Computing the number of certain Galois representations mod $p$

TL;DR

This paper computes lower bounds for the number of specific irreducible, odd Galois representations over _p for all primes up to 1999, leveraging the connection with mod p modular forms via Serre's Conjecture.

Contribution

It provides explicit lower bounds for the count of certain Galois representations for all primes up to 1999, extending understanding of their distribution.

Findings

Lower bounds computed for primes up to 1999

Establishes a link between Galois representations and modular forms

Advances knowledge on the enumeration of Galois representations

Abstract

Using the link between mod $p$ Galois representations of $\qu$ and mod $p$ modular forms established by Serre's Conjecture, we compute, for every prime $p\leq 1999$, a lower bound for the number of isomorphism classes of continuous Galois representation of $\qu$ on a two--dimensional vector space over $\fbar$ which are irreducible, odd, and unramified outside $p$.