Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field

Abstract

We give an upper bound for the trace of a Hecke operator acting on the space of holomorphic cusp forms with respect to certain congruence subgroups. Such an estimate has applications to the analytic theory of elliptic curves over a finite field, going beyond the Riemann hypothesis over finite fields. As the main tool to prove our bound on traces of Hecke operators, we develop a Petersson formula for newforms for general nebentype characters.

Final version; to appear in Algebra & Number Theory

References in corpus (1)

• Explicit calculations with Eisenstein series

Cited by in corpus (2)

• The fourth moment of Dirichlet $L$-functions along a coset and the Weyl bound
• Subconvexity bounds for twisted L-functions, II