# On the p-adic Stark conjecture at s=1 and applications

## TL;DR

This paper proves the p-adic Stark conjecture at s=1 for abelian extensions, relates it to Leopoldt's conjecture, and demonstrates its implications for the equivariant Tamagawa number conjecture and related conjectures.

## Contribution

It establishes the p-adic Stark conjecture at s=1 unconditionally for abelian extensions and links it to Leopoldt's conjecture and the ETNC, providing new evidence for these conjectures.

## Key findings

- Proved the p-adic Stark conjecture at s=1 for abelian extensions.
- Showed the conjecture is implied by Leopoldt's conjecture in certain cases.
- Established a descent theorem for the ETNC at s=1.

## Abstract

Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at s=1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F. We prove this conjecture unconditionally when E/Q is abelian. We also show that for certain non-abelian extensions E/F the p-adic Stark conjecture at s=1 is implied by Leopoldt's conjecture for E at p. Moreover, we prove that for a fixed prime p, the p-adic Stark conjecture at s=1 for E/F implies Stark's conjecture at s=1 for E/F. This leads to a `prime-by-prime' descent theorem for the `equivariant Tamagawa number conjecture' (ETNC) for Tate motives at s=1. As an application of these results, we provide strong new evidence for special cases of the ETNC for Tate motives and the closely related `leading term conjectures' at s=0 and s=1.