Variation of Selmer Groups

Cornell University, Ithaca NY

Investigators

Abstract

Abstract for award DMS-0400232 of Ramakrishna: The main purpose of this project is to continue an investigation of the PI of mod p Galois representations, p-adic Galois representations and their relations with problems in number theory and arithmetic algebraic geometry. A particular emphasis will be the understanding of how Selmer groups of various sorts change as more primes are added to the ramification set. In one context this variation of Selmer groups relates to how many `new at q' newforms there are congruent to a particular oldform. The PI's research is in Algebraic Number Theory. This field has its roots in the study of whole numbers and their various properties. A key tool in the PI's research is Galois theory. Galois theory is the study of symmetries of solutions of equations. The understanding of these symmetries shed slight on the the solutions themselves, a basic mathematical question. An example of such a symmetry is the fact that complex solutions to polynomial equations occur in complex conjugate pairs.