Doubly isogenous genus-2 curves with 𝐷₄-action

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation Pub Date : 2023-08-31 DOI:10.1090/mcom/3891

V. Arul, J. Booher, Steven R. Groen, Everett W. Howe, Wanlin Li, Vlad Matei, R. Pries, Caleb Springer

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Suppose <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C\">\n <mml:semantics>\n <mml:mi>C</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">C</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C prime\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>C</mml:mi>\n <mml:mo>′</mml:mo>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">C’</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are curves over a finite field <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-rational base points <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P\">\n <mml:semantics>\n <mml:mi>P</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">P</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper P prime\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>P</mml:mi>\n <mml:mo>′</mml:mo>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">P’</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, and let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\n <mml:semantics>\n <mml:mi>D</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D prime\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>D</mml:mi>\n <mml:mo>′</mml:mo>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">D’</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-<inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\n <mml:semantics>\n <mml:mn>2</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> maps on their Jacobians. We say that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper C comma upper P right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>C</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>P</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(C,P)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper C prime comma upper P prime right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>C</mml:mi>\n <mml:mo>′</mml:mo>\n </mml:msup>\n <mml:mo>,</mml:mo>\n <mml:msup>\n <mml:mi>P</mml:mi>\n <mml:mo>′</mml:mo>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(C’,P’)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are <italic>doubly isogenous</italic> if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper J a c left-parenthesis upper C right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>J</mml:mi>\n <mml:mi>a</mml:mi>\n <mml:mi>c</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>C</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Jac(C)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper J a c left-parenthesis upper C prime right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>J</mml:mi>\n <mml:mi>a</mml:mi>\n <mml:mi>c</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>C</mml:mi>\n <mml:mo>′</mml:mo>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Jac(C’)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are isogenous over <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper J a c left-parenthesis upper D right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>J</mml:mi>\n <mml:mi>a</mml:mi>\n <mml:mi>c</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>D</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Jac(D)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper J a c left-parenthesis upper D prime right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>J</mml:mi>\n <mml:mi>a</mml:mi>\n <mml:mi>c</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>D</mml:mi>\n <mml:mo>′</mml:mo>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">Jac(D’)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> are isogenous over <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. For curves of genus <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\n <mml:semantics>\n <mml:mn>2</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naïve heuristics predict, and we provide an explanation for this phenomenon.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics of Computation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/mcom/3891","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C C and C ′ C’ are curves over a finite field K K , with K K -rational base points P P and P ′ P’ , and let D D and D ′ D’ be the pullbacks (via the Abel–Jacobi map) of the multiplication-by- 2 2 maps on their Jacobians. We say that ( C , P ) (C,P) and ( C ′ , P ′ ) (C’,P’) are doubly isogenous if J a c ( C ) Jac(C) and J a c ( C ′ ) Jac(C’) are isogenous over K K and J a c ( D ) Jac(D) and J a c ( D ′ ) Jac(D’) are isogenous over K K . For curves of genus 2 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naïve heuristics predict, and we provide an explanation for this phenomenon.

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具有𝐷₄-作用的双重等均属-2曲线

我们研究了有限域上的曲线在多大程度上由它们的ζ函数和它们的某些覆盖的ζ功能表征。假设C C和C′C′是有限域K K上的曲线，具有K K-有理基点P P和P′P′，并且设D D和D′D′是乘-2 2映射在其雅可比上的回调（通过Abel–Jacobi映射）。如果J a C（C）Jac（C）和J a CK与J（D）Jac（D）和J（D′）Jac。对于自同构群包含八阶二面体群的亏格2 2的曲线，我们证明了双同构曲线对的数量大于天真启发式预测的数量，并对这一现象给出了解释。

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来源期刊

Mathematics of Computation 数学-应用数学

CiteScore

3.90

自引率

5.00%

发文量

审稿时长

7.0 months

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. Articles must be of significant computational interest and contain original and substantial mathematical analysis or development of computational methodology.