Action Report" adopted many of these recommendations, and claimed that "[t]his is the most comprehensive programme of work to tackle health inequalities ever undertaken in this country." It listed a range of new government policies including the introduction of a national minimum wage, higher benefits and pensions, and substantially increased spending on education, housing, urban regeneration, and health care. It also announced a number of specific initiatives including the "Sure Start" program (free child care, early education, and parent support for low income families), "Health Action Zones" (local strategies to improve health in deprived areas), and a series of anti-tobacco policies (including free nicotine replacement therapy for low income smokers) [START_REF]Reducing health inequalities: an action report[END_REF].

The second step followed in 2003 after the publication of the "Cross-Cutting Review of Health Inequalities", a systematic assessment by government of its progress in tackling health inequalities. This took as its starting point the two national health inequalities targets, suddenly announced by the Secretary of State for Health in 2001, to narrow the gap in life expectancy between areas and the difference in infant mortality across social classes by 10% in 2010. The Cross-Cutting Review tried to identify the most significant interventions that would support the delivery of these targets, by quantifying the contribution that the interventions would make to reducing inequalities in specific health outcomes [START_REF]Tackling Health Inequalities: Cross-Cutting Review[END_REF].

In response to this analysis the Department of Health published a revised strategy in 2003, entitled "Tackling health inequalities: a Program for Action". It had a foreword by the Prime Minister, Tony Blair, and set out the government's plans to achieve the two health inequalities targets by 2010. It reiterated the need to tackle the structural "upstream" determinants of health inequalities, but it had a stronger emphasis on "downstream" policies than the 1999 Action Report. Key interventions expected to contribute to closing the life expectancy gap were reducing smoking in manual social groups, managing other risks for coronary heart disease and cancer (poor diet and obesity, physical inactivity, hypertension), improving housing quality by tackling cold and dampness, and reducing accidents at home and on the road [START_REF]Tackling health inequalities: a Program for Action[END_REF].

The strategy was structured around the two over-all targets, and underpinned by 12 "headline indicators" (specific targets for intermediate outcomes) and 82 "departmental commitments" (specific actions by various governmental departments) which together were expected to ensure the timely delivery of the targets. The departmental commitments included further poverty reduction efforts, improved educational outcomes, expansion of the Sure Start scheme, expansion of smoking cessation services, improvement of primary care facilities in inner cities, and improved access to treatment for cancer and cardiovascular disease. Many of the departmental commitments were explicitly targeted towards low income groups or deprived areas, and most were quantified in terms of numbers of people to be reached and budgets to be allocated. The total budget exceeded £20 billion [START_REF]Tackling health inequalities: a Program for Action[END_REF].

The government's commitment to reducing health inequalities was matched by a commitment to critically review, revise and then re-review its policies (table 1). As a result of these frequent self-reflections it is relatively easy, even for outsiders, to gauge the results of the English strategy to reduce health inequalities.

Evaluation of the English strategy to reduce health inequalities Key results of the analysis are displayed in tables 2 and 3, and in figure 1.

1. Were the entry-points for policy adequately chosen, given the over-all targets? Most of the policies addressed one or more determinants of health inequalities, but as is evident from the tables only a few had direct relevance for achieving the life expectancy or infant mortality targets by the year 2010. A few policies related to underlying determinants of health inequalities, but effects on inequalities in life expectancy were unlikely either because any prevented deaths are relatively rare (as in the case of homelessness or fuel poverty), or because they will occur in a future far beyond 2010 (as in the case of child poverty, education and pathways to work). There were no policies addressing income inequality as such, or other important determinants of health inequalities like working conditions and excessive alcohol consumption. While some policies directly addressed the main causes of adult death, they aimed at improving average access to treatment, not inequalities in access (as in the case of improving access to cancer or coronary heart disease services). Only a few policies in the fields of prevention and primary care (e.g. road safety, smoking cessation, diet, flu vaccination, access to primary care) addressed entry-points which were directly relevant for achieving the life expectancy target in 2010. A similar story applies to the infant mortality target.

That the match between targets and policies was far from optimal, probably relates to the fact that the 1999 "Action Report" and the 2003 "Program for Action" were as much a product of scientific evidence as of political opportunity. The "Independent Inquiry into Inequalities in Health" had resulted in a huge "shopping list" from which policy-makers could freely choose [START_REF] Smith | The independent inquiry into inequalities in health[END_REF]. Many of the policies included in the program were part of strategies which the government already pursued for other reasons, e.g. because it had decided to invest in better education or to upgrade the funding level of the National Health Service upgraded to the European Union average. There were also no explicit and quantified links between departmental commitments, headline indicators and over-all targets, and the overall targets had been chosen without regard to the policies which they would require. The "Program for Action" therefore was only a loose attempt to link targets and policies.

Were the specific policies to address these entry-points adequately chosen?

Ideally the 1999 "Action Report" and the 2003 "Program for action" should have been based on evidence of the effectiveness of specific policies, but at the time this scientific evidencebase simply did not exist. This can easily be seen in the 1998 "Independent Inquiry" which laid the basis for the English strategy to reduce health inequalities but did not list a single evaluation study which showed that the proposed interventions could actually reduce inequalities in health. All the evidence was from observational studies showing plausible entry-points for intervention at most [START_REF]Independent Inquiry into Inequalities in Health (the Acheson report)[END_REF].

Although this has been commented on at the time, the lack of an adequate evidence-base seems to have been ignored by most of those involved in the development of this strategy, probably because it was felt that this extraordinary window of opportunity should be used for action on health inequalities, and that no time could be lost on evaluating policies before they were implemented on a large scale [START_REF] Macintyre | Using evidence to inform health policy: case study[END_REF]. Instead, attempts were made to assess the effects of these policies while they were implemented, but as a result most of these evaluations had a weak, non-experimental design.

Collecting evidence on effectiveness of policies to reduce health inequalities is difficult, because these are mostly complex interventions, and should not only be evaluated for their effectiveness but also for their differential effectiveness [START_REF] Mackenbach | How should interventions to reduce inequalities in health be evaluated?[END_REF]. It was only around 2005 that the need to collect evidence on differential effectiveness of policies clearly came into focus.

Reviews appearing in the second half of the decade then unambiguously showed the almost complete lack of existing evidence. "Umbrella reviews" of population tobacco control policies and policies on wider social determinants showed that where systematic reviews dealt with differential effects by socioeconomic status, only a minority actually found some weak evidence for inequalities reduction [START_REF] Main | Population tobacco control interventions and their effects on social inequalities in smoking: placing an equity lens on existing systematic reviews[END_REF][START_REF] Bambra | Tackling the wider social determinants of health and health inequalities: evidence from systematic reviews[END_REF].

It is therefore impossible to say whether the policies were adequately chosen. A scientifically rigorous answer would be "no", but for those who accept the urgency argument the answer could be "this would have to be determined during implementation".

3.

Were these policies successfully implemented on the scale needed for population-wide impacts?

Without any doubt, the English strategy was implemented on a large scale. As shown in table 2, many of the "departmental commitments" had large budgets, and some (such as the commitments towards child poverty reduction, social housing quality improvement, and expansion of the Sure Start scheme) had budgets in excess of £1 or even £10 billion (i.e., €1.5 or €15 billion at the former exchange rate) [START_REF]Tackling health inequalities: a Program for Action[END_REF]. This is by far the best-resourced of all the Western European strategies to reduce health inequalities which started during the decade. Remarkably, these "departmental commitments" were also mostly met (table 2).

Nevertheless, when one starts looking in more detail a number of problems come to light.

First, during the implementation of the 2003 "Program for Action" many changes were made to the original plans. To some extent these reflected "learning by doing": adapting the strategy to experiences in the field, and also to new insights which had emerged. For example, during the implementation of the strategy a stronger emphasis was laid on a limited number of "spearhead areas" (areas with a concentration of social and health deprivation), because it was felt that focusing on these areas was likely to generate better results. It then seemed logical to also rephrase the over-all life expectancy target in terms of differences between these spearhead areas and the national average. However, the speed with which policies were changed and evaluation targets adapted made partnership working and policy learning practically impossible [START_REF]Healthy Lives (the Marmot review)[END_REF].

Second, as mentioned above the "departmental commitments" were not based at all on an analysis of what would be needed to reduce health inequalities. Thanks to a "health inequalities intervention tool" which became available in 2007 (!) the scale at which policies need to be delivered gradually became clear. This revealed many examples of misalignment of "departmental commitments", "headline indicators" and overall targets. For example, even if the headline indicator for the "major killers" would have been met, the life expectancy target would not have been achieved [START_REF]Tackling Inequalities in Life Expectancy in Areas with the Worst Health and Deprivation[END_REF].

Do changes in intermediate outcomes suggest that the policies were effective?

That the "departmental commitments" were insufficient to achieve the targets became quite clear in the 2007 "Status Report". This concluded that while almost all departmental commitments were wholly or substantially achieved (table 2), the headline indicators and outcome targets showed a more disappointing picture (table 3). Some of the headline indicators showed reduced inequalities (less child poverty and narrowing inequalities in housing quality, educational attainment and uptake of flu vaccinations), but others suggested stable or even increased relative inequalities between socioeconomic groups (heart disease and cancer mortality, child road accidents, teenage pregnancies, smoking) [START_REF]Tackling health inequalities: 2007 Status Report of the Program for Action[END_REF].

That some of the "headline indicators" showed narrowing inequalities is a major achievement of the strategy. Unfortunately, however, no favorable changes were seen for some of the main drivers of inequalities in life expectancy or infant mortality, such as smoking. At the population level, smoking went down both in manual and in non-manual groups, and at a similar absolute rate of decline. For example, among men in the manual group the prevalence of smoking declined from 34% in 2001 to 28% in 2008, while it declined from 22% to 16% among men in the non-manual group [START_REF] Robinson | Smoking and drinking among adults 2008[END_REF]. As a result, while absolute inequalities in smoking remained stable, relative inequalities in smoking prevalence went up. Similarly, although absolute inequalities in mortality from cardiovascular disease declined, relative inequalities went up [START_REF]Tackling Health Inequalities: 10 Years On[END_REF].

Not included in the headline indicators, because it did not relate to government objectives, is income inequality. Inequalities in income and wealth in England have remained unchanged or even widened during 13 years of Labour government, although at a slower speed than during the previous decades [START_REF]An Anatomy of Economic Inequality in the UK. Report of the National Equality Panel[END_REF].

Mixed intermediate outcomes also emerge from the evaluation studies which have been conducted for a number of specific policies. Health Action Zones could loosely be evaluated but the authors concluded that these "made little impact in terms of measurable improvement in health outcomes during their short lifespan" [START_REF] Judge | Learning from Policy Failure? Health Action Zones in England[END_REF]. The evaluation of the Sure Start program concluded that "there are some signs that Local Sure Start Programs may be effective" but no health outcomes indicative of health inequalities reduction were observed [START_REF] Melhuish | the National Evaluation of Sure Start Research Team. Effects of fully-established Sure Start Local Programmes on 3-year-old children and their families living in England: a quasi-experimental observational study[END_REF]. Despite large-scale implementation of smoking cessation services in deprived areas the evaluation study showed that they only made a tiny contribution to reducing inequalities in smoking [START_REF] Bauld | Assessing the impact of smoking cessation services on reducing health inequalities in England: observational study[END_REF].

The latter study is particularly interesting because it showed how important the scale of implementation is for achieving population-wide impacts. Despite huge efforts only 2.3% of smokers in the lowest socioeconomic group were reached by the program and made a quit attempt, against 1.3% of smokers in the highest socioeconomic group. After one year, only 0.42% of all smokers in the lowest socioeconomic group had successfully quit, against 0.17% of all smokers in the highest socioeconomic group [START_REF] Bauld | Assessing the impact of smoking cessation services on reducing health inequalities in England: observational study[END_REF].

Did final health outcomes respond as specified in the over-all targets?

In view of these intermediate outcomes it cannot come as a surprise that the overall targets have not been met. This already became clear in the 2007 "Status Report" [START_REF]Tackling health inequalities: 2007 Status Report of the Program for Action[END_REF], and was confirmed by subsequent independent evaluations, including the 2009 "Tackling Health Inequalities: 10 Years On" review. This pointed at "significant improvements in the health of the population over the last 10 years, an improvement almost wholly shared by disadvantaged groups and areas, as measured by life expectancy and infant mortality". However, "[h]ealth inequalities between different groups and areas and the whole population [...] persist" and "[t]he current data [...] shows that the gap is no narrower than when the targets were first set" [START_REF]Tackling Health Inequalities: 10 Years On[END_REF].

Figure 1 shows the most recent update of progress towards the life expectancy and infant mortality targets. The life expectancy target, specified in terms of the gap between "spearhead areas" and the national average, does not show any sign of narrowing -if anything the gap has grown. Life expectancy has gone up in spearhead areas, but the national average has improved at a higher speed [START_REF]Mortality target monitoring (life expectancy and all-age all-cause mortality, overall and inequalities). Update to include data for 2008[END_REF]. The picture is not better for the gap in life expectancy between occupational classes -absolute inequalities in life expectancy have not narrowed since before Labour came to power in 1997, and even have widened for women [START_REF]Healthy Lives (the Marmot review)[END_REF]. Similar results are found for the gap in infant mortality [START_REF]Mortality target monitoring (infant mortality, inequalities) update to include data for 2008[END_REF] (figure 1).

One may ask whether it is not too early to assess the strategy's effect on inequalities in life expectancy. The time-line linking determinants to infant mortality may be short, but the time-line linking determinants to life expectancy may stretch over decades. That is theoretically true, but it does not stop life expectancy from changing rapidly -as can be seen in figure 1, life expectancy in "spearhead areas" increased by 2 to 3 years in the 10 years since the base-line for the targets was set. The "problem" is that life expectancy in nonspearhead areas (or in higher socioeconomic groups) has increased even more rapidly. It seems that the strategy has been unsuccessful in harnessing the underlying forces to help reducing, instead of widening, the gap.

One may also ask what would have happened in the absence of the English strategy to reduce health inequalities: would the gap in life expectancy or infant mortality have widened even more? Possibly: several European countries have witnessed a widening of inequalities in life expectancy over the past decades [START_REF] Mackenbach | Widening socioeconomic inequalities in mortality in six Western European countries[END_REF], and the "spontaneous" trend in England may therefore also have been towards a widening of the gap. As shown in table 3, the strategy has had some favourable effects which may have contributed to less widening. This attenuation is likely to have been modest in size, however, because an unambiguous narrowing of the gap for a headline indicator relevant for inequalities in life expectancy was only seen for influenza vaccinations (tables 2 and 3).

Finally, one may ask whether these targets were not simply "inspirational" goals, intended to provide a general policy direction, instead of success criteria for which those responsible for the strategy could be held accountable. From the documents published by and for the Department of Health, however, it is quite clear that they were not only intended to be success criteria, but have actually been used as such [START_REF]Tackling Inequalities in Life Expectancy in Areas with the Worst Health and Deprivation[END_REF][START_REF]Mortality target monitoring (life expectancy and all-age all-cause mortality, overall and inequalities). Update to include data for 2008[END_REF][START_REF]Mortality target monitoring (infant mortality, inequalities) update to include data for 2008[END_REF]. Although the Department of Health at the time was warned that achieving these targets might be difficult or impossible [START_REF] Bauld | Off target: a critical review of setting goals for reducing health inequalities in the United Kingdom[END_REF], it is reasonable to ask whether it would have been possible to justify the expense if the targets had been less ambitious.

Remarkably, the 2010 Marmot review argued that the targets were not ambitious enough. It criticized the life expectancy and infant mortality targets because "the current target does not capture the social gradient in health or the more complex patterning of health associated with other groups (for example, ethnic groups)" and "mortality [does] not reflect health status or other dimensions of well-being through the life course" [START_REF]Healthy Lives (the Marmot review)[END_REF]. This may be true, but would it not have been even more impossible to create and implement a strategy that would have achieved a 10% reduction of health inequalities across the whole gradient, or a 10% reduction of inequalities in healthy life expectancy, a new measure advocated by the Marmot review?

Reflections

This remarkable story contains a number of important lessons for all those engaged with studying and eventually reducing socioeconomic inequalities in health. The 13-year period started in high spirits, and promised to be a period of harvesting the practical results of two decades of research into the determinants of health inequalities. It has become clear, however, that reducing health inequalities is much more difficult than most researchers had foreseen [START_REF] Mackenbach | Has the English strategy to reduce health inequalities failed?[END_REF].

From the analysis presented above it appears that, in terms of its own targets, the English strategy has failed. It has had some partial successes, which should not be ignored, but it has unfortunately not been possible to reduce health inequalities as measured by inequalities in life expectancy or infant mortality. Other data, not reported here, show that other over-all health measures (such as measures of self-reported health) do not show a narrowing of inequalities either [START_REF]Healthy Lives (the Marmot review)[END_REF].

Why was this strategy not more successful? The analysis presented in the preceding paragraphs suggests the following clues. First, the English strategy did not address the correct entry-points: it spent resources on entry-points which were irrelevant for life expectancy or infant mortality, at least within the chosen time-frame, and it ignored relevant entry-points such as income inequality, working conditions, and excessive alcohol consumption. Second, it did not use effective policies: it had to rely on policies of unproven effectiveness, and to the extent that policies were evaluated during implementation most proved to be largely ineffective in reducing inequalities in health outcomes. Third, it was not delivered at a large enough scale: the scale required for achieving population-wide impacts was not determined in advance, and where reach was part of the evaluation as in the case of smoking cessation it proved to be insufficient to reduce inequalities at the population level. Are these failures caused by a lack of determination on the part of the government? Tony Blair's Labour government has been criticized for not being serious enough about tackling inequalities, and being indifferent to income inequalities [START_REF] Shaw | Health Inequalities and New Labour: how the promises compare with real progress[END_REF]. It has also been criticized for its neoliberal public health philosophy, which emphasized personal responsibility for behavior change at the expense of attention to underlying determinants [START_REF] Mckee | Choosing health? First choose your philosophy[END_REF]. But for those who are familiar with the situation in other Western European countries it is clear that determination is a relative issue -with all its weaknesses this Labour government probably was the most determined European government ever to tackle health inequalities.

The explanation should be sought elsewhere. First, the suboptimal choice of entry-points reflects the necessity for this (or any) government to match scientific evidence with political opportunity, and the lack of a democratic mandate to take more radical action. Labour had been elected on the basis of a party program that simply did not include a radical redistribution of income or wealth. Second, the choice of inadequate policies reflects the almost complete lack of scientific evidence on differential effectiveness. This in its turn was due to the relatively short time which had elapsed between the identification of the determinants of health inequalities (mostly in the 1980s and 1990s) and the urgent need of a newly elected and eager government to be advised on what it could do to tackle health inequalities. Third, the insufficient scale of implementation reflects a fundamental discrepancy between the necessary scale of change and the ability of state bureaucracies to change in response to new priorities [START_REF] Hunter | Getting to grips with health inequalities at last? Marmot review calls for renewed action to create a fairer society[END_REF]. This is painfully illustrated by the House of Commons' critique of the slowness of change within the National Health Service, where despite a resource allocation formula which prioritizes those in most need, disadvantaged areas after many years still receive much less than their full needs-based allocations [START_REF]Health Inequalities[END_REF].

If this analysis is correct, then the way forward includes the following ingredients. First, we need more advocacy to make sure that elected governments have a democratic mandate to make the necessary policy changes. Reducing health inequalities requires large-scale policy change in many fields, and this change will have to be articulated in political party programs.

For example, if a redistribution of income and wealth is necessary to reduce health inequalities, as many social epidemiologists and public health professionals believe, then they will have to convince the electorate that this should be done. Second, we need more research into differential effectiveness of policies. This research can only be done appropriately if policies are systematically tested out before they are widely implemented.

Third, the necessary scale of implementation can only be achieved by more focused policy efforts, based on careful alignment of targets, commitments and delivery. Strategies need to focus exclusively on strategic drivers of health inequalities, and make sure that the scale of implementation of policies matches the numbers of people which need to be reached and the degree of environmental or behavior change to be induced.

In conclusion, the health inequalities field in Europe has over the past decade entered the stage of systematic policy development and implementation. During these 13 years, England has led the way and set an extremely valuable example. In contrast to what other commentaries have suggested [START_REF] Bambra | A labour of Sisyphus? Public policy and health inequalities research from the Black and Acheson Reports to the Marmot Review[END_REF], there has been real progress since the Black Report.

When compared with the more recent strategies pursued in other countries, the English strategy still compares very favourably. For example, the Finnish strategy (summarized in table 4) addresses largely similar entry-points but contains a smaller number of specific policies. As in the English strategy, most of these policies are of unproven effectiveness. The Finnish strategy, however, has a much weaker implementation plan: there is no attempt at aligning the overall target with intermediate targets and delivery commitments, and there are no specific budget allocations [START_REF]National Action Plan to Reduce Health Inequalities 2008-2011[END_REF].

Unfortunately, the meagre results of the superior English strategy suggest that substantially reducing inequalities in over-all health is currently beyond our means. Even if we would have demonstrably effective policies (which we don't), we would probably be unable to deliver them on a sufficiently wide scale within a realistic window of political opportunity. For the foreseeable future, therefore, we need less ambitious aims, more focused approaches, and more rigorous evaluation.

Figure 1 .

 1 Figure 1. Did final health outcomes respond? Trends in the gaps in life expectancy and infant mortality since the targets were set.

Figure 1 .

 1 Figure 1. Did final health outcomes respond? Trends in the gaps in life expectancy and infant mortality since the targets were set.

Table 4 .

 4 The Finnish strategy to reduce health inequalities(2008)(2009)(2010)(2011) [a] Over-all target "To cut mortality inequalities between different vocational and educational groupings by one fifth by the year 2015" Policies General social policies: -poverty reduction (e.g. raising benefits) -comprehensive schools (e.g. health checks) -health promotion at vocational schools -support at transition from school to work -support for unemployed to get back into paid work -develop workplace health promotion -reduce homelessness Influencing lifestyles through policy -reduce excessive drinking (e.g. raise alcohol taxation) -reduce smoking (e.g. raise tobacco taxation) -promote healthy diet and exercise (e.g. work meals) Social welfare and health care -develop social work and primary health care -develop day care and child welfare clinics -develop rehabilitation for people at working age -services to support working ability of unemployed -equal services for older people -developing immigrant services Implementation strategy -Collaborative work within national government -Delegation of specific policies to municipalities -Voluntary participation of NGOs, businesses, etc. -No intermediate targets, no quantified commitments, no targeted budget -Comprehensive monitoring programme a. Source: ref. 7.

Source: refs. 26 and 27

Conpeting interests

None declared.

Source of funding

None.


Prof. Dr Dirk Zahn
email: dirk.zahn@chemie.uni-erlangen.de

Oliver Hochrein

On the Molecular Mechanisms of the acid-induced Dissociation of Hydroxy-Apatite in Water

Keywords: apatite dissociation, caries, molecular dynamics simulations Apatite dissociation  Caries  Molecular dynamics simulations

) surfaces of hydroxy-apatite ( Ca 10 (PO 4 ) 6 (OH) 2 ) in contact with aqueous solution. At neutral pH, the dissociation of ions is penalized by more than 150 kJ mol -1 giving rise to very stable apatite-water interfaces. This picture changes drastically with decreasing pH, as the protonation of phosphate and hydroxide ions lowers the free energy of calcium ions dissociation. Our simulations suggest the mechanism of acid-induced apatite decomposition to i) require a considerable degree of protonation of the apatite surface. The first ion dissociation step ii) involves calcium ions which electrostatic binding has been locally destabilized through phosphate and hydroxide protonation. The depletion of calcium ions embedding the anions then allows iii) the dissociation of the anionic species. Along this line, the protective role of fluoride in caries prevention is related to the stabilization of the calcium triangles embedding the OH -/F -ions.

First Author: Oliver Hochrein Order of Authors: Oliver Hochrein; Dirk Zahn Abstract: The enamel/saliva interface is mimicked by the comparably much simpler model of (001) surfaces of hydroxy-apatite ( Ca10(PO4)6(OH)2 ) in contact with aqueous solution. At neutral pH, the dissociation of ions is penalized by more than 150 kJ/mol giving rise to very stable apatite-water interfaces. This picture changes drastically with decreasing pH, as the protonation of phosphate and hydroxide ions lowers the free energy of calcium ions dissociation. Our simulations suggest the mechanism of acid-induced apatite decomposition to i) require a considerable degree of protonation of the apatite surface. The first ion dissociation step ii) involves calcium ions which electrostatic binding has been locally destabilized through phosphate and hydroxide protonation. The depletion of calcium ions embedding the anions then allows iii) the dissociation of the anionic species. Along this line, the protective role of fluoride in caries prevention is related to the stabilization of the calcium triangles embedding the OH-/F-ions.

Response to Reviewers:

The addressed typos and grammatical errors were corrected. Throughout the manuscript we now use the more precise notation Ca2+ or calcium ion.

Introduction

The enamel of human teeth predominantly consists of hydroxy-apatite (besides about 3 wt% proteins and lipoproteins and 2 wt% water). Therein, highly ordered crystal bundles of apatite-needles are arranged hierarchically in so-called enamel prisms. These prisms exhibit a parallel orientation to each other, and the surface of teeth is formed by the (001) basal planes [1]. It was shown from both experiments and calculations that the (001) planes of apatite are the most stable surfaces/interfaces with respect to both vacuum and aqueous phases [2]. Indeed, (001) surfaces are the predominant form of interfaces of teeth to the saliva [3].

While enamel is one of the hardest tissue materials known, it is quite vulnerable to aqueous solutions of low pH. Enamel damage via acid-induced apatite dissociation represents an every-day process, being part of the caries disease. The detailed mechanisms of caries are still not fully understood. Even for the simpler process of the acid-induced dissolution of nonbiogenic hydroxy-apatite no unique model is accepted. A common feature of the variety of mechanisms discussed so far is given by the protonation of the hydroxyl and phosphate ions of the apatite crystal surface as the initial step of the apatite decomposition [4].

Based a recent theoretical study in which we elucidated the hydration of different types of (001) surfaces of hydroxy-apatite [5], here we employ molecular dynamics simulations to explore the first steps of apatite dissociation. Within a systematic approach, ion dissociation is investigated for all accessible ionic species of the apatite surface. Along this line, the effect of the protonation of interfacial hydroxyl and phosphate ions is explored.

Simulation details

Our simulation model was chosen from a recent study dedicated to different types of (001) interfaces of apatite and water [5]. Using periodic boundary conditions, a sandwich structure of apatite slabs and water was obtained. Therein, apatite is mimicked by a block of 333 unit cells of monoclinic hydroxy-apatite [6]. By simple cutting of the unit cell (Fig. 1), the (001) and ( 1 00 ) surfaces of the apatite slices differ with respect to the occupation of the Ca2 ion positions. This is both counter-intuitive and energetically unfavorable. Here we use the preferred interface type for which 50% of the Ca2 positions are occupied at both faces of the apatite slab [5]. The latter is hydrated by 2394 water molecules giving rise to bulk water phases of about 2.5 nm thickness.

Empirical interaction potentials were adopted from earlier studies of apatite/calcium phosphate in aqueous solution [5,7]. Ewald summation is used to provide a proper account of the long-range Coulomb interactions. Within the constant-pressure constant-temperature ensemble, ambient conditions are applied. Therein, an anisotropic barostat is used in order to allow cell shape variations [8]. For the molecular dynamics simulations a time step of 2 fs was found to be appropriate.

For the potential of mean force calculations individual ions were separated from the apatite surface using the ion-surface distance as a model reaction coordinate. Probing distances of roughly 0 to 10 Å in intervals of 0.125 Å, the average constraint force is sampled from molecular dynamics simulation. For each constraint scenario, a few picoseconds of relaxation and sampling periods of about 25 ps were found to be adequate. Throughout this work we applied the constraint forces to individual ions, only. It should be pointed out, that this does rule out the separation of groups of ions from the apatite surface. Indeed, all other degrees of freedom are allowed to relax during the constant-temperature, constant-pressure molecular dynamics simulations.

Results

As a starting point to exploring the dissociation of the (001) apatite interface, we investigated the separation of individual hydroxide, phosphate and calcium ions (Ca1 and Ca2 types, see also Fig. 1) from the crystal surface. Unsurprisingly, the required free energy is rather large, taking a minimum of about 150 kJ mol -1 for the Ca1-type calcium ions. The strong endothermic nature of these dissociation steps reflects the extremely low solubility of apatite in aqueous solution of neutral and basic pH.

It is reasonable to expect the dissociation of apatite anions to be supported by hydroxide/phosphate protonation prior to the separation from the crystal surface. Unlike the hydroxide ions, only the oxygen atoms of the phosphate ions were found to act as hydrogen bond acceptors with respect to the aqueous solution. Thus only this ionic species appears as a first candidate to protonation from solution. Fig. 2 illustrates a protonated phosphate ion at the apatite-water interface after structural relaxation. The relaxed orientation of the hydrogen phosphate ions corresponds to a hydrogen bonded [HO .. HPO 4 ] 3-complex indicating the possibility of OH -protonation by the hydrogen phosphate ion. Thus, at the interface of apatite and acidic aqueous solution, we suggest hydroxide ion protonation to occur indirectly from via phosphate protonation.

In the course of proton uptake, the ionic charge is reduced and the Coulomb attraction to the crystal surface is lowered considerably. This scenario in principle holds both of the two anionic species. However, the phosphate ions exhibit only one oxygen atom that is accessible to protonation from the aqueous solution and we can hence only expect the formation of HPO 4 2-ions, whilst H 2 PO 4 -will only be formed at later stages of decomposing the apatite surface. Thus for the phosphate ions only a limited reduction of the Coulomb attraction to the crystal is possible. On the other hand, the charge of the hydroxide ions may be completely neutralized by protonation.

To explore the effect of protonation on anion dissociation we performed constraint molecular dynamics simulations and calculated the potential of mean force as a function of the ionapatite surface distance. This was done for OH -separation and OH -protonation and subsequent H 2 O separation from the interface. The energy profiles (Fig. 3) indicate a considerable lowering of the required energy, yet clearly show an endothermic character for both dissociation events. The phosphate ions, which center of mass lies below that of the hydroxide ions, are even stronger embedded (by calcium ions) into the apatite surface. Thus dissociation should be even more endothermic than observed for hydroxide ions.

As far as the initial step of ion dissociation is concerned, we hence conclude the stability of the anion sites of the apatite-water interface. In what follows, we therefore explore the separation of calcium ions from the apatite surface. Ca 2+ dissociation must be investigated for two separate cases, the Ca1 and the Ca2 type of calcium ion positions. The Ca1 species is coordinated by phosphate ions, only. We thus explored Ca 2+ separation from the apatite surface from the Ca1 position as a function of phosphate protonation in the first neighborhood of the calcium ion under investigation. The corresponding energy profiles (Fig. 3a) indicate the dissociation process to be endothermic up to two-fold protonation, whilst coordination by three hydrogen phosphate ions provides sufficient destabilization to make Ca 2+ dissociation exothermic.

In analogy to calcium ion separation from the Ca1 type lattice (surface) position, we then explored Ca 2+ displacement from the Ca2 type position. The Ca2 ions form a triangle embedding a hydroxide ion and are coordinated by three phosphate ions in the nearest neighborhood (see also Fig. 2). Strikingly, all four anions must be protonated to allow the dissociation of calcium ion from a Ca2 site (Fig. 4b). Thus a local excess charge of +4 must be created to cause exothermic Ca 2+ displacement.

An important issue of such local charge accumulation is given by the proton affinities of the corresponding anions. Even at neutral pH, the protonation of single PO 4 3-and OH -ions of the apatite surface is intuitive. The collective protonation of all four anions coordinating a Ca2 type calcium ion however requires acidic conditions.

It should also be pointed out that the protonation of the hydroxide ion is a crucial prerequisite to Ca2 dissociation. Thus replacing OH -by F -ion at the apatite surface (e.g. by application of fluorinated tooth paste) helps to avoid the dissociation of Ca2 type ions. Fluoride ions exhibit much lower proton affinities (pKB = 10.8, weak base) compared to hydroxide (pkB=0, strong base). In contrast to hydroxide ion protonation, HF formation is much less feasible and so is the formation of an excess +4 charge in the neighborhood of Ca2 type calcium ions. While fluoride incorporation into hydroxy-apatite is also known to improve the mechanical hardness, the protective role of fluoride for caries prevention may therefore be attributed to chemical considerations: Ca 3 F units represent a particularly stable motive of apatite with respect to ion dissociation into solution.

Conclusions

On the basis of our simulations we can hence suggest a conclusive mechanism of the first step of hydroxyapatite dissociation in aqueous solution. Acidic conditions strongly promote the dissolution process, as a central prerequisite to ion dissociation is given by a considerable degree of protonation of the apatite surface. Focusing on the ideal single crystalline (001) surface of apatite, the hydroxide and phosphate ions were found to be rather inert to dissociation. This may be attributed to the embedding calcium ions which must be removed from the apatite surface prior to anion dissociation. The separation of calcium ions requires strong repulsive Coulomb interactions which imply the accumulation of an excess +3 and +4 charge in the nearest neighborhood of Ca1 and Ca2 type calcium ions, respectively. For the Ca2 type ions, such coordination may be suppressed by replacing the neighboring hydroxide ion by a fluoride ion. Thus, the apatite surface may be stabilized by the formation of Ca 3 F motives. Therein, each of the Ca2 type calcium ions is protected with respect to ion dissociation into solution.

Figure captions

Fig. 1 Unit cell of hydroxyapatite (monoclinic modification). The overall model system comprises a sandwich of 333 unit cells of hydroxy-apatite which (001) / (00 ) surfaces are in contact with water. Periodic boundaries are applied in all directions.

Along the c direction the water phase extends over about 2.5 nm which was shown to be sufficient to allow bulk water phase of more than 1 nm thickness [5]

Figure 4b

Click here to download line figure: Fig4b.doc

Fig. 2 Fig. 3

 23 Fig. 2Close-up of a of the apatite-water interface (solvent not shown) highlighting the local neighborhood of a hydroxide ion at the crystal surface. The hydroxide ions are embedded by triangles of calcium ions (Ca2 positions, see also Fig.1). Next to the calcium triangle, a single phosphate ion was protonated. The relaxed orientation of the hydrogen phosphate ions corresponds to a hydrogen bonded [HO .. HPO 4 ] 3- complex. While not directly accessible to hydrogen bonding with the solvent, the OH -ions can hence accept a proton indirectly via phosphate protonation

Figure 1

 1 Figure 1Click here to download high resolution image

Figure 2 Figure 3

 23 Figure 2 Click here to download high resolution image

Acknowledgments

The authors wish to acknowledge R. Kniep for fruitful discussions.