Harris–Venkatesh plus Stark

Abstract: The class number formula describes the behavior of the Dedekind zeta function at $s = 0$ and $s = 1$. The Stark and Gross conjectures extend the class number formula, describing the behavior of Artin $L$-functions and $p$-adic $L$-functions at $s = 0$ and $s = 1$ in terms of units. The Harris–Venkatesh conjecture describes the residue of Stark units modulo $p$, giving a modular analogue to the Stark and Gross conjectures while also serving as the first verifiable part of the broader conjectures of Venkatesh, Prasanna, and Galatius. In this talk, I will draw an introductory picture, formulate a unified conjecture combining Harris–Venkatesh and Stark for weight one modular forms, and describe the proof of this in the imaginary dihedral case.

number theory

Audience: researchers in the topic

University of Arizona Algebra and Number Theory Seminar