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|A relative trace formula comparison for the global Gross-Prasad conjecture for orthogonal groups|
The global Gross-Prasad conjecture (really its refinement by Ichino and Ikeda) is a remarkable conjectural formula generalizing Waldspurger's formula for the central value of a Rankin-Selberg $L$ function. I will explain a relative trace formula approach to this conjecture, akin in spirit to the successful comparison for unitary groups. The approach relies on a somewhat strange matching of orbits, and on two local conjectures of smooth transfer and fundamental lemma type, which I will formulate. If time permits, I will discuss some recent evidence for these local identities in some low rank cases.
|Class Groups, Congruences, and Cup Products|
The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the p-part of the class group of the p-th cyclotomic field to congruences of Bernoulli numbers mod p. For p and N prime with N = 1 mod p, a similar result of Calegari and Emerton relates the rank of the p-part of the class group of Q(N^1/p) to whether or not a certain quantity (Merel's number) is a p-th power mod N. We study this rank by building off of an idea of Wake and Wang-Erickson, namely to relate elements of the class group to the vanishing of certain cup products in Galois cohomology. Using this idea, we prove new bounds on the rank in terms of similar p-th power conditions, and we give exact characterizations of the rank for small p. This talk will aim to explain the interplay between ranks of class groups, explicit congruences, and cup products in Galois cohomology This is joint work with Karl Schaefer.