Central $L$ values of congruent number elliptic curves

TL;DR

This paper expresses the central L-values of certain congruent number elliptic curves as squares of CM values of theta functions, extending classical formulas and aiding both theoretical understanding and computational methods.

Contribution

It generalizes classical formulas of Gauss by representing L-values of congruent number elliptic curves as squares of theta function CM values.

Findings

L(E_n,1) expressed as square of theta function CM values

Generalization of Gauss's classical formulas

Implications for theory and computation of elliptic curves

Abstract

Let $E_n$ be the congruent number elliptic curve $y^2=x^3-n^2x$, where $n$ is square-free and not divisible by primes $p\equiv 3\pmod 4$. In this paper, we prove that $L(E_n,1)$ can be expressed as the square of CM values of some simple theta functions, generalizing two classical formulas of Gauss. Our result is meaningful in both theory and practical computation.