# Serre's constant of elliptic curves over the rationals

## TL;DR

This paper introduces Serre's constant for elliptic curves over rationals, providing criteria for non-surjective Galois representations and classifying possible Galois image combinations, with results contingent on Serre's Uniformity Conjecture.

## Contribution

It defines Serre's constant, characterizes non-surjective Galois representations, and classifies infinite occurrences of certain Galois image configurations for non-CM elliptic curves over rationals.

## Key findings

- Determined all Serre's constants for infinitely many non-CM elliptic curves (conditionally).
- Classified possible Galois representation combinations occurring infinitely often.
- Conjectured all potential configurations of Serre's constant and Galois images.

## Abstract

Let $E$ be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer $A(E)$, that we call the {\it Serre's constant associated to $E$}, that gives necessary conditions to conclude that $\rho_{E,m}$, the mod m Galois representation associated to $E$, is non-surjective. In particular, if there exists a prime factor $p$ of $m$ satisfying ${\rm val}_p(m) > {\rm val}_p(A(E))>0$ then $\rho_{E,m}$ is non-surjective. {Conditionally under Serre's Uniformity Conjecture, w}e determine all the Serre's constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod $p$ Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over $\mathbb{Q}$, and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of ${\rm GL}_2(\mathbb{Z}_p)$. Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre's constant.