On a Gross conjecture over imaginary quadratic fields

TL;DR

This paper proves a version of Gross's conjecture related to the leading terms of Artin L-series for certain abelian extensions over imaginary quadratic fields, specifically for primes split in the base field and not dividing 6.

Contribution

It establishes the validity of Gross's conjecture for all such extensions and primes, extending previous partial results in the area.

Findings

Gross conjecture holds for all abelian extensions over imaginary quadratic fields under specified conditions.

The proof applies to all rational primes split in the base field not dividing 6.

The result confirms the conjecture in a broad class of cases.

Abstract

Let $k$ be an imaginary quadratic number field, and $F/k$ a finite abelian extension of Galois group $G$. We show that a Gross conjecture concerning the leading terms of Artin $L$-series holds for $F/k$ and all rational primes which are split in $k$ and which do not divide $6$.