A Shimura-Shintani correspondence for rigid analytic cocycles

Abstract: In their paper Singular moduli for real quadratic fields: a rigid analytic approach, Darmon and Vonk introduced rigid meromorphic cocycles, i.e. elements of $H^1(\mathrm{SL}_2(\mathbb{Z}[1/p]),\mathcal{M}^\times)$ where $\mathcal{M}^\timesx$ is the multiplicative group of rigid meromorphic functions on the $p$-adic upper-half plane. Their values at RM points belong to narrow ring class fields of real quadratic fiends and behave analogously to CM values of modular functions on $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}$. In this talk I will present some progress towards developing a Shimura-Shintani correspondence in this setting.

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic