Abelian varieties over finite fields and their groups of rational points

IF 0.9 1区 数学 Q2 MATHEMATICS
Stefano Marseglia, Caleb Springer
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First, when <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> End</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math> is locally Gorenstein, we show that the group structure of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math> is determined by <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> End</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math>. In fact, the same conclusion is attained if <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> End</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math> has local Cohen–Macaulay type at most <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>, under the additional assumption that <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> is ordinary or <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>q</mi></math> is prime, although the conclusion is not true in general. Second, the description in the Gorenstein case is used to characterize cyclic isogeny classes in terms of conductor ideals. Third, going in the opposite direction, we characterize squarefree isogeny classes of abelian varieties with <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi></math> rational points in which every abelian group of order <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>N</mi></math> is realized as a group of rational points. Finally, we study when an abelian variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> over <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub></math> and its dual <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>A</mi></mrow><mrow><mo>∨</mo></mrow></msup></math> satisfy or fail to satisfy several interrelated properties, namely <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi><mi>≅</mi><mo> ⁡<!--FUNCTION APPLICATION--></mo><msup><mrow><mi>A</mi></mrow><mrow><mo>∨</mo></mrow></msup></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo><mi>≅</mi><mo> ⁡<!--FUNCTION APPLICATION--></mo><msup><mrow><mi>A</mi></mrow><mrow><mo>∨</mo></mrow></msup><mo stretchy=\"false\">(</mo><msub><mrow><mi mathvariant=\"double-struck\">𝔽</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy=\"false\">)</mo></math>, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> End</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo>\n<mo>=</mo><mi> End</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>∨</mo></mrow></msup><mo stretchy=\"false\">)</mo></math>. In the process, we exhibit a sufficient condition for <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi>\n<mo>≇</mo> <msup><mrow><mi>A</mi></mrow><mrow><mo>∨</mo></mrow></msup></math> involving the local Cohen–Macaulay type of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> End</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo stretchy=\"false\">(</mo><mi>A</mi><mo stretchy=\"false\">)</mo></math>. In particular, such an abelian variety <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>A</mi></math> is not a Jacobian, or even principally polarizable. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"31 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra & Number Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/ant.2025.19.521","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

Over a finite field 𝔽q, abelian varieties with commutative endomorphism rings can be described by using modules over orders in étale algebras. By exploiting this connection, we produce four theorems regarding groups of rational points and self-duality, along with explicit examples. First, when End (A) is locally Gorenstein, we show that the group structure of A(𝔽q) is determined by End (A). In fact, the same conclusion is attained if End (A) has local Cohen–Macaulay type at most 2, under the additional assumption that A is ordinary or q is prime, although the conclusion is not true in general. Second, the description in the Gorenstein case is used to characterize cyclic isogeny classes in terms of conductor ideals. Third, going in the opposite direction, we characterize squarefree isogeny classes of abelian varieties with N rational points in which every abelian group of order N is realized as a group of rational points. Finally, we study when an abelian variety A over 𝔽q and its dual A satisfy or fail to satisfy several interrelated properties, namely AA, A(𝔽q)A(𝔽q), and End (A) = End (A). In the process, we exhibit a sufficient condition for A A involving the local Cohen–Macaulay type of End (A). In particular, such an abelian variety A is not a Jacobian, or even principally polarizable.

有限域上的阿贝尔变分及其有理点群
在有限域𝔽q上,具有交换自同态环的阿贝尔变可以用在阶上的模来描述。通过利用这种联系,我们提出了关于有理点群和自对偶的四个定理,并给出了明确的例子。首先,当End (A)是局部Gorenstein时,我们证明了A(𝔽q)的群结构是由End (A)决定的。实际上,如果End (A)是局部Cohen-Macaulay型,在A为普通或q为素数的附加假设下,也得到了相同的结论,尽管结论一般不成立。其次,用Gorenstein案例中的描述来描述导体理想条件下的循环等源类。第三,在相反的方向上,我们描述了具有N个有理点的阿贝尔变体的无平方同基因类,其中每个N阶阿贝尔群都被实现为一组有理点。最后,研究了一个阿贝尔变量A /𝔽q及其对偶A在什么情况下∨满足或不满足几个相互关联的性质,即A(𝔽q)≠A²A∨(𝔽q),以及End (A)= End (A∨)。在此过程中,我们给出了a≇a∨涉及End的局部Cohen-Macaulay型的一个充分条件。特别地,这样一个阿贝尔变量a不是雅可比矩阵,甚至不是主要可极化的。
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来源期刊
CiteScore
1.80
自引率
7.70%
发文量
52
审稿时长
6-12 weeks
期刊介绍: ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms. The policies of ANT are set by the editorial board — a group of working mathematicians — rather than by a profit-oriented company, so they will remain friendly to mathematicians'' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. All electronic content becomes free and open access 5 years after publication.
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