Two-dimensional mod p Galois representations attached to modular forms

Опубликовано 6 сентября 2016, 17:40

Classical newforms are cusp forms on congruence subgroups of SL(2,Z) that are eigenvectors for the Hecke operators. These modular forms give rise to two-dimensional representations of the absolute Galois group of the rational field. Conversely, if one starts with a semisimple two-dimensional representation of this Galois group over a finite field, the representation should arise from a newform if a mild necessary condition (involving complex conjugation) is satisfied. When the representation is irreducible, Serre's conjecture (which is essentially a theorem) predicts the modularity and specifies that possible weights and levels of newforms that give rise to the representation. When the representation is reducible, the set of weights and levels that give rise to it is apparently more complicated to describe. I will discuss the simplest possible examples of this phenomenon, where the representation is about as uncomplicated as possible, the weight is 2 and the level is either a prime number or the product of two distinct primes.