Class number formulas via 2-isogenies of elliptic curves

TL;DR

This paper establishes new formulas linking class numbers of quadratic imaginary fields to character sums derived from 2-isogenies of elliptic curves, extending classical Dirichlet results.

Contribution

It introduces novel relations between class numbers and elliptic curve isogenies, providing a new perspective on class number formulas.

Findings

Derived explicit formulas connecting class numbers and elliptic curve isogenies.

Extended classical Dirichlet results to elliptic curve contexts.

Provided computational evidence supporting the new relations.

Abstract

A classical result of Dirichlet shows that certain elementary character sums compute class numbers of quadratic imaginary number fields. We obtain analogous relations between class numbers and a weighted character sum associated to a 2-isogeny of elliptic curves.