On $p$-adic solutions to KZ equations, ordinary crystals, and $p^s$-hypergeometric solutions
Abstract
We consider the KZ connection associated with a family of hyperelliptic curves of genus $g$ over the ring of $p$-adic integers $\mathbb{Z}_p$. Then the dual connection is the Gauss-Manin connection of that family. We observe that the Gauss-Manin connection has an ordinary $F$-crystal structure and its unit root subcrystal is of rank $g$. We prove that all local flat sections of the KZ connection annihilate the unite root subcrystal, and the space of all local flat sections of the KZ connection is a free $\mathbb{Z}_p$-module of rank $g$. We also consider the reduction modulo $p^s$ of the unit root subcrystal for any $s\geq 1$. We prove that its annihilator is generated by the so-called $p^s$-hypergeometric flat sections of the KZ connection. In particular, that means that the reduction modulo $p^s$ of an arbitrary local flat section of the KZ connection over $\mathbb{Z}_p$ is a linear combination of the $p^s$-hypergeometric flat sections.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.19318
- arXiv:
- arXiv:2406.19318
- Bibcode:
- 2024arXiv240619318V
- Keywords:
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- Mathematics - Number Theory;
- Mathematical Physics;
- Mathematics - Algebraic Geometry
- E-Print:
- 12 pages, references updated, a remark added