Bas Edixhoven

A rigid analytic Gross-Zagier formula and arithmetic applications

Let f be a newform of weight 2 and squarefree level N. Its Fourier coefficients generate a ring Of whose fraction field Kf has finite degree over Q. Fix an imaginary quadratic field K of discriminant prime to N, corresponding to a Dirichlet character E. The L-series L(f /K, s) = L(f, s)L(f 0 E, s) of f over K has an analytic continuation to the whole complex plane and a functional equation relating L(f/K, s) to L(f/K, 2 s). Assume that the sign of this functional equation is 1, so that L(f/K, s) vanishes to even order at s = 1. This is equivalent to saying that the number of prime factors of N which are inert in K is odd. Fix any such prime, say p. The field K determines a factorization N = N+Nof N by taking N+, resp. Nto be the product of all the prime factors of N which are split, resp.

Special Points on the Product of Two Modular Curves

We prove, assuming the generalized Riemann hypothesis for imaginary quadratic fields, the following special case of a conjecture of Oort, concerning Zarsiski closures of sets of CM points in Shimura varieties. Let X be an irreducible algebraic curve in C2, containing infinitely many points of which both coordinates are j-invariants of CM elliptic curves. Suppose that both projections from X to C are not constant. Then there is an integer m ≥ 1such that X is the image, under the usual map, of the modular curve Y20(m). The proof uses some number theory and some topological arguments.

On the Manin constants of modular elliptic curves

For M a positive integer, let X 0 (M) Q be the modular curve over Q classifying elliptic curves with a given cyclic subgroup of order M and let J 0(M)Q be the jacobian of X 0(M)Q. An elliptic curve E over Q is said to be modular if it is an isogeny factor (isogenics over Q) of some J 0(M)Q; the smallest M for which this happens is then called the level of E. The Shimura-Taniyama conjecture states that every elliptic curve E over Q is modular, and that the level of E equals the conductor of E. A modular elliptic curve of level M is called strong if there exists a closed immersion E ↪ J 0(M)Q. It follows from the multiplicity one principle for modular forms that such an immersion is unique up to sign.

Serre’s Conjecture

The aim of the first section is to state Serre’s conjecture and to tell what is presently known about it, without proof. We start by recalling what modular forms are. Then we recall the result, due to Deligne, that to a mod p modular form one can associate a mod p Galois representation. After that we state Serre’s conjecture and what we know about it. In Section 2 we will see which cases of it are actually needed in order to prove, following Wiles, that all semi-stable elliptic curves over ℚ are modular. In the last two sections we will sketch the proofs in those cases. These notes follow, to some extent, the lectures given by Dick Gross during the conference.

On the André-Oort Conjecture for Hilbert Modular Surfaces

In order to state the conjecture mentioned in the title, we need to recall some terminology and results on Shimura varieties; as a general reference for these, we use [20, Sections 1–2]. So let S:= Resc/Grnc be the algebraic group over R obtained by restriction of scalars from C to R of the multiplicative group. For V an 1[8-vector space, it is then equivalent to give an R-Hodge structure or an action by S on it. A Shimura datum is a pair(G X)withGa connected reductive affine algebraic group over Q, andXa G(R)-conjugacy class in the set of morphisms of algebraic groups Hom(S, GR.), satisfying the three conditions of [20, Def. 1.4] (i.e., the usual conditions (2.1.1-3) of [13]). These conditions imply thatXhas a natural complex structure (in fact, the connected components are hermitian symmetric domains), such that every representation ofGon a Q-vector space defines a polarizable variation of Hodge structure onX.For(G X)a Shimura datum, andKa compact open subgroup of G(Af), we let ShK(G, X)((C) denote the complex analytic varietyG(Q)\(X x G(A f )/ K)which has a natural structure of quasi-projective complex algebraic variety, denoted Sill G2 that maps X1to X2; for K1and K2 compact open subgroups of Gi(Af) and G2 (AO withf (KOcontained in K2, such anfinduces a morphism Sh(f)from ShK, (G1, Xi)c to ShK2(G2, X2)c.

COMPARISON OF INTEGRAL STRUCTURES ON SPACES OF MODULAR FORMS OF WEIGHT TWO, AND COMPUTATION OF SPACES OF FORMS MOD 2 OF WEIGHT ONE. WITH APPENDICES BY JEAN-FRANÇOIS MESTRE AND GABOR WIESE

Two integral structures on the Q-vector space of modular forms of weight two on X_0(N) are compared at primes p exactly dividing N. When p=2 and N is divisible by a prime that is 3 mod 4, this comparison leads to an algorithm for computing the space of weight one forms mod 2 on X_0(N/2). For p arbitrary and N>4 prime to p, a way to compute the Hecke algebra of mod p modular forms of weight one on Gamma_1(N) is presented, using forms of weight p, and, for p=2, parabolic group cohomology with mod 2 coefficients. Appendix A is a letter from Mestre to Serre, of October 1987, where he reports on computations of weight one forms mod 2 of prime level. Appendix B reports on an implementation for p=2 in Magma, using Steins modular symbols package, with which Mestres computations are redone and slightly extended.

Computational aspects of modular forms and Galois representations : how one can compute in polynomial time the value of Ramanujan's tau at a prime

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujans tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoofs algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujans tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.

Modular forms on Schiermonnikoog

Preface Contributors 1. Modular forms Bas Edixhoven, Gerard van der Geer and Ben Moonen 2. On the basis problem for Siegel modular forms with level Siegfried Bocherer, Hidenori Katsurada and Rainer Shulze-Pillot 3. Mock theta functions, weak Maass forms, and applications Kathrin Bringmann 4. Sign changes of coefficients of half integral weight modular forms Jan Hendrik Bruinier and Winfried Kohnen 5. Gauss map on the theta divisor and Greens functions Robin de Jong 6. A control theorem for the images of Galois actions on certain infinite families of modular forms Luis Dieulefait 7. Galois realizations of families of Projective Linear Groups via cusp forms Luis Dieulefait 8. A strong symmetry property of Eisenstein series Bernhard Heim 9. A conjecture on a Shimura type correspondence for Siegel modular forms, and Harders conjecture on congruences Tomoyoshi Ibukiyama 10. Peterssons trace formula and the Hecke eigenvalues of Hilbert modular forms Andrew Knightly and Charles Li 11. Modular shadows and the Levy-Mellin -adic transform Yuri I. Manin and Matilde Marcolli 12. Jacobi forms of critical weight and Weil representations Nils-Peter Skoruppa 13. Tannakian categories attached to abelian varieties Rainer Weissauer 14. Torellis theorem from the topological point of view Rainer Weissauer 15. Existence of Whittaker models related to four dimensional symplectic Galois representations Rainer Weissauer 16. Multiplying modular forms Martin H. Weissman 17. On projective linear groups over finite fields Gabor Wiese.

Algebraic Stacks Whose Number of Points over Finite Fields is a Polynomial

The aim of this article is to investigate the cohomology (l-adic as well as Betti) of schemes, and more generally of certain algebraic stacks, that are proper and smooth over the integers and have the property that there exists a polynomial P with rational coefficients such that for all prime powers q the number of points over the field with q elements is P(q). We prove that for all prime numbers l the l-adic etale cohomology is a direct sum of Tate twists of the trivial representation. Our main tools here are Behrends Lefschetz trace formula and l-adic Hodge theory. In the last section we investigate the Hodge structure on the Betti cohomology. The motivation for this article comes from applications to certain moduli stacks of curves.

On the computation of the coefficients of a modular form

We give an overview of the recent result by Jean-Marc Couveignes, Bas Edixhoven and Robin de Jong that says that for l prime the mod l Galois representation associated to the discriminant modular form Δ can be computed in time polynomial in l. As a consequence, Ramanujan’s τ(p) for prime numbers p can be computed in time polynomial in logp. The mod l Galois representation occurs in the Jacobian of the modular curve X1(l), whose genus grows quadratically with l. The challenge therefore is to do the necessary computations in time polynomial in the dimension of this Jacobian. The field of definition of the l2 torsion points of which the representation consists is found via a height estimate, obtained from Arakelov theory, combined with numerical approximation. The height estimate implies that the required precision for the approximation grows at most polynomially in l.