Symmetric correspondences with decomposable minimal equation

TL;DR

This paper investigates symmetric correspondences with decomposable minimal equations on smooth projective curves, demonstrating how they induce Jacobian decompositions and providing explicit examples with specific genus and isogeny components.

Contribution

It introduces new examples of curves with symmetric correspondences leading to Jacobian decompositions, expanding understanding of algebraic curve structures.

Findings

Constructed families of curves with prescribed genus and Jacobian decomposition.

Demonstrated minimal equations of degree ll+1 for these correspondences.

Showed the Jacobian decomposes into at least 2^ll isogeny components.

Abstract

We study symmetric correspondences with completely decomposable minimal equation on smooth projective curves $C$. The Jacobian of $C$ then decomposes correspondingly. For all positive integers $g$ and $\ell$, we give series of examples of smooth curves $C$ of genus $n^\ell (g-1) +1$ with correspondences satisfying minimal equations of degree $\ell+1$ such that the Jacobian of $C$ has at least $2^\ell$ isogeny components.