A basis for the space of modular forms
A combinatorial interpretation of Serre's conjecture on modular Galois representations
We state a conjecture concerning modular absolutely irreducible odd 2-dimensional representations of the absolute Galois group over finite fields which is purely combinatorial (without using modular forms) and proof that it is equivalent to Serre’s strong conjecture. The main idea is to replace modular forms with coefficients in a finite field of characteristic , by their counterparts in the theory of modular symbols.
A larger GL 2 large sieve in the level aspect
In this paper we study the orthogonality of Fourier coefficients of holomorphic cusp forms in the sense of large sieve inequality. We investigate the family of GL 2 cusp forms modular with respect to the congruence subgroups Γ1(q), with additional averaging over the levels q ∼ Q. We obtain the orthogonality in the range N ≪ Q 2−δ for any δ > 0, where N is the length of linear forms in the large sieve.
A local large sieve inequality for cusp forms
We prove a large sieve type inequality for Maass forms and holomorphic cusp forms with level greater or equal to one and of integral or half-integral weight in short interval.
A quadratic form with prime variables associated with Hecke eigenvalues of a cusp form
Let be a normalized primitive holomorphic cusp form of even integral weight for the full modular group , and denote its th Fourier coefficient by . We consider the hybrid problem of quadratic forms with prime variables and Hecke eigenvalues of normalized primitive holomorphic cusp forms, which generalizes the result of D. Y. Zhang, Y. N. Wang (2017).
A relation between Fourier coefficients of holomorphic cusp forms and exponential sums
A relative trace formula proof of the Petersson trace formula
A stationary phase formula for exponential sums over and applications to GL(3)-Kloosterman sums
A test for identifying Fourier coefficients of automorphic forms and application to Kloosterman sums.
An application of the projections of automorphic forms
Analytic continuation of representations and estimates of automorphic forms.
Anticyclotomic main conjectures.
Asymptotic formulae for partition ranks
Using an extension of Wright's version of the circle method, we obtain asymptotic formulae for partition ranks similar to formulae for partition cranks which where conjectured by F. Dyson and recently proved by the first author and K. Bringmann.
Asymptotic formulas for the coefficients of certain automorphic functions
We derive asymptotic formulas for the coefficients of certain classes of weakly holomorphic Jacobi forms and weakly holomorphic modular forms (not necessarily of integral weight) without using the circle method. Then two applications of these formulas are given. Namely, we estimate the growth of the Fourier coefficients of two important weak Jacobi forms of index 1 and non-positive weights and obtain an asymptotic formula for the Fourier coefficients of the modular functions for all integers k,l...
Automorphic forms, hyperfunction cohomology, and period functions.