Finding All Solutions of qKZ Equations in Characteristic $p$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908742393528320 |
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| author | Mukhin, Evgeny Varchenko, Alexander |
| author_facet | Mukhin, Evgeny Varchenko, Alexander |
| contents | In [J. Lond. Math. Soc. 109 (2024), e12884, 22 pages, arXiv:2208.09721], the difference qKZ equations were considered modulo a prime number $p$ and a family of polynomial solutions of the qKZ equations modulo $p$ was constructed by an elementary procedure as suitable $p$-approximations of the hypergeometric integrals. In this paper, we study in detail the first family of nontrivial examples of the qKZ equations in characteristic $p$. We describe all solutions of these qKZ equations in characteristic $p$ by demonstrating that they all stem from the $p$-hypergeometric solutions. We also prove a Lagrangian property (called the orthogonality property) of the subbundle of the qKZ bundle spanned by the $p$-hypergeometric sections. This paper extends the results of [arXiv:2405.05159] on the differential KZ equations to the difference qKZ equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_06326 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finding All Solutions of qKZ Equations in Characteristic $p$ Mukhin, Evgeny Varchenko, Alexander Mathematical Physics Algebraic Geometry Number Theory In [J. Lond. Math. Soc. 109 (2024), e12884, 22 pages, arXiv:2208.09721], the difference qKZ equations were considered modulo a prime number $p$ and a family of polynomial solutions of the qKZ equations modulo $p$ was constructed by an elementary procedure as suitable $p$-approximations of the hypergeometric integrals. In this paper, we study in detail the first family of nontrivial examples of the qKZ equations in characteristic $p$. We describe all solutions of these qKZ equations in characteristic $p$ by demonstrating that they all stem from the $p$-hypergeometric solutions. We also prove a Lagrangian property (called the orthogonality property) of the subbundle of the qKZ bundle spanned by the $p$-hypergeometric sections. This paper extends the results of [arXiv:2405.05159] on the differential KZ equations to the difference qKZ equations. |
| title | Finding All Solutions of qKZ Equations in Characteristic $p$ |
| topic | Mathematical Physics Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2503.06326 |