Abstract: In this paper we compute dimension formulas for rings of Siegel modular forms of genus g = 3 Let denote the main congruence subgroup of level two, the Hecke subgroup of level two and the full modular group We give the dimension formulas for genus g = 3 for the above mentioned groups and determine the graded ring of modular forms with respect to

Abstract: © Université Bordeaux 1, 1995, tous droits réservés. L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: In an earlier paper (math.NT/9906019) we showed that any integral unimodular lattice L of rank n which is not isometric with Z^n has a characteristic vector of norm at most n-8. [A "characteristic vector" of L is a vector w in L such that 2|(v,w-v) for all v in L; it is known that the characteristic vectors all have norm congruent to n mod 8 and comprise a coset of 2L in L.] Here we use modular forms and the classification of unimodular lattices of rank <24 to find all L whose minimal characteristic vectors have norm n-8. Along the way we also obtain congruences and a lower bound on the kissing number of unimodular lattices with minimal norm 2. We then state and prove analogues of these results for self-dual codes, and relate them directly to the lattice problems via "Construction A".

Abstract: We give a geometrical construction of the canonical automorphic factor for the Jacobi group and construct new vector valued modular forms from Jacobi forms by differentiating them with respect to toroidal variables and then evaluating at zero.

Abstract: The Weil-Taniyama conjecture states that every elliptic curve E/Q of conductor N can be parametrized by modular functions for the congruence subgroup T0(N) of the modular group r = F5L(2,Z). Equivalently, there is a nonconstant map ip from the modular curve Xq(N) to E. We present here a method of computing the degree of such a map for arbitrary N. Our method, which works for all subgroups of finite index in I" and not just r0(Ar), is derived from a method of Zagier published in 1985; by using those ideas, together with techniques which have recently been used by the author to compute large tables of modular elliptic curves, we are able to derive an explicit and general formula which is simpler to implement than Zagier's. We discuss the results obtained, including a table of degrees for all the modular elliptic curves of conductors up to 200.

Abstract: The main purpose of this paper is to examine certain well-known modular equations of degree 5 from a very intuitive angle. Unlike the classical treatment of this subject, which requires parametrization of certain key quantities, we will give direct proofs to these equations by recasting them as theta function identities which are quite interesting in their own right. Ramanujan discovered an astounding number of modular equations. Since he did not supply proofs to these identities, we do not know his methods. In this paper, we will see that in the study of the modular identities of degree 5, the following well-known identities (see [1, p. 262, Entries 10(iv) and (v)]) are of great significance: 00 (1l ) 92 (q)-q22(q5) = 4ql (1+ q2n)(1 + q5n)(1 q5n)2(l + ql0n-5 n=1 00 (2) 93(q) -_ &(q5) = 4q fJ(1 + q2n-1)(1 + q5n)(1 q5n)2(1 + q1f) n=1 In combination with another identity (7), they yield many very striking theta function identities. In particular, in the course of our discussions, we will provide simple direct proofs to the following two identities: 00 2 (q) 2 (q ) _2 (q)t2(q5) q 5 (q)2 (q2 ) 8q fJ(i q2n)(1 qlOfn)2, n=1 2 0(q) _ 3 (q) _ 9(q) 2 ?? (1-q2n)5 V 2(q 5) V 3(q 5) V 4(q5) 11 (1-ql~n) According to Bruce Berndt (see [1, p. 285]), direct proofs of these two identities have not been given. We assume the reader is familiar with the basic properties of the theta functions covered in [2, Chapters 21 and 22]. Received by the editors August 29, 1993. 1991 Mathematics Subject Classification. Primary 33D 10; Secondary 1 1 B65.

Abstract: The equation of Fermat has undoubtedly had a far greater influence on the development of mathematics than anyone could have imagined. After 1847 most serious mathematical approaches to the problem followed the line introduced by Kummer. This approach involved a detailed analysis of the ideal class groups of cyclotomic fields.

Abstract: We study the Donaldson invariants of simply connected $4$-manifolds with $b_+=1$, and in particular the change of the invariants under wall-crossing. We assume the conjecture of Kotschick and Morgan about the shape of the wall-crossing terms (which Oszva\'th and Morgan are now able to prove), and are determine a generating function for the wall-crossing terms in terms of modular forms. As an application we determine all the Donaldson invariants of the projective plane in terms of modular forms. The main tool are the blowup formulas, which are used to obtain recursive relations.

Abstract: We continue our study of Yoshida's lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.

Abstract: We examine the electric-magnetic duality for a U(1) gauge theory on a general four manifold which generates the SL(2, Z) group. The partition functions for such a theory transforms as a modular form of specific weight. However, in the canonical approach, we show that S-duality for the abelian theory, like T-duality, is generated by a canonical transformation leading to a modular invariant partition function.

Abstract: Under certain assumptions, we prove a conjecture of Mazur and Tate describing a relation between the modular symbol attached to an elliptic curve with split multiplicative reduction atp, and itsp-adic period. We generalize this relation to modular forms of weight 2 with coefficients not necessarily in .

Abstract: The residue theorem is employed to obtain new identities amongpthe powers of theta constants with rational characteristics. The technique is then used to derive some known identities of Ramanujan.

Abstract: © Annales de l’institut Fourier, 1995, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: This is the first of two interconnected parts: Part I contains the geometric theory of generalized modular forms and their connections with the cooperation algebra for elliptic cohomology, E�� ∗E�� , while Part II is devoted to the more algebraic theory associated with Hecke algebras and stable oper- ations in elliptic cohomology. We investigate the structure of the stable operation algebra E�� ∗ E�� by first determining the dual cooperation algebra E�� ∗E�� . A major ingredient is our identification of the cooperation algebra E�� ∗E�� with ar ing of general- ized modular forms whoses exact determination involves understanding certain integrality conditions; this is closely related to a calculation by N. Katz of the ring of all 'divided congruences' amongst modular forms. We relate our present work to previous constructions of Hecke operators in elliptic cohomology. We also show that a well known operator on modular forms used by Ramanujan, Swinnerton-Dyer, Serre and Katz cannot extend to a stable operation.

Abstract: In doing p-adic analysis on spaces of classical modular functions and forms, it is convenient and traditional to broaden the notion of “modular form” to a class called “overconvergent p-adic modular forms.” Critical for the analysis of the p-adic Banach spaces composed of this wider class of forms is the “Atkin U -operator”, which is completely continuous and whose spectral theory (still not very well understood) seems to be the key to a good deal of arithmetic. The part of the spectrum of U corresponding to eigenvalues which are p-adic units is somewhat more understood, thanks to the work of Hida. As for the rest of the spectrum, it is surprising how fragmentary our information is (although recent work of Coleman, resolving in part some prior conjectures of ours, has improved the situation). We have begun an experimental search for nonclassical (but “overconvergent”) eigenfunctions in a fairly simple way. We take a number of classical modular functions which are p-adically overconvergent (e.g., j, 1/j,...), and try to find their p-adic “U -eigenfunction expansion.” There is a straightforward computational procedure to approximate such eigenfunction expansions, even though, on a theoretical level, we do not even know that the “expansions” that our algorithm produces converge in any sense, or even settle down numerically. Experimentally, they seem to, and they produce candidate Fourier expansions. In our computations, we specialize principally to p = 5. The same eigenfunctions (produced by our algorithm) seem to occur as terms in each of the eigenfunction expansions we have calculated. This leads us to suspect that we have in fact encountered all the 5-adic eigenfunctions

Abstract: Although much is known about the partition function, little is known about its parity. For the polynomials D ( x ) : = ( D x 2 + 1 ) / 24 , where D ≡ 23 ( mod 24 ) , we show that there are infinitely many m (resp. n ) for which p ( D ( m ) ) is even (resp. p ( D ( n ) ) is odd) if there is at least one such m (resp. n ). We bound the first m and n (if any) in terms of the class number h ( − D ) . For prime D we show that there are indeed infinitely many even values. To this end we construct new modular generating functions using generalized Borcherds products , and we employ Galois representations and locally nilpotent Hecke algebras.

Abstract: © Foundation Compositio Mathematica, 1995, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Abstract: The goal of this paper is to discuss a Newton-Hodge inequality for modular forms. More precisely, for a prime number p and an integer N prime to p we consider the characteristic polynomial of the Hecke operator Up on the space Sk-^-2 (^i (^\" N)) of cusp forms for the congruence subgroup Fi (j»\" N) of SL'z (Z). The main theorem bounds the Newton polygon of this polynomial from below by an explicit polygon denned in terms of the genus and number of cusps of the modular curve Xi (N). The main technique is a motivic variation of theorems of Mazur, Ogus, Illusie and Nygaard on the Katz conjecture (according to which the Newton polygon of Frobenius on crystalline cohomology is bounded in terms of dimensions of Hodge cohomology groups) and a computation of these Hodge groups using logarithmic schemes. We get new information because the relevant Hodge nitration is not of type (k + 1,0), (0, k + 1) as usual, but rather of type (k + 1,0), (k, 1), ..., (1, k), (0, k + 1).

Abstract: In this paper, we shall give a new relation between the arithmetic of quaternion algebras and modular forms; we shall express the type numberT q, N of a split order of type (q, N) as the sums of dimensions of some subspaces of the space of cusp forms of weight 2 with respect to Γ0(qN) which are common eigenspaces of Atkin-Lehner's involutions.