p-adic monodromy and mod p unlikely intersections, II

2025/11/30 by Ruofan Jiang, Jiang, Ruofan
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Advanced Mathematical Identities #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2512.00687

Abstract

We study ordinary abelian schemes in characteristic p and their moduli spaces from the perspective of char p Mumford--Tate, log Ax--Lindemann, and geometric André--Oort conjectures (abbreviated as \MTT_p, logAL_p and geoAO_p). In this paper, we achieve multiple goals: (A) establish the implication MT_p⇔ logAL_p ⇒ geoAO_p, and show that they all follow from the Tate conjecture for abelian varieties. The equivalence MT_p⇔ logAL_p is exploited from both sides, which enables us to \noindent(B) develop a representation theory approach to logAL_p and geoAO_p by first establishing many cases of MT_p via classical techniques, and (C) develop an algebraization approach to \MTT_p that transcends the limitation of classical methods. In particular, we introduce ``crystalline Hodge loci'', a rigid analytic geometric object that encodes the essential information needed for proving logAL_p, while being very approachable via (integral and relative) p-adic Hodge theory. This enables us to prove logAL_p for compact Tate-linear curves with unramified p-adic monodromy. As an application, we establish \MTT_p for many abelian fourfolds of p-adic Mumford type.

Citations

• Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds
• Splitting of almost ordinary abelian surfaces in families and the S-integrality conjectures
• Reductions of abelian surfaces over global function fields
• The Mumford--Tate conjecture for products of abelian varieties
• Altered local uniformization of Berkovich spaces

Related

• p-adic monodromy and mod p unlikely intersections, I
• A survey around the Hodge, Tate and Mumford-Tate conjectures for abelian varieties
• The Mumford--Tate conjecture for products of abelian varieties
• The p-adic monodromy group of abelian varieties over global function fields of characteristic p
• On the Mumford-Tate conjecture for abelian varieties with reduction conditions