[Paper Review] Isogenous decomposition of the Jacobian of generalized Fermat curves

Mariela Carvacho, Rub ́ En A. Hidalgo+1 more|arXiv (Cornell University)|Jul 10, 2015

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper provides an isogenous decomposition of the Jacobian variety of generalized Fermat curves of type $(p,n)$, where $p$ is prime, into products of Jacobians of cyclic $p$-gonal curves. Using group actions and algebraic geometry techniques, the authors derive explicit equations for these $p$-gonal curves and show that the Jacobian is isogenous to a product of elliptic curves or higher-dimensional Jacobians, offering new families of Riemann surfaces with such decompositions.

ABSTRACT

A closed Riemann surface $S$ is called a generalized Fermat curve of type $(p,n)$, where $p,n \geq 2$ are integers, if it admits a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms so that $S/H$ is an orbifold of genus zero with exactly $n+1$ cone points, each one of order $p$. It is known that $S$ is a fiber product of $(n-1)$ classical Fermat curves of degree $p$ and, for $(p-1)(n-1)>2$, that it is a non-hyperelliptic Riemann surface. In this paper, assuming $p$ to be a prime integer, we provide a decomposition, up to isogeny, of the Jacobian variety $JS$ as a product of Jacobian varieties of certain cyclic $p$-gonal curves. Explicit equations for these $p$-gonal curves are provided in terms of the equations for $S$. As a consequence of this decomposition, we are able to provide explicit positive-dimensional families of closed Riemann surfaces whose Jacobian variety is isogenous to the product of elliptic curves.

Motivation & Objective

• To determine whether the Jacobian variety of a generalized Fermat curve admits an isogenous decomposition into lower-genus Jacobians.
• To provide explicit equations for the $p$-gonal curves whose Jacobians appear in the decomposition when $p$ is prime.
• To construct positive-dimensional families of Riemann surfaces whose Jacobians are isogenous to products of elliptic curves, addressing open questions on such decompositions.
• To extend known results on hyperelliptic and Fermat curves to a broader class of non-hyperelliptic Riemann surfaces with abelian automorphism groups.

Proposed method

• The authors use the group action of $H riangleleft ext{Aut}(S)$, where $H o ext{Aut}(S)$ is isomorphic to $\mathbb{Z}_p^n$, to analyze the quotient $S/H$ and its orbifold structure.
• They apply the Kani-Rosen theorem on decompositions of Jacobians of curves with group actions to derive isogeny decompositions of $JS$.
• For each subgroup $H_i$ of $H$, the quotient $S/H_i$ is analyzed to determine its genus and signature, leading to the identification of associated $p$-gonal curves.
• Explicit algebraic equations for the $p$-gonal curves are constructed from the defining equations of the generalized Fermat curve via monodromy and group action considerations.
• The method relies on the classification of $r$-tuples of branch points and their associated monodromy data to define irreducible components of the associated algebraic curves $E_{(a,\alpha)}$.
• The decomposition is validated using the theory of principally polarized abelian varieties and Poincaré’s complete reducibility theorem.

Experimental results

Research questions

• RQ1 Can the Jacobian of a generalized Fermat curve of type $(p,n)$ be decomposed up to isogeny into products of Jacobians of lower-genus curves when $p$ is prime?
• RQ2 What are the explicit equations of the $p$-gonal curves whose Jacobians appear in such a decomposition?
• RQ3 Are there positive-dimensional families of Riemann surfaces whose Jacobians are isogenous to products of elliptic curves, beyond known hyperelliptic cases?
• RQ4 How does the automorphism group structure of the generalized Fermat curve influence the isogenous decomposition of its Jacobian?

Key findings

• For $p=2$, the Jacobian of a genus 5 Riemann surface with $\mathbb{Z}_2^4$ action is isogenous to a product of 4 elliptic curves, forming a two-dimensional family.
• A one-dimensional family of genus 17 Riemann surfaces with $\mathbb{Z}_2^5$ action has Jacobians isogenous to products of elliptic curves.
• A genus 49 Riemann surface with $\mathbb{Z}_2^6$ action has a Jacobian isogenous to a product of 24 elliptic curves.
• For $p=3$, a one-dimensional family of genus 10 Riemann surfaces with $\mathbb{Z}_3^3$ action and quotient of genus zero with three cone points of order 3 has Jacobian isogenous to a product of 5 elliptic curves and one 5-dimensional Jacobian.
• For $p=3$, a two-dimensional family of genus 65 Riemann surfaces with $\mathbb{Z}_3^4$ action and quotient of genus zero with five cone points of order 3 has Jacobian isogenous to a product of 40 elliptic curves and five 5-dimensional Jacobians.
• The Jacobian of the classical Fermat curve $F_8$ of genus 21 is isogenous to $E_2^{21}$, where $E_2: v^2 = u^4 + 1$, confirming a conjecture in the literature.