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

TL;DR

The paper develops and analyzes the toric analogue of Grothendieck’s section conjecture by introducing the toric fundamental gerbe and its Galois sections. It proves the toric section conjecture for abelian varieties over $p$-adic fields and provides strong evidence for hyperbolic curves over such fields, showing how toric data can mirror classical section phenomena while being more tractable in key cases. A central theme is the relationship between toric and étale data, mediated by Brauer obstructions and Picard-theoretic dualities, with global consequences for Selmer groups and number-field questions. The work also develops a robust Tannakian and gerbe-theoretic framework (fundamental gerbes, pre-cactuses, and cactuses) to reinterpret toric Galois sections as quasi-fibre functors, linking geometric objects to representation-theoretic structures and enabling new approaches to long-standing conjectures.

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 étale cover. It is an extension of the étale fundamental group scheme by a projective limit of tori. Grothendieck's section conjecture for the étale 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.

Theorems & Definitions (109)

• Conjecture 1.1: Grothendieck
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Conjecture 1.5
• Proposition 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Definition 2.1: borne-vistoli