$p$-adic properties of motivic fundamental lines (Kato's conjecture is (probably) false for (not so) trivial reasons)

TL;DR

This paper investigates the compatibility of $p$-adic fundamental lines with specializations in $p$-adic families of Galois representations, confirming some conjectures while highlighting limitations and suggesting modifications for broader cases.

Contribution

It proves the compatibility of $p$-adic fundamental lines with certain specializations and identifies cases where compatibility fails, proposing adjustments to existing conjectures.

Findings

Compatibility holds for specific $p$-adic families of Galois representations.

Fundamental lines are not compatible with arbitrary characteristic zero specializations.

Highlights the need to modify conjectures using completed cohomology.

Abstract

We prove the conjectured compatibility of $p$-adic fundamental lines with specializations at motivic points for a wide class of $p$-adic families of $p$-adic Galois representations (for instance, the families which arise from $p$-adic families of automorphic representations of the unit group of a quaternion algebra or of a totally definite unitary groups) and deduce the compatibility of the Equivariant Tamagawa Number Conjectures for them. However, we also show that fundamental lines are not compatible with arbitrary characteristic zero specializations with values in a domain in general. This points to the need to modify some conjectures on $p$-adic variation of special values using completed cohomology.