Let p and q be distinct prime numbers with \(q\equiv 1\pmod {12}\). Let N be a positive integer that is coprime to pq. We prove a formula relating the Hasse–Weil zeta function of the modular curve \(X_0(qN)_{\mathbb {F}_q}\) to the Ihara zeta function of the p-isogeny graph of supersingular elliptic curves defined over \(\overline{\mathbb {F}_q}\) equipped with a \(\Gamma _0(N)\)-level structure. When \(N=1\), this recovers a result of Sugiyama.

中文翻译：

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关于超奇异同源图和模曲线的zeta函数

令p和q是不同的质数\(q\equiv 1\pmod {12}\)。令N为与pq互质的正整数。我们证明了将模曲线\(X_0(qN)_{\mathbb {F}_q}\)的 Hasse–Weil zeta 函数与定义于\上的超奇异椭圆曲线p同源图的 Ihara zeta 函数联系起来的公式(\overline{\mathbb {F}_q}\)配备\(\Gamma _0(N)\)级结构。当\(N=1\)时，这恢复了Sugiyama的结果。