Elliptic curves over a finite field with a specified subgroup and the trace formula
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909509139562496 |
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| author | Katsuoka, Tadahiro |
| author_facet | Katsuoka, Tadahiro |
| contents | 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 $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elliptic curves over a finite field with a specified subgroup and the trace formula Katsuoka, Tadahiro Number Theory 11G20, 11F72, 14G15, 14H52 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 $p$. |
| title | Elliptic curves over a finite field with a specified subgroup and the trace formula |
| topic | Number Theory 11G20, 11F72, 14G15, 14H52 |
| url | https://arxiv.org/abs/2412.18181 |