Abstract
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.
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Notes
Note that the left-hand side of the isomorphism does not depend on the prime p. On the right-hand side, while the prime p appears in the notation, it is in fact independent of p. This is because the divisor group is defined in terms of the set of vertices of the graph \(X_p^q(N)\), which is independent of p. The prime p is only relevant when we define the edges of the graph.
References
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Acknowledgements
We thank the anonymous referee for very helpful comments and suggestions. The authors’ research is supported by the NSERC Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096.
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Lei, A., Müller, K. On the zeta functions of supersingular isogeny graphs and modular curves. Arch. Math. 122, 285–294 (2024). https://doi.org/10.1007/s00013-023-01937-z
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DOI: https://doi.org/10.1007/s00013-023-01937-z