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<p>Ihara and Birch obtained a formula expressing the sum of powers of the traces of elliptic curves over a fixed finite field of characteristic <i>p</i> in terms of the traces of Hecke operators for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{SL}_2(\mathbb {Z})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Generalizing the theorems of Ihara and Birch, for a finite abelian group <i>A</i> whose order is coprime to <i>p</i>, Kaplan and Petrow gave a formula for statistical description of powers of the traces of elliptic curves which contain subgroups isomorphic to <i>A</i>. In this paper, we generalize the theorems of Ihara, Birch, and Kaplan–Petrow to the case where the order of <i>A</i> is divisible by <i>p</i>.</p>

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

  • 期刊论文

  • 发表日期：

    2025年12月17日

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  • Tadahiro Katsuoka

摘要

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 \(\textrm{SL}_2(\mathbb {Z})\) SL 2 ( 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.