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Decomposition of jacobians of generalized Fermat curves

Gary Martinez-Nunez

TL;DR

This paper develops a group-action–driven decomposition of Jacobians for generalized Fermat curves $X_{(n,p)}$ with $p$ prime, extending Humbert–Edge results and the group-algebra framework to a broad recursive family. By analyzing the $E_{(n,p)}$-action, it derives an explicit isogeny decomposition of $J(X_{(n,p)})$ into Jacobians of étale quotients $X_{T}/H$, with factor dimensions $ rac{(n-|T|-1)(p-1)}{2}$ and multiplicities governed by combinatorial counts ${n+1\choose |T|}\dfrac{(p-1)^{n-|T|}-(-1)^{n-|T|}}{p}$. It shows that for $p\ge 5$ none of the factors are Prym–Tyurin varieties, and provides a lower bound on the number of elliptic components in the $(n,3)$ case, linking to Ekedahl–Serre questions. The results unify and extend earlier decompositions for Humbert–Edge curves and offer explicit factor-counting for small $n$, contributing to the understanding of completely decomposable Jacobians in families of curves with large automorphism groups.

Abstract

We give a decomposition of the jacobian variety of a generalized Fermat curve. This extends a result obtained by Auffarth, Lucchini-Arteche and Rojas on Humbert-Edge curves, which are a particular case of generalized Fermat curves. (A counting on the number of factor has been added)

Decomposition of jacobians of generalized Fermat curves

TL;DR

This paper develops a group-action–driven decomposition of Jacobians for generalized Fermat curves with prime, extending Humbert–Edge results and the group-algebra framework to a broad recursive family. By analyzing the -action, it derives an explicit isogeny decomposition of into Jacobians of étale quotients , with factor dimensions and multiplicities governed by combinatorial counts . It shows that for none of the factors are Prym–Tyurin varieties, and provides a lower bound on the number of elliptic components in the case, linking to Ekedahl–Serre questions. The results unify and extend earlier decompositions for Humbert–Edge curves and offer explicit factor-counting for small , contributing to the understanding of completely decomposable Jacobians in families of curves with large automorphism groups.

Abstract

We give a decomposition of the jacobian variety of a generalized Fermat curve. This extends a result obtained by Auffarth, Lucchini-Arteche and Rojas on Humbert-Edge curves, which are a particular case of generalized Fermat curves. (A counting on the number of factor has been added)
Paper Structure (12 sections, 34 theorems, 78 equations)

This paper contains 12 sections, 34 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. jacobian varieties
  4. Generalized Fermat curves
  5. A decomposition of jacobians of generalized Fermat curves
  6. A decomposition of the jacobian of generalized Fermat curves of type $(n,k)$
  7. Decomposition by jacobians of étale quotients
  8. About Humbert-Edge curves
  9. On the Prym-Tyurin property
  10. On the Ekedahl-Serre questions
  11. On the number of factors of each dimension for small $n$
  12. Counting elliptic curves in the decomposition

Key Result

Theorem 1.1

ALAR21 Let $X_{n}$ be a Humbert-Edge curve of type $n\geq 3$ and genus $g_{n}$. Then we have the following decomposition of $J(X_{n})$: where $\pi_{T}:X_{n} \to X_{T}$ is the natural projection and $\varphi$ is an isogeny. Moreover,

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 48 more