The section conjecture for the toric fundamental group over $p$-adic fields

TL;DR

This paper explores the toric section conjecture over p-adic fields, showing its implications for the original conjecture, and proves it for abelian varieties, providing evidence for hyperbolic curves.

Contribution

It establishes the toric section conjecture for abelian varieties over p-adic fields and links its resolution to the classical section conjecture, offering new insights and reduction strategies.

Findings

Proves the toric section conjecture for abelian varieties over p-adic fields.

Shows that resolving the toric section conjecture simplifies the original conjecture.

Provides strong evidence for the conjecture's validity for hyperbolic curves over p-adic fields.

Abstract

The toric fundamental group is the Tannaka dual of a category of vector bundles which become direct sums of line bundles on a finite \'etale cover. It is an extension of the \'etale fundamental group scheme by a projective limit of tori. Grothendieck's section conjecture for the \'etale fundamental group implies the analogous statement for the toric fundamental group. We call this the toric section conjecture. We prove that a resolution of the toric section conjecture would reduce the original one to particular cases about which more is known, mainly due to J. Stix. We prove that abelian varieties over $p$-adic fields satisfy the toric section conjecture, and give strong evidence that it holds for hyperbolic curves over $p$-adic fields, too.