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arXiv · 2412.18181

Elliptic curves over a finite field with a specified subgroup and the trace formula

Tadahiro Katsuoka
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Abstract

Ihara and Birch obtained a formula expressing the sum of powers of the traces of elliptic curves over a fixed finite field of characteristic $p$ in terms of the traces of Hecke operators for $\mathrm{SL}_2(\mathbb{Z})$. Generalizing the theorems of Ihara and Birch, for a finite abelian group $A$ whose order is coprime to $p$, Kaplan and Petrow gave a formula for statistical description of powers of the traces of elliptic curves which contain subgroups isomorphic to $A$. In this paper, we generalize the theorems of Ihara, Birch, and Kaplan--Petrow to the case where the order of $A$ is divisible by $p$.

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Tadahiro Katsuoka. 2024-12-24. Elliptic curves over a finite field with a specified subgroup and the trace formula. https://arxiv.org/abs/2412.18181

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