Congruences of algebraic $L$-functions of motives

TL;DR

This paper develops a framework for studying congruences between algebraic parts of special $L$-function values of motives, establishing interpolation properties and formulating refined conjectures linking special values, cohomology, and $p$-adic families.

Contribution

It introduces a new framework for analyzing congruences of algebraic $L$-values, including interpolation properties and refined conjectures on the Equivariant Tamagawa Number Conjecture.

Findings

Algebraic local Euler factors satisfy interpolation in $p$-adic families.

Existence of algebraic $p$-adic $L$-functions in broad $p$-adic families.

Refined conjectures linking special values, cohomology, and Hecke algebras.

Abstract

We develop a framework to investigate conjectures on congruences between the algebraic part of special values of $L$-functions of congruent motives. We show that algebraic local Euler factors satisfy precise interpolation properties in $p$-adic families of motives and that algebraic $p$-adic $L$-functions exist in quite large generality for $p$-adic families of automorphic motives. We formulate two conjectures refining (and correcting) the currently existing formulation of the Equivariant Tamagawa Number Conjecture with coefficients in Hecke algebras and pointing out the links between conjectures on special values and completed cohomology.