# Hypergeometric Properties of Genus 3 Generalized Legendre Curves

## TL;DR

This paper explores the connection between period integrals and Frobenius traces of genus 3 generalized Legendre curves, demonstrating their computation via hypergeometric functions over classical and finite fields, extending known phenomena from elliptic curves.

## Contribution

It introduces a novel approach linking period integrals and Frobenius traces for genus 3 curves using hypergeometric functions, expanding the understanding of such relationships beyond elliptic curves.

## Key findings

- Both properties are computable via hypergeometric functions.
- The phenomenon observed in elliptic curves extends to genus 3 curves.
- Provides a new computational framework for genus 3 curves.

## Abstract

Inspired by a result of Manin, we study the relationship between certain period integrals and the trace of Frobenius of genus 3 generalized Legendre curves. We show that both of these properties can be computed in terms of "matching" classical and finite field hypergeometric functions, a phenomenon that has also been observed in elliptic curves and many higher dimensional varieties.