p-Curvature and Non-Abelian Cohomology
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912819651280896 |
|---|---|
| author | Lam, Yeuk Hay Joshua Litt, Daniel |
| author_facet | Lam, Yeuk Hay Joshua Litt, Daniel |
| contents | Let $X\to S$ be a smooth projective morphism. Katz proved the Grothendieck-Katz $p$-curvature conjecture for the Gauss-Manin connection on the $i$-th cohomology of $X/S$: if its $p$-curvature vanishes mod $p$ for infinitely many $p$, then the action of $π_1(S,s)$ on $H^i(X_s, \mathbb{Z})$ factors through a finite group. We prove a non-abelian analogue of this statement: if the $p$-curvature of the isomonodromy foliation on the moduli of flat bundles of rank $r$ on $X/S$ vanishes mod $p$ for infinitely many $p$, then the action of $π_1(S,s)$ on the rank $r$ integral characters of $π_1(X_s)$ factors through a finite group. We deduce many new cases of the Bost/Ekedahl--Shepherd-Barron--Taylor conjecture.
The proofs rely on a non-abelian version of Katz's formula, and a non-abelian version of the Hodge index theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_07933 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | p-Curvature and Non-Abelian Cohomology Lam, Yeuk Hay Joshua Litt, Daniel Algebraic Geometry Number Theory Let $X\to S$ be a smooth projective morphism. Katz proved the Grothendieck-Katz $p$-curvature conjecture for the Gauss-Manin connection on the $i$-th cohomology of $X/S$: if its $p$-curvature vanishes mod $p$ for infinitely many $p$, then the action of $π_1(S,s)$ on $H^i(X_s, \mathbb{Z})$ factors through a finite group. We prove a non-abelian analogue of this statement: if the $p$-curvature of the isomonodromy foliation on the moduli of flat bundles of rank $r$ on $X/S$ vanishes mod $p$ for infinitely many $p$, then the action of $π_1(S,s)$ on the rank $r$ integral characters of $π_1(X_s)$ factors through a finite group. We deduce many new cases of the Bost/Ekedahl--Shepherd-Barron--Taylor conjecture. The proofs rely on a non-abelian version of Katz's formula, and a non-abelian version of the Hodge index theorem. |
| title | p-Curvature and Non-Abelian Cohomology |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2601.07933 |