Resonance Sums, Shifted Convolutions, and Bounds towards the Square-Root Cancellation Hypothesis

The square-root cancellation hypothesis, in its original form, concerns cancellation in certain GL(1) sums with applications to the distribution of zeros of L-functions associated with GL(2) cusp forms. Building on Ye’s work on a varying GL(2) cusp form and my work (jointly with Ye and Gillespie) on the Rankin-Selberg convolution of two GL(2) cusp forms, both allowed to move, I will discuss a variant in which only one form is permitted to vary. This leads naturally to shifted convolution sums and new analytic challenges. I will outline my methods and preliminary ideas in this setup and discuss how these fit into the broader concept of the square root cancellation hypothesis.