Class Numbers via 3-Isogenies and Elliptic Surfaces

TL;DR

This paper establishes a connection between character sums from 3-isogenies on elliptic surfaces over finite fields and the class number of imaginary quadratic fields, extending known results from 2-isogenies.

Contribution

It introduces a higher-dimensional analog linking character sums of 3-isogenies to class numbers, expanding the scope of existing class number formulas.

Findings

Character sum related to 3-isogenies equals class number of ()

Extends 2-isogeny class number formulas to 3-isogenies

Provides new insights into elliptic surfaces over finite fields

Abstract

We show that a character sum attached to a family of 3-isogenies defined on the fibers of a certain elliptic surface over $\mathbb{F}_p$ relates to the class number of the quadratic imaginary number field $\Q(\sqrt{-p})$. In this sense, this provides a higher-dimensional analog of some recent class number formulas associated to 2-isogenies of elliptic curves.