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