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On Transformation properties of hypergeometric motives and Diophantine equations

Ariel Pacetti

TL;DR

This work develops an arithmetic framework for hypergeometric transformation formulas by assigning hypergeometric motives to Gauss' ${}_2F_1$ and analyzing their monodromy, rigidity, and pullbacks. It provides criteria to decide when two such motives are isomorphic, connecting these criteria to finite hypergeometric sums and Jacobi sums, and proves arithmetic analogues of classical transformations via explicit covers of the projective line. Through degree-2 maps $\\pi_1$ and $\\pi_2$, it relates exponent triples in Diophantine problems (à la Darmon–Frey) to hypergeometric motives, offering a pathway to derive Diophantine consequences from motivic transformations. Overall, the paper bridges classical hypergeometric identities with arithmetic geometry, enabling new tools to investigate Diophantine equations through motivic transformations and covers.

Abstract

Over the last two hundred years different transformation formulas for Gauss' hypergeometric function ${}_2F_1$ were discovered. The goal of the present article is to study their arithmetic analogue for the underlying hypergeometric motive. As an application, we show how these transformation properties can be used in the study of some Diophantine equations.

On Transformation properties of hypergeometric motives and Diophantine equations

TL;DR

This work develops an arithmetic framework for hypergeometric transformation formulas by assigning hypergeometric motives to Gauss' and analyzing their monodromy, rigidity, and pullbacks. It provides criteria to decide when two such motives are isomorphic, connecting these criteria to finite hypergeometric sums and Jacobi sums, and proves arithmetic analogues of classical transformations via explicit covers of the projective line. Through degree-2 maps and , it relates exponent triples in Diophantine problems (à la Darmon–Frey) to hypergeometric motives, offering a pathway to derive Diophantine consequences from motivic transformations. Overall, the paper bridges classical hypergeometric identities with arithmetic geometry, enabling new tools to investigate Diophantine equations through motivic transformations and covers.

Abstract

Over the last two hundred years different transformation formulas for Gauss' hypergeometric function were discovered. The goal of the present article is to study their arithmetic analogue for the underlying hypergeometric motive. As an application, we show how these transformation properties can be used in the study of some Diophantine equations.

Paper Structure

This paper contains 9 sections, 17 theorems, 77 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Hypergeometric motives
  3. Finite hypergeometric sums
  4. Special values
  5. Kummer's $S_4$ transformations
  6. Hypergeometric relations and covers
  7. Applications to Diophantine equations
  8. The map $\pi_1$:
  9. The map $\pi_2$:

Key Result

Proposition 2.1

Let $H$ be a normal subgroup of $G$ and let $\rho:H \to {\rm GL}_n(L)$ be an irreducible representation of $H$. If the representation extends to an $n$-dimensional representation of $G$, then the extension is unique up to a twist by a character of $G/H$.

Theorems & Definitions (49)

  • Example 1
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1: continued
  • Example 2
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 39 more