q-deformation with (φ, Γ) structure of the de Rham cohomology of the Legendre family of elliptic curves

2020/06/22 by Ryotaro Shirai, Shirai, Ryotaro
Mathematics · #14F30 #14F40 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2006.12310

openalex publication_date 2020/06/22 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

In the late '60s, B. Dwork studied a Frobenius structure compatible with the classical hypergeometric differential equation with parameters ((1)/(2),(1)/(2) ; 1 ) by analyzing behavior of solutions of the differential equation under Frobenius transformation. Recently, P. Scholze conjectured the existence of q-de Rham cohomology groups for any ℤ-scheme. In this paper, we give a Frobenius structure compatible with the q-hypergeometric differential equation with parameters (q^\frac12,q^\frac12;q) by showing a q-analogue of some results of Dwork. This construction gives a q-deformation with (φ,Γ)-structure over ℤ_p[[q-1]][[λ]] of the de Rham cohomology of the p-adic Legendre family of elliptic curves which has Frobenius structure and connection.

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