Second order modular forms
S-extremal strongly modular lattices
S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only finitely many s-extremal strongly modular lattices of even minimum.
Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications
For an odd and squarefree level N, Kohnen proved that there is a canonically defined subspace [...] S κ + 1 2 n e w ( N ) ⊂ S κ + 1 2 ( N ) , and S κ + 1 2 n e w ( N ) and S 2 k n e w ( N ) are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product [...] a g ( m ) a g ( n ) ¯ of two arbitrary Fourier coefficients of a Hecke eigenform g of halfintegral weight and of level 4N in terms of certain cycle integrals of the corresponding form f of integral weight. To this...
Simple zeros of degree 2 -functions
We prove that the complete -functions of classical holomorphic newforms have infinitely many simple zeros.
Small eigenvalues of Laplacian for
Solving diophantine equations
Solving Ramanujan's differential equations for Eisenstein series via a first order Riccati equation
Some computations with Hecke rings and deformation rings. With an appendix by Amod Agashe and William Stein.
Some eta-identities arising from theta series.
Some relations between the analytical modular forms and Maass waveforms for
Some Remarks on Odd Maass Wave Forms (and a Correction to [1]).
Some results on algebraic independence. (Quelques résultats d'indépendance algébrique.)
Special values of the standard zeta functions for elliptic modular forms.
Stickelberger elements and modular parametrizations of elliptic curves.
Sums Involving the Values at Negative Integers of L-Functions of Quadratic Characters.
Sums of Powers of Cusp Form Coefficients.
Sums of Powers of Cusp Form Coefficients. II.
Superelliptic equations arising from sums of consecutive powers
Using only elementary arguments, Cassels solved the Diophantine equation (x-1)³ + x³ + (x+1)³ = z² (with x, z ∈ ℤ). The generalization (with x, z, n ∈ ℤ and n ≥ 2) was considered by Zhongfeng Zhang who solved it for k ∈ 2,3,4 using Frey-Hellegouarch curves and their corresponding Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solutions for k = 5 have x = z = 0, and that there are no solutions for k = 6. The chief innovation...
Sur la multiplication complexe des fonctions elliptiques
Sur la nature arithmétique des valeurs de fonctions modulaires