Constructing abelian varieties from rank 2 Galois representations

IF 1.3 1区 数学 Q1 MATHEMATICS Compositio Mathematica Pub Date : 2024-03-07 DOI:10.1112/s0010437x23007728
Raju Krishnamoorthy, Jinbang Yang, Kang Zuo
{"title":"Constructing abelian varieties from rank 2 Galois representations","authors":"Raju Krishnamoorthy, Jinbang Yang, Kang Zuo","doi":"10.1112/s0010437x23007728","DOIUrl":null,"url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$U$</span></span></img></span></span> be a smooth affine curve over a number field <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$K$</span></span></img></span></span> with a compactification <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$X$</span></span></img></span></span> and let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline4.png\"><span data-mathjax-type=\"texmath\"><span>${\\mathbb {L}}$</span></span></img></span></span> be a rank <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$2$</span></span></img></span></span>, geometrically irreducible lisse <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$\\overline {{\\mathbb {Q}}}_\\ell$</span></span></img></span></span>-sheaf on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$U$</span></span></img></span></span> with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$E\\subset \\overline {\\mathbb {Q}}_{\\ell }$</span></span></img></span></span>, and has bad, infinite reduction at some closed point <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$x$</span></span></img></span></span> of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$X\\setminus U$</span></span></img></span></span>. We show that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline11.png\"><span data-mathjax-type=\"texmath\"><span>${\\mathbb {L}}$</span></span></img></span></span> occurs as a summand of the cohomology of a family of abelian varieties over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline12.png\"><span data-mathjax-type=\"texmath\"><span>$U$</span></span></img></span></span>. The argument follows the structure of the proof of a recent theorem of Snowden and Tsimerman, who show that when <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline13.png\"><span data-mathjax-type=\"texmath\"><span>$E=\\mathbb {Q}$</span></span></img></span></span>, then <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline14.png\"><span data-mathjax-type=\"texmath\"><span>${\\mathbb {L}}$</span></span></img></span></span> is isomorphic to the cohomology of an elliptic curve <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240307072459819-0092:S0010437X23007728:S0010437X23007728_inline15.png\"><span data-mathjax-type=\"texmath\"><span>$E_U\\rightarrow U$</span></span></img></span></span>.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Compositio Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/s0010437x23007728","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

Let Abstract Image$U$ be a smooth affine curve over a number field Abstract Image$K$ with a compactification Abstract Image$X$ and let Abstract Image${\mathbb {L}}$ be a rank Abstract Image$2$, geometrically irreducible lisse Abstract Image$\overline {{\mathbb {Q}}}_\ell$-sheaf on Abstract Image$U$ with cyclotomic determinant that extends to an integral model, has Frobenius traces all in some fixed number field Abstract Image$E\subset \overline {\mathbb {Q}}_{\ell }$, and has bad, infinite reduction at some closed point Abstract Image$x$ of Abstract Image$X\setminus U$. We show that Abstract Image${\mathbb {L}}$ occurs as a summand of the cohomology of a family of abelian varieties over Abstract Image$U$. The argument follows the structure of the proof of a recent theorem of Snowden and Tsimerman, who show that when Abstract Image$E=\mathbb {Q}$, then Abstract Image${\mathbb {L}}$ is isomorphic to the cohomology of an elliptic curve Abstract Image$E_U\rightarrow U$.

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从秩 2 伽罗瓦表示构建无常变体
让 $U$ 是一条在数域 $K$ 上的光滑仿射曲线,其紧凑性为 $X$;让 ${mathbb {L}}$ 是一个在 $U$ 上的秩为 2$、几何上不可还原的 lisse $/overline {{\mathbb {Q}}_\ell$ 舍夫,其环状行列式扩展为一个积分模型、在某个固定数域 $E\subset \overline {\mathbb {Q}}_{\ell }$中都有弗罗贝尼斯迹,并且在 $X\setminus U$ 的某个闭点 $x$ 上有坏的、无限的还原。我们证明 ${mathbb {L}}$ 是作为 U$ 上的无性变体族的同调之和出现的。斯诺登和齐默尔曼的论证沿用了他们最近证明的一个定理的结构,即当 $E=\mathbb {Q}$ 时,${/mathbb {L}}$ 与椭圆曲线 $E_U\rightarrow U$ 的同调同构。
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来源期刊
Compositio Mathematica
Compositio Mathematica 数学-数学
CiteScore
2.10
自引率
0.00%
发文量
62
审稿时长
6-12 weeks
期刊介绍: Compositio Mathematica is a prestigious, well-established journal publishing first-class research papers that traditionally focus on the mainstream of pure mathematics. Compositio Mathematica has a broad scope which includes the fields of algebra, number theory, topology, algebraic and differential geometry and global analysis. Papers on other topics are welcome if they are of broad interest. All contributions are required to meet high standards of quality and originality. The Journal has an international editorial board reflected in the journal content.
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