p-adic and mod p Galois Representations, and Generalized Breuil-Mezard Conjectures

University Of Arizona, Tucson AZ

Investigators

Abstract

The relationship between p-adic Galois representations and their reductions modulo p has been central to number theory in recent years. For instance, the method pioneered by Wiles to prove that every elliptic curve defined over the rational numbers arises from a modular form -- and hence to prove Fermat's Last Theorem -- relies crucially on the study of certain deformation spaces of p-adic Galois representations with specified reduction modulo p. For representations of the absolute Galois group of the p-adic numbers, a conjecture of Breuil and Mezard -- now a theorem in many cases, due to Breuil and Mezard, to the PI, and to Kisin -- predicts the structure of these deformation spaces. The PI proposes to study generalizations of the Breuil-Mezard conjecture to finite extensions of the p-adics. The investigator further proposes to determine the reduction modulo p of certain specific classes of p-adic representations. Each case in which a generalized Breuil-Mezard conjecture is proved should yield a modularity theorem, by a method of Kisin. In many cases this work will yield other information of arithmetic interest, such as the shape of the mod p representation attached to a modular form. The PI's research is in number theory, one of the oldest branches of mathematics. At heart, number theory is the study of whole number solutions to equations, although sophisticated modern techniques can sometimes give the appearance of being far-removed from this goal. In recent decades, number theory has had revolutionary applications to the fields of cryptography (creating codes) and cryptanalysis (breaking codes). For instance, the communications of many cellular telephones are protected by a cryptosystem based on elliptic curves, one of the primary objects of study in the PI's field. The PI is deputy director of Canada/USA Mathcamp, a summer program for mathematically talented high school students, and believes it is essential that there be research-active mathematicians participating in such endeavors.