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Some cases of Serre's uniformity problem

Posted in
Speaker: 
Pedro João Lemos
Affiliation: 
MPIM
Date: 
Wed, 2018-01-31 16:30 - 17:30
Location: 
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

Given a number field $K$, Serre's uniformity question asks whether there exists a
constant $C_K$ (depending only on $K$) such that if $E/K$ is an elliptic curve
without complex multiplication, then $\bar{\rho}_{E,p}$ is surjective for every
prime $p>C_K$. In this talk, building on previous work by Darmon and Merel,
I will show that Serre's uniformity question has a positive answer for the family
of elliptic curves defined over the rationals, without complex multiplication and
admitting a non-trivial cyclic isogeny. I will then present some more recent
work I have been conducting on other families of elliptic curves.

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