Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups; the CM method

TL;DR

This paper proves torsion congruences between abelian p-adic L-functions related to automorphic representations of definite unitary groups, advancing non-commutative Iwasawa theory and its applications to motives like CM elliptic curves.

Contribution

It establishes torsion congruences for general definite unitary groups and provides explicit results for cases n=1 and n=2, with implications for CM motives.

Findings

Proved torsion congruences for abelian p-adic L-functions in definite unitary groups.

Derived explicit results for n=1 and n=2 cases.

Explored implications for motives such as CM elliptic curves.

Abstract

In this work we prove the so-called "torsion congruences" between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results in the special cases of $n=1$ and $n=2$. In both of these cases we also explain their implications for some particular "motives", as for example elliptic curves with complex multiplication.