What question did this study set out to answer?

This research aims to express the Frobenius action on de Rham cohomology using p-adic gamma values in hypergeometric contexts.

April 1, 2026Open Access

An analogue of Ogus’s theorem for certain hypergeometric curves

Key Points

This research aims to express the Frobenius action on de Rham cohomology using p-adic gamma values in hypergeometric contexts.
Analyzed Frobenius action on de Rham cohomology for hypergeometric curves
Utilized a transformation formula for hypergeometric functions
Reduced cases to CM elliptic curves where known results apply.
Provided new examples of p-adic gamma values related to Frobenius actions
Confirmed the applicability of expressions known from specific cases, like Fermat curves and CM elliptic curves.

Abstract

For a family whose Picard–Fuchs equation is given by a certain hypergeometric differential equation, we express the Frobenius action on the de Rham cohomology as products of p-adic gamma values. Such an expression is known in specific cases, namely Fermat curves and CM elliptic curves, but remains unknown in general. The main result of this paper provides new examples. The idea of the proof is to reduce the statement to the case of CM elliptic curves by using a transformation formula for hypergeometric functions.

An analogue of Ogus’s theorem for certain hypergeometric curves

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Abstract

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