A bound for the image conductor of a principally polarized abelian variety with open Galois image

Proceedings of the American Mathematical Society, Series B Pub Date : 2020-01-20 DOI:10.1090/bproc/131

Jacob Mayle

{"title":"A bound for the image conductor of a principally polarized abelian variety with open Galois image","authors":"Jacob Mayle","doi":"10.1090/bproc/131","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a principally polarized abelian variety of dimension <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g\">\n <mml:semantics>\n <mml:mi>g</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">g</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> over a number 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>. Assume that the image of the adelic Galois representation of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is an open subgroup of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G upper S p Subscript 2 g Baseline left-parenthesis ModifyingAbove double-struck upper Z With caret right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>G</mml:mi>\n <mml:mi>S</mml:mi>\n <mml:msub>\n <mml:mi>p</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>2</mml:mn>\n <mml:mi>g</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">Z</mml:mi>\n </mml:mrow>\n <mml:mo stretchy=\"false\">^</mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">GSp_{2g}(\\hat {\\mathbb {Z}})</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Then there exists a positive integer <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\">\n <mml:semantics>\n <mml:mi>m</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">m</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> so that the Galois image of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is the full preimage of its reduction modulo <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\">\n <mml:semantics>\n <mml:mi>m</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">m</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. The least <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m\">\n <mml:semantics>\n <mml:mi>m</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">m</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> with this property, denoted <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m Subscript upper A\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>m</mml:mi>\n <mml:mi>A</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">m_A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, is called the <italic>image conductor</italic> of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Jones [Pacific J. Math. 308 (2020), pp. 307–331] recently established an upper bound for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m Subscript upper A\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>m</mml:mi>\n <mml:mi>A</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">m_A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, in terms of standard invariants of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, in the case that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\">\n <mml:semantics>\n <mml:mi>A</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">A</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is an elliptic curve without complex multiplication. In this paper, we generalize the aforementioned result to provide an analogous bound in arbitrary dimension.</p>","PeriodicalId":106316,"journal":{"name":"Proceedings of the American Mathematical Society, Series B","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2020-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/bproc/131","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

Let A A be a principally polarized abelian variety of dimension g g over a number field K K . Assume that the image of the adelic Galois representation of A A is an open subgroup of G S p 2 g ( Z ^ ) GSp_{2g}(\hat {\mathbb {Z}}) . Then there exists a positive integer m m so that the Galois image of A A is the full preimage of its reduction modulo m m . The least m m with this property, denoted m A m_A , is called the image conductor of A A . Jones [Pacific J. Math. 308 (2020), pp. 307–331] recently established an upper bound for m A m_A , in terms of standard invariants of A A , in the case that A A is an elliptic curve without complex multiplication. In this paper, we generalize the aforementioned result to provide an analogous bound in arbitrary dimension.

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具有开伽罗瓦象的主极化阿贝尔变象的象导体的界

设A A是数域K K上的维g g的主极化阿贝尔变换。假设A的亚历伽罗瓦表示的象是GSp 2g (Z ^) GSp_{2g}(\hat {\mathbb {Z}})的开子群。则存在正整数m m，使得a a的伽罗瓦像是其约化模m m的完整原像。具有这种性质的最小的m m记为m A m_A，称为A A的像导体。Jones [Pacific J. Math. 308 (2020)， pp. 307-331]最近在A A为椭圆曲线且没有复数乘法的情况下，根据A A的标准不变量建立了m A m_A的上界。在本文中，我们推广了上述结果，给出了任意维上的一个类似界。

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来源期刊

Proceedings of the American Mathematical Society, Series B

CiteScore

1.60

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0.00%

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