{"title":"费马曲线的海森堡覆盖层","authors":"Debargha Banerjee, Loïc Merel","doi":"10.4153/s0008414x24000476","DOIUrl":null,"url":null,"abstract":"<p>For <span>N</span> integer <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$\\ge 1$</span></span></img></span></span>, K. Murty and D. Ramakrishnan defined the <span>N</span>th Heisenberg curve, as the compactified quotient <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$X^{\\prime }_N$</span></span></img></span></span> of the upper half-plane by a certain non-congruence subgroup of the modular group. They ask whether the Manin–Drinfeld principle holds, namely, if the divisors supported on the cusps of those curves are torsion in the Jacobian. We give a model over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbf {Z}[\\mu _N,1/N]$</span></span></img></span></span> of the <span>N</span>th Heisenberg curve as covering of the <span>N</span>th Fermat curve. We show that the Manin–Drinfeld principle holds for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$N=3$</span></span></img></span></span>, but not for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$N=5$</span></span></img></span></span>. We show that the description by generator and relations due to Rohrlich of the cuspidal subgroup of the Fermat curve is explained by the Heisenberg covering, together with a higher covering of a similar nature. The curves <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$X_N$</span></span></img></span></span> and the classical modular curves <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$X(n)$</span></span></img></span></span>, for <span>n</span> even integer, both dominate <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$X(2)$</span></span></img></span></span>, which produces a morphism between Jacobians <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$J_N\\rightarrow J(n)$</span></span></img></span></span>. We prove that the latter has image <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline10.png\"/><span data-mathjax-type=\"texmath\"><span>$0$</span></span></span></span> or an elliptic curve of <span>j</span>-invariant <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline11.png\"/><span data-mathjax-type=\"texmath\"><span>$0$</span></span></span></span>. In passing, we give a description of the homology of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline12.png\"/><span data-mathjax-type=\"texmath\"><span>$X^{\\prime }_{N}$</span></span></span></span>.</p>","PeriodicalId":501820,"journal":{"name":"Canadian Journal of Mathematics","volume":"69 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Heisenberg covering of the Fermat curve\",\"authors\":\"Debargha Banerjee, Loïc Merel\",\"doi\":\"10.4153/s0008414x24000476\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For <span>N</span> integer <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline1.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\ge 1$</span></span></img></span></span>, K. Murty and D. Ramakrishnan defined the <span>N</span>th Heisenberg curve, as the compactified quotient <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$X^{\\\\prime }_N$</span></span></img></span></span> of the upper half-plane by a certain non-congruence subgroup of the modular group. They ask whether the Manin–Drinfeld principle holds, namely, if the divisors supported on the cusps of those curves are torsion in the Jacobian. We give a model over <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathbf {Z}[\\\\mu _N,1/N]$</span></span></img></span></span> of the <span>N</span>th Heisenberg curve as covering of the <span>N</span>th Fermat curve. We show that the Manin–Drinfeld principle holds for <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$N=3$</span></span></img></span></span>, but not for <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline5.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$N=5$</span></span></img></span></span>. We show that the description by generator and relations due to Rohrlich of the cuspidal subgroup of the Fermat curve is explained by the Heisenberg covering, together with a higher covering of a similar nature. The curves <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline6.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$X_N$</span></span></img></span></span> and the classical modular curves <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline7.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$X(n)$</span></span></img></span></span>, for <span>n</span> even integer, both dominate <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline8.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$X(2)$</span></span></img></span></span>, which produces a morphism between Jacobians <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline9.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$J_N\\\\rightarrow J(n)$</span></span></img></span></span>. We prove that the latter has image <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline10.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$0$</span></span></span></span> or an elliptic curve of <span>j</span>-invariant <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline11.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$0$</span></span></span></span>. In passing, we give a description of the homology of <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240604071203237-0536:S0008414X24000476:S0008414X24000476_inline12.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$X^{\\\\prime }_{N}$</span></span></span></span>.</p>\",\"PeriodicalId\":501820,\"journal\":{\"name\":\"Canadian Journal of Mathematics\",\"volume\":\"69 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-05-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Canadian Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4153/s0008414x24000476\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Canadian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4153/s0008414x24000476","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
对于 N 个整数$\ge 1$,K. Murty 和 D. Ramakrishnan 定义了第 N 条海森堡曲线,即上半平面通过模数群的某个非共轭子群的紧凑商 $X^{\prime}_N$。他们提出了马宁-德林费尔德原理是否成立的问题,即如果这些曲线的尖顶上支持的除数在雅各布中是扭转的,那么马宁-德林费尔德原理是否成立。我们给出了在 $\mathbf {Z}[\mu _N,1/N]$ 上的第 N 条海森堡曲线作为第 N 条费马曲线的覆盖的模型。我们证明马宁-德林菲尔德原理在 $N=3$ 时成立,但在 $N=5$ 时不成立。我们证明,费马曲线的尖顶子群的生成器和罗尔利希关系的描述可以用海森堡覆盖和一个性质类似的更高覆盖来解释。对于 n 偶整数,曲线 $X_N$ 和经典模态曲线 $X(n)$都支配着 $X(2)$,这就产生了雅各布数 $J_N\rightarrow J(n)$ 之间的形态。我们将证明后者的图象为 $0$ 或 j 不变的椭圆曲线为 $0$。顺便描述一下 $X^{prime }_{N}$ 的同调。
For N integer $\ge 1$, K. Murty and D. Ramakrishnan defined the Nth Heisenberg curve, as the compactified quotient $X^{\prime }_N$ of the upper half-plane by a certain non-congruence subgroup of the modular group. They ask whether the Manin–Drinfeld principle holds, namely, if the divisors supported on the cusps of those curves are torsion in the Jacobian. We give a model over $\mathbf {Z}[\mu _N,1/N]$ of the Nth Heisenberg curve as covering of the Nth Fermat curve. We show that the Manin–Drinfeld principle holds for $N=3$, but not for $N=5$. We show that the description by generator and relations due to Rohrlich of the cuspidal subgroup of the Fermat curve is explained by the Heisenberg covering, together with a higher covering of a similar nature. The curves $X_N$ and the classical modular curves $X(n)$, for n even integer, both dominate $X(2)$, which produces a morphism between Jacobians $J_N\rightarrow J(n)$. We prove that the latter has image $0$ or an elliptic curve of j-invariant $0$. In passing, we give a description of the homology of $X^{\prime }_{N}$.