On Saito-Kurokawa descent for congruence subgroups

The conjecture made by H. Saito and N. Kurokawa states the existence of a “lifting” from the space of elliptic modular forms of weight 2k?2 (for the full modular group) to the subspace of the space of Siegel modular forms of weightk (for the full Siegel modular group) which is compatible with the action of Hecke operators. (The subspace is the so called “Maaß spezialschar” defined by certain identities among Fourier coefficients). This conjecture was proved (in parts) by H. Maaß, A.N. Andrianov and D. Zagier. The purpose of this paper is to prove a generalised version of the conjecture for cusp forms of odd squarefree level.     

The first moment of Salié sums

The main objective of this article is to study the asymptotic behavior of Salié sums over arithmetic progressions. We deduce from our asymptotic formula that Salié sums possess a bias towards being positive. The method we use is based on the Kuznetsov formula for modular forms of half-integral weight. Moreover, in order to develop an explicit formula, we are led to determine an explicit orthogonal basis of the space of modular forms of half-integral weight.     

Congruences for modular forms of weights two and four

We prove a conjecture of Calegari and Stein regarding mod p congruences between modular forms of weight four and the derivatives of modular forms of weight two.     

Variation of Heegner points in Hida families

Given a weight two modular form f with associated p-adic Galois representation V _f , for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of V _f by taking the Kummer images of Heegner points on the modular abelian variety attached to f. We show that these classes can be interpolated as f varies in a Hida family and construct an Euler system of big Heegner points for Hida’s universal ordinary deformation of V _f . We show that the specialization of this big Euler system to any form in the Hida family is nontrivial, extending results of Cornut and Vatsal from modular forms of weight two and trivial character to all ordinary modular forms, and propose a horizontal nonvanishing conjecture for these cohomology classes. The horizontal nonvanishing conjecture implies, via the theory of Euler systems, a conjecture of Greenberg on the generic ranks of Selmer groups in Hida families.     

Modularity of Galois traces of class invariants

Zagier showed that the Galois traces of the values of j-invariant at CM points are Fourier coefficients of a weakly holomorphic modular form of weight 3/2 and Bruinier–Funke expanded his result to the sums of the values of arbitrary modular functions at Heegner points. In this paper, we identify the Galois traces of real-valued class invariants with modular traces of the values of certain modular functions at Heegner points so that they are Fourier coefficients of weight 3/2 weakly holomorphic modular forms.     

Coefficients of half-integral weight modular forms

In this paper, we study the distribution of the coefficients a(n) of half-integral weight modular forms modulo odd integers M. As a consequence, we obtain improvements of indivisibility results for the central critical values of quadratic twists of L-functions associated with integral weight newforms established in Ono and Skinner (Fourier coefficients of half-integral weight modular forms modulo ?, Ann. of Math. 147 (1998) 453-470). Moreover, we find a simple criterion for proving cases of Newman's conjecture for the partition function.     

Generalized Kac-Moody algebras, automorphic forms and Conway's group I

The moonshine properties imply that the twisted denominator identities coming from the action of the monster group on the monster algebra define modular forms. In this paper, we motivate the conjecture that the action of an extension of Conway's simple group Co₁ on the fake monster algebra gives rise to automorphic forms of singular weight on Grassmannians. We prove the conjecture for elements with square-free level and nontrivial fixpoint lattice.     

Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series

Zagier [23] proved that the generating functions for the traces of level 1 singular moduli are weight 3/2 modular forms. He also obtained generalizations for “twisted traces”, and for traces of special non-holomorphic modular functions. Using properties of Kloosterman-Salié sums, and a well known reformulation of Salié sums in terms of orbits of CM points, we systematically show that such results hold for arbitrary weakly holomorphic and cuspidal half-integral weight Poincaré series in Kohnen’s Γ₀(4) plus-space. These results imply the aforementioned results of Zagier, and they provide exact formulas for such traces.     

p-adic periods and modular symbols of elliptic curves of prime conductor

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.Oblatum 24-XI-1993 & 8-VI-1994     

Maass relations for Saito–Kurokawa lifts of higher levels

The Ramanujan Journal - It is known that among Siegel modular forms of degree 2 and level 1 the only functions that violate the Ramanujan conjecture are Saito–Kurokawa lifts of modular forms...     

Some applications of smooth bilinear forms with Kloosterman sums

We revisit a recent bound of I. Shparlinski and T. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet L-functions.     

Binary forms as sums of two squares and Châtelet surfaces

The representation of integral binary forms as sums of two squares is discussed and applied to establish the Manin conjecture for certain Chatelet surfaces over ℚ.     

Vertex Coloring of Graphs by Total 2-Weightings

An assignment of weights to the edges and the vertices of a graph is a vertex-coloring total weighting if adjacent vertices have different total weight sums. Of interest in this paper are vertex-coloring total weightings with weight set of cardinality two, a problem motivated by the conjecture that every graph has such a weighting using the weights 1 and 2. Here we prove the existence of such weightings for certain families of graphs using any two different real weights. A related problem where all vertices have unique weight sums is also discussed.     

Critical slope p-adic L-functions of CM modular forms

For ordinary modular forms, there are two constructions of a p-adic L-function attached to the non-unit root of the Hecke polynomial, which are conjectured but not known to coincide. We prove this conjecture for modular forms of CM type, by calculating the critical-slope L-function arising from Kato’s Euler system and comparing this with results of Bellaïche on the critical-slope L-function defined using overconvergent modular symbols.     

Dimension variation of classical and p-adic modular forms

A quadratic bound is obtained for a conjecture of Gouvêa-Mazur on arithmetic variation of dimensions of classical and p-adic modular forms. Oblatum 22-VIII-1996 & 8-X-1997     

Self-replication and Borwein-like algorithms

We investigate certain Eisenstein congruences, as predicted by Harder, for level p paramodular forms of genus 2. We use algebraic modular forms to generate new evidence for the conjecture. In doing this, we see explicit computational algorithms that generate Hecke eigenvalues for such forms.     

On type numbers of split orders of definite quaternion algebras

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.     

Arithmetic properties of shimura sums related to several modular forms

The paper is concerned with Shimura sums related to modular forms with multiplicative coefficients which are products of Dedekind η-functions of various arguments. Several identities involving Shimura sums are established. The type of identity obtained depends on the splitting of primes in certain imaginary quadratic number fields.     

A counterexample to the Gouvêa–Mazur conjecture

Gouvêa and Mazur made a precise conjecture about slopes of modular forms. Weaker versions of this conjecture were established by Coleman and Wan. In this Note, we exhibit examples contradicting the full conjecture as it currently stands. To cite this article: K. Buzzard, F. Calegari, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Minimal lifts of dihedral 2-dimensional Galois representations

In this paper we study the connection between odd dihedral 2-dimensional modulo p Galois representations and modular forms with complex multiplication. More precisely, we prove that, for every such representation satisfying some explicit conditions, there exists a modular form giving rise to it of the type given by Serre’s conjecture (in its strong version) with the additional property of having complex multiplication.