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Central $L$ values of congruent number elliptic curves

Xuejun Guo, Dongxi Ye, Hongbo Yin

TL;DR

This work expresses the central values $L(E_n,1)$ of congruent number elliptic curves $E_n$ as squares of CM values of theta functions, extending Gauss’s classical formulas to all square-free $n$ with $p\equiv3\pmod{4}$ avoidance. The authors build a CM framework using a Hecke character over $K=\mathbb{Q}(i)$, analyze associated ring class fields, and rewrite $L(E_n,1)$ as a CM trace of theta-values via Eisenstein series. A key factorization lemma then decomposes these CM theta values into unary-theta data, enabling explicit square formulas for $L(E_n,1)$ and giving practical avenues for computing congruent-number statuses and Tate–Shafarevich group sizes. The paper also discusses mock Heegner zeros of theta functions, linking zeros to vanishing of $L(E_n,1)$ and presenting computational criteria based on congruence classes mod $8$ and CM symmetries. Overall, the results yield both theoretical insight into CM values and concrete, computable criteria for congruent-number questions and related arithmetic invariants.

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.

Central $L$ values of congruent number elliptic curves

TL;DR

This work expresses the central values of congruent number elliptic curves as squares of CM values of theta functions, extending Gauss’s classical formulas to all square-free with avoidance. The authors build a CM framework using a Hecke character over , analyze associated ring class fields, and rewrite as a CM trace of theta-values via Eisenstein series. A key factorization lemma then decomposes these CM theta values into unary-theta data, enabling explicit square formulas for and giving practical avenues for computing congruent-number statuses and Tate–Shafarevich group sizes. The paper also discusses mock Heegner zeros of theta functions, linking zeros to vanishing of and presenting computational criteria based on congruence classes mod and CM symmetries. Overall, the results yield both theoretical insight into CM values and concrete, computable criteria for congruent-number questions and related arithmetic invariants.

Abstract

Let be the congruent number elliptic curve , where is square-free and not divisible by primes . In this paper, we prove that 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.
Paper Structure (7 sections, 27 theorems, 94 equations)

This paper contains 7 sections, 27 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. The Hecke character $\tilde{\chi}_{n}$
  3. The class fields $H_{\mathfrak{m}}$ and $H_{4n}$
  4. Eisenstein series and the theta functions
  5. Factorization lemma
  6. $L(E_{n},1)$ as a square
  7. Mock Heegner zeros of theta functions

Key Result

Theorem 1.1

Let $m=\prod_{i=1}^s p_i$ with each prime $p_i\equiv1\pmod{4}$, $b$ an even integer such that $b^2\equiv -1\pmod{m^2}$, $\tau_m=\frac{b+i}{2 m^2}$, $\tau'_{m}=\frac{b+m^2+i}{2 m^2}$. Then In particular, $L(E_n,1)=0$ if and only if $\theta_{\chi_{n}}\left(\tau_n\right)=0$ when $n$ is odd, or $\theta_{\chi_{n/2}}(\tau'_{n/2})=0$ when $n$ is even.

Theorems & Definitions (50)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • ...and 40 more