Vol. 12, No. 10, 2018

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

Ian Petrow

Vol. 12 (2018), No. 10, 2471–2498
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.

Keywords
traces of Hecke operators, modular curves over a finite field, elliptic curves over a finite field, Petersson formula for newforms, Tsfasman–Vlăduţ–Zink theorem
Mathematical Subject Classification 2010
Primary: 11F25
Secondary: 11F11, 11F72, 11G20, 14G15
Milestones
Received: 6 May 2018
Revised: 22 July 2018
Accepted: 23 August 2018
Published: 1 February 2019
Authors
Ian Petrow
Departement Mathematik
ETH Zürich
Zürich
Switzerland