Bayou Arithmetic Research Days

In this talk we will study how CM fields are distributed according to the Galois group of their Galois closures, when ordered by discriminant. In particular, we will show that there are certain families of CM fields which have positive natural density in the set of all CM fields and moreover, that these families are the only ones that have positive density. We then apply these results to study a famous conjecture of Colmez that relates Faltings heights of CM abelian varieties to special values at s = 0 of logarithmic derivatives of Artin L-functions. Specifically, we will show that conditional on a mild upper bound for the growth of certain counting functions for number fields (which is known to hold in many cases), the Colmez conjecture is true for 100% of CM fields. This is joint work with Riad Masri and Frank Thorne.

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.