Number Theory Seminar – University of Copenhagen

Number Theory Seminar

Cody Lee Gunton (University of Arizona) will give a Number Theory Seminar titled "Néron component groups and crystalline representations."

Abstract: Let A_K be an abelian variety over a finite extension K of the p-adic numbers. Attached to A_K are its Néron model A, a smooth group defined over the integers O of K, and its p-adic Tate module T, a representation of the absolute Galois group of K. The reduction of A modulo the prime ideal of O may not be connected; let Phi be its component group. There is a subtle and important relationship between the geometry of A and the Galois representation T; for instance, it is a result of Coleman and Iovita that one can detect from T whether or not A is proper. In this talk I will present my work, which builds on work of Kim-Marshall in the case of K unramified, showing that the p-power torsion in Phi can be calculated from the cohomology of T if A is semiabelian. Along the way, I will give examples to highlight issues relating to ramification in K, and will explain how these issues can be addressed using p-adic uniformization and recent work in torsion p-adic Hodge theory.