## Algebra & Number Theory

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

Ian Petrow

#### 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.

#### Article information

Source
Algebra Number Theory, Volume 12, Number 10 (2018), 2471-2498.

Dates
Revised: 22 July 2018
Accepted: 23 August 2018
First available in Project Euclid: 14 February 2019

https://projecteuclid.org/euclid.ant/1550113229

Digital Object Identifier
doi:10.2140/ant.2018.12.2471

Mathematical Reviews number (MathSciNet)
MR3911137

Zentralblatt MATH identifier
07026824

#### Citation

Petrow, Ian. Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field. Algebra Number Theory 12 (2018), no. 10, 2471--2498. doi:10.2140/ant.2018.12.2471. https://projecteuclid.org/euclid.ant/1550113229

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