Twist-minimal trace formula for holomorphic cusp forms

Kieran Child

doi:10.1007/s40993-021-00302-9

Abstract

We derive an explicit formula for the trace of an arbitrary Hecke operator on spaces of twist-minimal holomorphic cusp forms with arbitrary level and character, and weight at least 2. We show that this formula provides an efficient way of computing Fourier coefficients of basis elements for newform or cusp form spaces. This work was motivated by the development of a twist-minimal trace formula in the non-holomorphic case by Booker, Lee and Strömbergsson, as well as the presentation of a fully generalised trace formula for the holomorphic case by Cohen and Strömberg.

Highlights

Modular forms play a central role in much of modern number theory
In this paper we derive a formula for the trace of Hecke operators on spaces of so-called ‘twist-minimal’ forms (Theorem 1), and demonstrate that this is sufficient to recover the Fourier coefficients of basis elements of cusp form spaces (Theorem 2)
We give an explicit expression for the trace of these operators on subspaces of modular forms, and we show that these traces can be used to generate basis elements

Summary

Introduction

Modular forms play a central role in much of modern number theory. Their properties have been crucial in establishing the modularity theorem (in turn proving Fermat’s last theorem, see [1]), and the Fourier coefficients of modular forms provide explicit data for problems such as the congruent number problem, representations of integers by quadratic forms, and classifications of Galois representations (see [2] for an overview of several applications). The sieving of the formula to twist-minimal spaces results in much simpler expressions to compute these spaces, and allows us to calculate a smaller number of twist-minimal spaces in order to recover a large number of different cusp form spaces at once This efficiently provides explicit data for aforementioned problems on representation of integers and Galois representations. Following work by others in [9] and [10], this formula is available in full generality for traces of any Hecke operator on spaces of any level and character This formula can be used to find Fourier coefficients for basis elements of cusp form spaces (as done in [11], with a newform sieve, for the generation of LMFDB data), our formula contains much simpler expressions, and is more efficient for this task in the case that the level N is not square-free.

Preliminary theory

Deriving the trace formula

Factorising over n

Examples

Dimension formula