L-value criterion for congruent numbers

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A positive rational number $q$ is congruent if there are positive rational numbers $a,b,c$ with $a^2+b^2=c^2$ and $ab/2=q$; thus $q$ is the area of a right triangle with rational side lengths. For $n\in\mathbb N$ let $E_n/\mathbb Q$ be the elliptic curve $y^2=x^3-n^2x$, including its point at infinity. Its Hasse--Weil $L$-function is defined initially for $\operatorname{Re}s>3/2$ by \[ L(E_n,s)=\prod_{p\text{ good}}(1-a_pp^{-s}+p^{1-2s})^{-1} \prod_{p\text{ bad}}(1-a_pp^{-s})^{-1}. \] Here $p$ ranges over primes, good or bad according to the minimal model of $E_n$; at a good prime $a_p=p+1-\#E_n(\mathbb F_p)$, and at a bad prime $a_p$ is $1$, $-1$ or $0$ according as reduction is split multiplicative, nonsplit multiplicative or additive. The value $L(E_n,1)$ is taken using analytic continuation. Omitting finitely many Euler factors as in the slides does not change whether this value is zero, since those factors are finite and nonzero at $s=1$.

Statement recorded in the supplied source.

For every $n\in\mathbb N$, is $n$ congruent if and only if $L(E_n,1)=0$? [S2]

Short English statement

For each congruent-number elliptic curve, does vanishing of its $L$-function at one exactly detect the existence of a rational right triangle of area $n$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8900010 (AMR-088-0010). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Christian Wuthrich, Open problems in number theory, lecture slides (December 2011) slides headed Conjecture (part of Birch and Swinnerton-Dyer conjecture). https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf.