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The maximum or minimum number of rational points on curves of genus three over finite fields

We show that for all finite fields F_q, there exists a curve C over F_q of genus 3 such that the number of rational points on C is within 3 of the Serre-Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over F_q.


Introduction
More than half a century ago, André Weil proved a formula for the number of rational points, N (C), on a smooth projective algebraic curve C of genus g over a finite field F q . This formula, along with his proof of what is referred to as the Riemann hypothesis for curves, provides upper (resp. lower) bounds on the maximum (resp. minimum) number of rational points possible There are many cases in which the Weil upper and lower bounds cannot be attained. Some are trivial: for example when the bound is not an integer. Also, when the field size, q, is small with respect to the genus, g, the lower bound will be negative and thus cannot be attained. In [21], Serre made a non-trivial improvement to the Weil bound (which we will refer to hereafter as the Serre-Weil bound): and introduced the explicit formulae method to provide better bounds for large genus. Since then there has been considerable interest in determining the actual maximum and minimum. (cf. [21], [22], [23], [24], [27], [20] [28], [3], [9], [10], [11], [12], [13], [15], [16], [2]) In the present paper we are concerned with the following question which was posed in [22]: for which genus, g, is the difference between the upper bound and the actual maximum N q (g) bounded as q varies? For genus 1 and any q, the difference is either or 0 or 1 ( [29]). For genus 2, Serre determined N q (2) for all q, and showed that the difference from the Serre-Weil bound is always less than or equal to 3 ( [21]); for genus 3, he determined the maximum for q ≤ 25 ( [23]). The present paper is devoted to showing that for genus 3 and all q, either the maximum or the minimum is within 3 of the Serre-Weil upper or lower bound.
The techniques involved in the proof of the main theorem include Serre's theory of hermitian modules as well as "glueing" of polarizations on abelian varieties. The theory of hermitian modules is detailed in the Appendix. This theory provides an equivalence of categories between abelian varieties over F q which are isogenous to a product of copies of an ordinary elliptic curve, and torsion-free modules of finite type over a ring which is defined in terms of the Frobenius of the elliptic curve. A polarization on the abelian variety then translates to a hermitian form on the module. Thus the classification of hermitian forms over rings can be used to determine the existence or non-existence of the corresponding polarized abelian variety.
To determine whether there is a curve whose number of points is close to the Weil bound, we first make a list of the possible zeta functions for such a curve as in [13]. The numerator of the zeta function of a curve is given by the characteristic polynomial of Frobenius acting on the Jacobian of the curve. From Tate's theorem we know that the isogeny type of an abelian variety over a finite field is determined by the characteristic polynomial of Frobenius. For each zeta function, we investigate the corresponding isogeny type using the equivalence of categories provided by Serre's theory. Since not all Jacobians of curves we consider are isogenous to the product of one elliptic curve with itself, we are naturally led to consider the glueing of polarizations on abelian varieties of different isogeny types. In some cases, the glueing is possible; in others, it is not. When it is not, we obtain improvements on the upper bounds. In all cases we are able to conclude by making use of the Torelli theorem in dimension 3 that there exists a curve whose number of points is within 3 of the Serre-Weil upper or lower bound.
Serre invoked the theory of hermitian modules to treat the genus 2 case in [22] and [23], but did not give details. In [24], the full proof of the genus 2 case is given, including more explanation of the equivalence of categories. More recent work on this subject was done by Everett Howe ([6]), who translated the notion of principal polarization working with Deligne's more general equivalence of categories. The notion of the glueing of polarizations on abelian varieties has also been employed by several authors recently for various purposes ( [4], [7]).
The proof of the main theorem of this paper is divided into cases which correspond naturally to the existence or non-existence of indecomposable hermitian forms over certain rings of a given discriminant. In each case we treat finite fields F q with q of a special form, which leads to a number of interesting diophantine problems.
The paper is organized as follows: Section 2 contains the statement of the main theorems of the paper. Section 3 contains a description of how the theory of hermitian modules will be used in the proofs of the main theorems and also recalls the short list of possible zeta function for curves with the number of points under consideration. Section 4 contains the proofs of the main theorems.
Acknowledgements I would like to thank J-P. Serre for his letters on hermitian modules which inspired this project, and for his generous help and time spent working on it. This paper is based on work supported by a Mentoring Grant from the Association for Women in Mathematics. I would also like to thank Jim Milne and René Schoof for helpful discussions and Fritz Beukers for the reference to Apéry's paper.

Statement of Results
Let q = p e , with p prime, e ≥ 1. By a curve over the finite field F q , we mean a smooth, projective, absolutely irreducible curve. For such a curve, C, let g = g(C) denote the genus, and N = N (C) denote the number of rational points over F q .
Definition For fixed g and q, let N q (g) (resp. M q (g)) denote the maximum (resp. minimum) of N (C) as C runs through all curves of genus g over F q .
Throughout we write q = p e uniquely in the form where x is the largest integer whose square is less than or equal to q, and a is an integer such that −x ≤ a ≤ x.
Furthermore, there exists a curve C of genus g(C) = 3 over F q such that Note 2.1 For example if we write q = 5 in the form q = x 2 + x + 3, with x = 1 and a = 3, then x is not the greatest integer part of the square root of q and a does not satisfy the inequality a ≤ x; so instead we write q in the form q = x 2 + 1, a = −1. Nor do we write q = 9 in the form q = x 2 + x + 3, since it would not satisfy a ≤ x.
Furthermore, there exists a genus 3 curve C over F q such that Note 2.2 For example q = 3 is not of the form q = x 2 + 2 with 2 ≤ x; instead it is of the form q = x 2 + x + 1. Together, Theorems 1 and 2 can be used to establish the following result.
Theorem 3 For any prime power q = p e , there exists a curve C of genus g(C) = 3 over F q such that In other words, Theorem 3 says that for all q, at least one of the following holds:

Hermitian modules
In the Appendix, Serre gives an equivalence between the following two categories: the category of abelian varieties over F q which are isogenous over F q to a product of copies of E, where E is an ordinary elliptic curve over F q ; and the category of torsion-free R d -modules of finite type, where R d = Z[π], π is the Frobenius of E, and d = a 2 − 4q, where #E(F q ) = q + 1 − a. The equivalence holds under the assumption that d is the discriminant of an imaginary quadratic field. In that case, R d is equal to the ring of integers in the field; thus it is a Dedekind domain, and the modules under consideration are projective. If d is not the discriminant of an imaginary quadratic field, then R d is an order in the ring of integers. Although the equivalence of categories does not necessarily hold under those conditions, we will still use the same notation and make use of the functor S given in the Appendix. In most cases there will be no conflict with the notation from Section 2 since we will let a = m; in all other cases, we will use the notation d ′ . The condition that E be ordinary is equivalent to requiring that a be prime to p. Throughout the paper we will use the notation E m to denote an elliptic curve with q + 1 + m points over the field F q .
In Section 5 of the Appendix, polarizations on abelian varieties are translated into positive definite hermitian forms on R-modules, and the polarization is principal if and only if the hermitian form has discriminant 1. The Jacobian of a curve has a canonical principal polarization which corresponds to the theta divisor. For an absolutely irreducible curve, the theta divisor is irreducible, so the canonical polarization corresponds to an indecomposable hermitian module with discriminant 1.
We will use this correspondence in two directions. In cases where we can show that there is no indecomposable hermitian module of discriminant 1, we can conclude that no curve of that type exists. In cases where we find an indecomposable hermitian module of discriminant 1, we can use the theorem of Torelli in its precise form (see the Appendix to [13]) to conclude that a curve of that type (or of the opposite type) exists.

Zeta functions
In each case, we identify the zeta function of the curve we are searching for. The zeta function determines the isogeny type of the Jacobian of the curve.

Definition 1 A curve has zeta function of type
where {α i ,ᾱ i } is the family of g conjugate pairs of eigenvalues of Frobenius acting on the Jacobian of the curve over F q . Fact 3.2 (Defect 1) We recall from [23] that a curve whose number of points is equal to q + gm must have g ≤ 2.
Fact 3.3 (Defect 2) We recall from [24] or [13] that if the fractional part of 2 √ q, , then a curve whose number of points is equal to q + gm − 1 is of type [m, m, ..., m − 2].

Proofs of Theorems
This section is organized as follows. We will prove the theorems from Section 2 in the order they were stated. For each possible zeta function to be treated via the equivalence of categories, we must check the condition that the trace be relatively prime to the characteristic, and deal with special cases where this fails.
To begin, we notice that except for one special case (q = 2), none of the fields in Theorems 1 and 2 have characteristic 2. Fact 4.1 If q = p e , e > 1, then p = 2 if q is of any of the following forms: Proof: In case i, if q were a power of 2 then x would be odd, and so which implies q ≡ 2 (mod 8). Thus q = 2, e = 1 is the only possibility.
In case ii, x would be even, and we would have which implies q ≡ 2 (mod 4). In cases iii. and iv, x 2 + x is always even, so q is odd.

Proposition 1
If q is of the form q = p e = x 2 + x + a, a = 1 or a = 3, a ≤ x, then m is prime to p unless q = 3.
Proof: a=1. Write q = p e = x 2 + x + 1 with m = 2x + 1. Suppose p divides m. Then implies that p divides (x 2 − 1). If p divides (x + 1), then p divides x, which is impossible. So p divides (x − 1). Thus p also divides (x + 2) and 3. But Nagell and Ljunggren (see [19], for example) have shown that the only solution to is p = 7, e = 3, x = 18. So no odd power of 3, e ≥ 3, is of the form x 2 + x + 1. So q = 3 is the only exception. Indeed in that case m = 3.
Then p e − 3m = x(x− 5) is divisible by p. We cannot have p divides x since then we would also have p divides (x + 1). So in fact, p divides (x − 5). In addition, 2p e − mx = x + 6 is divisible by p. Thus we must have p = 11. However, there are no solutions to the equation 11 e = x 2 + x + 3 since there are no solutions modulo 5.
Proposition 2 If q is of the form q = p e = x 2 + x + a, a = 1 or a = 3, a ≤ x, then m − 2 is prime to p unless q = 343 or possibly p = 5.
implies that p divides 3x + 2 and x + 3 and x − 4. So p = 7. If e = 1, then p = 7 does not divide 2x − 1. If e is odd, e ≥ 3, then due to the result of Nagell and Ljunggren cited above, the only solution is p = 7, e = 3, x = 18. In that case, p = 7 divides 35, so q = 343 is an exception. a=3. Write q = p e = x 2 + x + 3 is divisible by p, and so is 8x − 4, which implies that p divides 15. If p = 3, then the only powers of p, e odd, which are of the form x 2 + x + 3 are 3 and 3 5 = 243. This can be proved (as Serre pointed out to me) by Skolem's ℓ-adic method with ℓ = 5 (in [26], Skinner attempted to give an ℓ-adic proof with ℓ = 11, but his argument is incomplete). Neither q = 3 nor q = 243 satisfy the divisibility condition "3 divides 2x − 1". It would not be necessary to exclude p = 5 in the hypotheses if there are no solutions to the equation 5 e = x 2 + x + 3 with e ≥ 3, e odd and 5 dividing 2x − 1. I have checked with the help of pari that there are no solutions for e < 1600.

q=3
To prove Theorem 1, first consider the case q = 3. If q = 3, then m = 3, and the explicit formula bound is N (C) ≤ 10. In fact, a curve C of genus 3 and defect 3 over F 3 with N (C) = 10 = q + 1 + 3m − 3 was given in [24] with the equation Its zeta funtion is of type [m, m, m − 3].
Proposition 3 If q is of the form q = p e = x 2 + x + a, a = 1 or a = 3 with a ≤ x, then a defect 2 curve of genus 3 is of type [m, m, m − 2].
Proof: By Fact 3.3, it suffices to check that the fractional part of 2 For a = 1, this implies x = 0, which is not possible. For a = 3, this implies x ≤ 3, which does not occur for any q. (See Note 2.1).
Proposition 4 There is no indecomposable rank 2, Hermitian module of dis- Proof: Since the class number of R d is 1, projective R d modules are free, so we can express the module P and the Hermitian form H as a matrix whose ij th entry is H(e i , e j ) where {e i } is a basis for P over R d . A change of basis will give an equivalent matrix. We will work with the matrix in a reduced form. A rank 2 Hermitian form in "reduced" form can be written as implies that λ, µ ∈ Z, and we assume that e 1 is chosen with λ minimal so that λ ≤ µ.
We are interested in Hermitian forms which are positive definite, so λ > 0. The proof of the proposition relies on the following lemma: Lemma 1 If d is square-free, satisfies d ≡ 1 (mod 4) and R d has class number one, then we can find a basis for P over R d such that the matrix for a Hermitian form in reduced form satisfies Proof: If αᾱ λ 2 is too large, we can replace α by α + λr for any r ∈ R d by replacing the basis element e 2 by e ′ 2 = e 2 +re 1 . As a complex number, and |λ| = λ, so it suffices to show that r can be chosen so that So it is enough to show that for every z ∈ C, there exists an element r ∈ R d such that the distance squared from z to r is less than the bound.
We look for the point in the complex plane which is furthest from a lattice point, or in other words, the smallest radius so that circles centered at the lattice points will cover the plane. It suffices to consider points in the right triangle with vertices (0, 0), 2 ), since we can then extend the argument by symmetry to the rest of the fundamental domain. We look for the point ( 1 2 , a) in the triangle which is equidistant from the two lattice points, (0, 0) and ( 1 2 , we find Calculating the distance squared from this point to the origin we find exactly the bound stated in the lemma, and this point is the furthest possible distance away from the closest lattice point. c.q.f.d.
But then 3µ = 2 + αᾱ ≤ 9 implies that µ = 3 and αᾱ = 7. Again this is impossible since 7 is not the norm of an element of R −11 .
Note 4.1.2 Proposition 4 can also be deduced from the results of Otremba (see [18]). In that paper, she proves (via the mass formula) that some hermitian forms are "alone in their genus"; i.e. any lattice which is locally isomorphic to the given one is globally isomorphic. See especially (for rank 2) p.9 and p.13. These computations imply Proposition 4, provided one checks that every lattice with discriminant 2 is in the same genus as the decomposable one (the 1,2 lattice, in Otremba's notation).

Glueing criteria
The following section is translated from [25]. Suppose that B and C are two polarized abelian varieties over a perfect field k b : B → B * and c : C → C * .
Suppose that the polarizations have the same degree, n, and that n is prime to the characteristic, p, of k (if the characteristic is = 0). Denote by N b and N c the kernels of b and c; these are finiteétale group schemes of order n 2 . We will identify them with theirk points. The Galois group G = Gal(k/k) operates on these groups. Let µ be the group of roots of unity, written additively. According to Mumford, [ [14], p.227] the polarizations b and c define non-degenerate, alternating bilinear forms on N b and N c , with values in µ, and compatible with the action of G. We will denote them by (x, y) → x, y . Let f be a map f : N b → N c verifying the following conditions: To the data (B, b, C, c, f ) given above, we can now associate an abelian variety, A, isogenous to B × C, equipped with a polarization a of degree 1 as follows: Let B × C have the polarization which is the product of the polarizations b and c. The kernel of B × C → B * × C * is N b × N c . Let F be the subgroup of this kernel which is the graph of the isomorphism f . Property (2) above shows that F is totally isotropic; property (1) shows that F is stable by the action of G, i.e. that it is a finiteétale k-subgroup of B × C. According to Mumford [14], the polarization on B × C passes to the quotient by F . In other words, if we Comparing the degrees, we see that deg(a) = 1, so a is principal. This is how we obtain a polarized abelian variety A by "glueing" (B, b) to (C, c) via f .

q = 3 continued
The general framework for the glueing of polarized abelian varieties can be applied in both backwards and forward directions. The following theorem works in the backwards direction ("unglueing").
Theorem 4 Suppose q = p e , q = 3, q = 343, p = 5 and that q is of the So all abelian varieties in this proof are ordinary. Due to Fact 4.1, we also know that 2 is prime to the characteristic, so the group schemes we work with will beétale. Thus we will identify finite group schemes with their F q -points. In the cases at hand, we have d = −3 or d = −11, so in particular, d is the discriminant of an imaginary quadratic field of class number one.
By the equivalence of categories given in the Appendix, it suffices to show that the indecomposable principal polarization on A induces an indecomposable polarization on E m ×E m of degree 2, where E m is an elliptic curve with q+1−m points over F q .
Let Then since Q( √ d) has class number one,  Together (b, c) is the induced polarization on B × C, with kernel N b × N c , and so we must have embedded diagonally. We want to show that N b has order 4.
By the definition of B and C, we know that B ∩ C is killed by 2, so it is contained in C [2], which has order 4. In addition, B ∩ C is stable under the action of Frobenius, so it is an So B ∩ C is an F 4 -vector space of order less than or equal to 4. B ∩ C cannot be trivial, or else A would be split and the polarization would be decomposable. So B ∩ C must have order 4 and the polarization (b, c) has degree 4. Since B ∩ C is embedded diagonally into N b × N c , it follows that N b has order 4.
In addition, B is orthogonal to C with respect to the skew-symmetric bilinear pairing , defined by the polarization of A on the ℓ-adic Tate module associated to A. This follows from the fact that φ is hermitian with respect to the pairing, since Thus for elements b ∈ T l B and c ∈ T l C of the ℓ-adic Tate modules associated to B and C, we have This shows that they are orthogonal since Z ℓ (1) is torsion-free. Thus B ∩ C is maximal isotropic with respect to the pairing. Finally, the polarization b is indecomposable, else λ would not be. Thus we have established that N q (3) ≤ q + 1 + 3m − 3 for such q. To finish the proof of Theorem 1, we must show that there exists a curve C of genus g(C) = 3 over F q such that |N (C) − (q + 1)| = 3m − 3.
Proposition 5 Let q = p e = x 2 + x + a, a = 1 or a = 3, with a ≤ x, and q = 3, q = 3 5 . Then there exists a curve C of genus g(C) = 3 over F q with zeta function of type ±[m − 1, m − 1, m − 1].
We first show that (m − 1) is prime to p and To check that (m − 1) is prime to p, write m − 1 = 2x. By Fact 4.1 we know that p = 2. So if p divides (m − 1) then p must divide x. For p e = x 2 + x + 1 this is impossible. If p e = x 2 + x + 3 and p divides x, then p = 3. As explained in Proposition 2 above, the only powers of 3, e odd, of the form 3 e = x 2 + x + 3 are 3 and 3 5 = 243. Now we can apply the theory of Hermitian modules. By Theorem 8.2 in [5], there exists an indecomposable, positive definite, unimodular Hermitian module over R d ′ of rank 3. Applying the functor S from the Appendix to this module, we obtain an abelian variety A isogenous to E m−1 × E m−1 × E m−1 with an indecomposable principal polarization. Then by the Torelli theorem (cf. [13], Appendix), there exists a genus 3 curve X over F q whose Jacobian is isomorphic either to A or to the quadratic twist of A; hence X is of type Corollary 2 For q = 7 or q = 13, we have N q (3) = q + 1 + 3m − 3.
Proof: This was noted in [24]. It follows from Proposition 5 and the fact that a Frobenius with the opposite sign is not possible because the corresponding curve would have a negative number of points. For the polarization on C = E m−3 we take 3 times the canonical polarization.
To glue B to C we must find an isomorphism of the kernels of b and c as group schemes, and since the order is not prime to the characteristic, it is not enough to consider F q -points. The kernel of B is isomorphic to the 3-torsion of E m . So both kernels are of (étale, local) type: Z/3Z × µ 3 (twisted), and there is only one choice for the pairing up to sign.
Note 4.1.5 An explicit search for a genus 3 curve over F 243 with the maximum or minimum number of points currently seems out of reach. Even counting the points on each of the approximately 243 6 homogeneous plane quartics in the variables x 2 , y 2 , z 2 using the naive method would take 3 40 steps. A computation of this size is currently infeasible. A more fruitful approach may be to generate the equations of the elliptic curves E m and E m−3 , which we can easily do, and to try to glue them together explicitly.

Proof of Theorem 2
It remains to show that there exists a curve C over F q such that |N (C)−(q+1)| = 3m − 2.
Then with e > 1, since by [1] there exist at most two solutions to such an equation and in this case they are e = 1, x = 1 and e = 3, x = 5. The second solution does not satisfy the congruence condition and the first solution does not matter since we do not write q = 3 in the form x 2 + 2 (See Note 2.2).

q=2
When q = 2, a curve C of genus 3 and defect 2 over F 2 was given in [24]. Its equation is: and it has 7 points, which is defect 2 since m = 2. It has zeta function of type This is possible because and it is the only zeta function possible for this case. It is a twist of the Klein curve which becomes isomorphic to the Klein curve over the field F 2 7 . It is interesting to note that when q = 2, the zeta function type [m, m, m− 2] is not possible because the curve would have 7 points over F 2 but only 1 point over F 8 . Alternatively, we can show that the glueing of the supersingular elliptic curves is not possible by examining the group schemes in question and showing there is no isomorphism between them.
The zeta function type [m, m + √ 3 − 1, m − √ 3 − 1] is not possible because the curve would have 7 points over F 2 but only 5 points over In both cases, the module is indecomposable because all the values taken by the Hermitian form are divisible by 2, so 1 is not represented. But if the form were equivalent to 1 0 0 2 , then 1 would be represented. The next theorem shows that in this case it is possible to glue two abelian varieties together to obtain the Jacobian of a genus 3 curve with the desired property.
Theorem 5 Suppose q = x 2 + j, j = 1 or j = 2 with j satisfying j ≤ x, and suppose q = 2. Then there exists an abelian variety A over F q , isogenous to E m × E m × E m−2 , with an indecomposable principal polarization.
Proof: By Fact 4.1, the characteristic of the field is not equal to 2, and by Proposition 7, there exist abelian varieties B = E m × E m and C = E m−2 .
j=1. Let b be the polarization on B corresponding to the positive definite indecomposable Hermitian module Let c be two times the canonical polarization on C. It has kernel equal to the 2-torsion of C. We proceed by calculating the kernel of b and then glueing it to the kernel of c. The order of (the F q points of) these group schemes is 4, which is prime to the characteristic. By Mumford's criteria, we need to find an isomorphism of the Galois modules which is an anti-isometry with respect to the pairings. If necessary, we will replace C by an isogenous elliptic curve.
is the λ-torsion of E m , and λ = 1 + i. The λ-torsion of the elliptic curve is contained in the 2-torsion, since 2 = (1 + i)(1 − i). The Frobenius of E m acts on the 2-torsion of E m by fixing one of the three non-trivial points and by exchanging the other two. This can be seen by looking at the 2 × 2 matrix which represents the action of Frobenius on the ℓ-adic Tate module when ℓ = 2. The characteristic polynomial of Frobenius is t 2 + mt + q, where −m is the trace of the matrix and q is the determinant. In this case, q = x 2 + 1 and m = 2x, with x even since q is odd. Thus we have m ≡ 0 (mod 4). So the number of points on the elliptic curve over Similarly, N (E m−2 ) = q + 1 + m − 2 ≡ 0 (mod 4).
In both cases, the trace is even and the determinant is odd, so the possibilities for the matrix of Frobenius (mod 2) are: and considering all the possibilities for the entries (mod 4). So we only need to chose C to be an elliptic curve E m−2 with Frobenius fixing one of its 2-torsion points and switching the other two. It suffices to take an elliptic curve with endomorphism ring Z[π] ⊂ R ′ such that It follows from Deuring that such a curve exists. By chosing C in this way we can find an isomorphism of C [2] with the kernel of b which respects the action of Frobenius. Finally, the Galois module isomorphism must be an anti-isometry with respect to the pairings. The pairings take values ±1. j=2 The proof is almost the same except in this case λ = √ 2 and the polarization on B is given by the matrix
Corollary 3 If q = x 2 + j, j = 1 or j = 2 with j satisfying j ≤ x, and q = 2, then there exists a curve of type ±[m, m, m − 2].
Proof: This follows from Theorem 5 and the fact that (see [17]): if A is a principally polarized indecomposable abelian variety over an algebraically closed field, of dimension 3, then A is the Jacobian of a curve. The "precise Torelli" theorem in the Appendix to [13] allows you to descend from the algebraically closed field to any field at the cost of quadratic twist, as is explained there.
Example An example of a genus 3 curve corresponding to this type of glueing was found by van der Geer and van der Vlugt. It can be described as the curve over F 27 obtained by taking the fiber product of where α 3 + 2α 2 + 1 = 0. It is a defect 2 curve with 56 points (instead of 58 as stated in [3]).

Proof of Theorem 3
Let q = p e be a power of a prime. We divide the proof into two cases: e even and e odd. e even. First suppose e = 2r is even. Then Ibukiyama has shown [8] that, for p an odd prime, there exists a curve C of genus 3 over F p such that Thus for all q an even power of an odd prime, there exists a genus 3 curve attaining either the Weil maximum or the Weil minimum. Now suppose that p = 2, so that q = 2 2r . Then m = 2 r+1 , and (m − 1) is prime to p. If we let d ′ = (m − 1) 2 − 4q, then d ′ = −2 r+2 + 1, and so we see that for r ≥ 1, By Theorem 8.2 in [5], there exists an indecomposable, positive definite, unimodular Hermitian module over R d ′ of rank 3. Applying the functor S from the Appendix to this module, we obtain an abelian variety isogenous to E m−1 × E m−1 × E m−1 with an indecomposable principal polarization. Then by the Torelli theorem, (see the Appendix to [13]) there exists a genus 3 curve over F q which is of type For example, over F 4 , the equation of the Klein curve is given in [24] to show that e odd. Now suppose that e is odd. As usual, write with −x ≤ a ≤ x. We divide the proof according to the value of d = m 2 − 4q. If On pose R = Z[X]/(X 2 − aX + q). Vu les hypothèses ci-dessus, R est l'anneau des entiers du corps quadratique imaginaire K = Q( √ d), dans lequel p est décomposé.
Par contre, si g = 2, ou si g = 3 et C est hyperelliptique, on a Aut(L) = Aut(C). des choses connues. Plaçons-nous d'abord sur un corps algébriquement clos k (on prendra ensuite pour k une clôture algébrique du corps fini k). Soit A une variété abélienne sur k, de dimension n. Si l est un nombre premier = caract. k, on sait ce qu'est le l-ième module de Tate T l (A) de A: c'est la limite projective des points de l m -division de A. C'est un Z l -module libre de rang 2n. Ce module "contrôle" les isogénies A ′ → A de degré une puissance de l, au sens suivant: une telle isogénie correspond bijectivementà un sous Z l -module d'indice fini de T l (A) (à savoir l'image de T l (A ′ ) dans T l (A)). Dans notre cas (k fini et k clôture algébrique de k), on voit en outre que les k-isogénies correspondent aux sousmodules de T l (A) qui sont stables par l'action de Galois, i.e. par le Frobenius π: puisqu'on a supposé A ∈ Ab(a, q), cela veut dire que le sous-module en question est stable par R. Finalement, on voit que les k-isogénies A ′ → A qui sont de degré une puissance de l sont classifiées par les sous R l -modules de T l (A), où R l = R ⊗ Z l . Dans le cas particulier A = E, on constate que T l (E) est un R l -module libre de rang 1 (regarder les rangs!). Si A est de la forme A L , on constate aussi que T l (A) = T l (E) ⊗ R L = T l (E) ⊗ L l , où le produit tensoriel est pris sur R l et L l désigne Z l ⊗ L = R l ⊗ R L. Il est alors immédiat que les R l -sous-modules de T l (A) d'indice fini sont de la forme T l (E) ⊗ L ′ où L ′ est un sous-R-module de L d'indice une puissance de l, et l'on a alors A ′ = E ⊗ L ′ . Autrement dit, l'énoncé (3.6) est vrai si le degré de l'isogénie est de la forme l m avec l = p.
On est donc ramenéà regarder le cas où le degré est une puissance de la caractéristique p, Evidemment, tout revient ici encoreà contrôler les isogénies. Pour une variété abélienne quelconque, cela peut se faire au moyen d'un module de Dieudonné convenable. Heureusement, les variétés abéliennes qui nous intéressent ici sont ordinaires, et cela simplifie beaucoup la situation. En effet, pour une telle variété A, on peut définir un "p-ième module de Tate" T p (A) qui a exactement les mêmes propriétés que les T l ,à savoir: c'est un Z p -module libre de rang 2n = 2. dim(A), et il contrôle les p-isogénies comme ci-dessus. Ce module est somme directe de deux modules de rang n: La partieétale T p (A) e est définie comme la limite projective des points de p m -division de A; la partie infinitésimale T p (A) i peut se définir comme le Hom (dans la catégorie des groupes formels) de G m dans A, ou bien comme le Z pdual de T p (A * ) e , où A * est la duale de A. On démontre que l'on a les mêmes propriétés que ci-dessus pour les T l . Noter que l'anneau R p = Z p ⊗ R estégal a Z p × Z p (les deux facteursétant caractérisés par le fait que π donne une unité dans le premier, et unélément de l'idéal maximal dans le second). Cette décomposition de R p est compatible avec la décomposition en deux morceaux de T p (A), ainsi qu'avec le fait que toute p-isogénie se décompose en une isogénié etale et une isogénie radicielle. Bref, tout marche très bien, et l'on arrive ainsì a démontrer (3.6) pour les p-isogénies.