Central $L$ values of congruent number elliptic curves

Abstract

Let $E_n$ be the congruent number elliptic curve $y^2=x^3-n^{2}x$, where $n$ is square-free and not divisible by primes $p\equiv 3(\mathrm{mod}4)$. 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.

19 pages