# Number Theory Abstracts, Term 1 2018-2019

#### S

- Siksek, Samir
Title:

**On the asymptotic Fermat conjecture**

Abstract: The asymptotic Fermat conjecture states that for a number field K there is a constant B_K such that for primes p \ge B_K the only K-rational points on the Fermat curve X^p+Y^p+Z^p=0, up to the obvious symmetries, are (x:y:z)=(1,-1,0), or (x:y:z)=(1,\zeta,\zeta^2) where \zeta=\exp(2\pi i/3). In this talk we survey joint work with Nuno Freitas, and with Haluk Sengun, on the asymptotic Fermat conjecture. In particular we prove AFC for the family of number fields K=\Q(\zeta_{2^r})^+.

#### W

- Weiss, Ariel
Irreducibility of Galois representations associated to low weight Siegel modular forms

Abstract: If f is a cuspidal modular eigenform of weight k>1, Ribet showed that its associated p-adic Galois representation is irreducible for all primes. More generally, it is a hard open problem to show that the p-adic Galois representation attached to a cuspidal automorphic representation of GL(n) is irreducible. In this talk, I will discuss this conjecture for low weight Siegel modular forms, and prove that if such a form is not CAP or endoscopic, then its associated p-adic Galois representation is irreducible for 100% of primes p.