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arXiv · 2505.13133

Central $L$ values of congruent number elliptic curves

Xuejun Guo·Dongxi Ye·Hongbo Yin
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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.

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Xuejun Guo, Dongxi Ye, Hongbo Yin. 2025-05-19. Central $L$ values of congruent number elliptic curves. https://arxiv.org/abs/2505.13133

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