Images of Pseudo-Representations and Coefficients of Modular Forms modulo p

TL;DR

This paper studies the images of 2D Galois representations over semi-local rings and applies these results to modular forms mod p, proving new density bounds for their Fourier coefficients and classifying special forms related to cyclotomic and CM forms.

Contribution

It provides a detailed description of the images of Galois representations and establishes uniform lower bounds on the density of non-zero Fourier coefficients of modular forms mod p, identifying a special subspace of forms.

Findings

Density of non-zero coefficients is positive for non-zero forms.

Existence of a uniform positive lower bound for coefficient density for most forms.

Identification of a special subspace of modular forms related to cyclotomic and CM forms.

Abstract

We describe the image of general families of two-dimensional representations over compact semi-local rings. Applying this description to the family carried by the universal Hecke algebra acting on the space of modular forms of level $N$ modulo a prime $p$, we prove new results about the coefficients of modular forms mod $p$. If $f=\sum_{n=0}^\infty a_n q^n$ is such a form, for which we can assume without loss of generality that $a_n=0$ if $(n,Np)>1$, calling $\delta(f)$ the density of the set of primes $\ell$ such that $a_\ell \neq 0$, we prove that $\delta(f)>0$ provided that $f$ is not zero (and if $p=2$, not a multiple of $\Delta$). More importantly, we prove, when $p>2$, a {\it uniform} version of this result, namely that there exists a constant $c>0$ depending only on $N$ and $p$ such that $\delta(f)>c$ for all forms $f$ except for those in an explicit subspace of infinite codimension of the space of all modular forms mod $p$ of level $N$. Forms in this subspace, called {\it special} modular forms mod $p$, are proved to be closely related to certain classes of modular forms mod $p$ previously studied by the author, Nicolas and Serre, called cyclotomic and CM modular forms mod $p$.