## STNB2019 (33rd edition)

### Serre's constant of elliptic curves over the rationals

#### Presenters

Enrique González-Jiménez

#### Abstract

Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of
this talk is to define a positive integer A(E), that we call the Serre’s constant associated to E, that gives necessary
conditions to conclude that $ρ_{E,m}$ , the mod m Galois representation associated to E, is non-surjective. In particular,
if there exists a prime factor p of m satisfying $val_p (m) > val_p (A(E))$ then $ρ_{E,m}$ is non-surjective. We 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 nonsurjective subgroups of $GL_2 (\mathbb{Z}_p )$. Finally, we
conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre’s constant. This is joint work with Harris B. Daniels.

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