Computational Algebra Seminar: Combes -- Computing mod p Selmer groups

Speaker: Lewis Combes
Title: Computing mod p Selmer groups
Time & Place: 15:00 - 16:00, Thursday 20 Feb, SMRI Seminar Room

Abstract:
Selmer groups arise naturally in computational problems, such as determining the rank
of an elliptic curve. p-adic Galois representations also have associated Selmer groups; 
the Bloch-Kato conjecture says these ranks are controlled by the order of vanishing of 
an L-function. While an analogue of this picture is expected to exist for mod p Galois 
representations, very little is concretely known. We present a method to compute 
Selmer groups associated to mod p Galois representations, using class field theory. 
We present data collected on the distribution of ranks of mod 2 Selmer groups, as well 
as some interesting examples mod 3. Finally, we speculate on appropriate analogues of 
the main constructions in the Bloch-Kato conjecture in the mod p setting.