Average values of Rankin-Selberg L-functions for GL(2)

I will discuss some techniques to estimate central values of Rankin-Selberg
L-functions for GL(2) in the self-dual setting, and in particular how a
trick from the spectral theory of automorphic forms can be used to reduce
the problem to a suitable application of the approximate functional
equation method. This work is motivated both by applications to bounding
(or else showing systematic growth of) Mordell-Weil ranks of abelian
varieties in certain abelian extensions of CM fields (not conditional on
the conjecture of Birch and Swinnerton-Dyer), as well as to folklore
conjectures for higher-rank groups which might eventually be accessible by
similar techniques. I will explain both of these motivations if time
permits.