Spans of Hecke points on modular curves
Spans of Hecke points on modular curves
复制标题
模曲线上 Hecke 点的跨度
DOI：
--
复制
发表时间：
2001
期刊：
影响因子：
--
通讯作者：
B. Poonen
中科院分区：
文献类型：
--
作者：
B. Poonen
本文修正了文献中描述模曲线上点的象在Hecke对应下的跨距的秩的一个定理。设X是Q上的模曲线，与同余子群Γ0（N），Γ1（N）或Γ（N）之一相关联.假设X的亏格至少为2。在将x取为x −∞类的映射下，用它在雅可比矩阵J中的像来标识X，其中∞ ∈ X（Q）表示通常的尖点。设Jtors表示J（Q）的挠子群.对任意不整除N的素数p，X上的Hecke对应Tp导出J的自同态τp.最后，设ZTp（x）表示通过将Tp应用于x而获得的X（Q）的p + 1个点在J（Q）中的Z-跨度。本文的主要结果是定理2，它与下面的结论相矛盾。陈述1（[Si 2]中的定理0.4）。设x ∈ X（Q）是非尖点、非CM点。然后，对于p足够大，秩ZTp（x）= { p，如果x ∈ Jtors。p + 1，否则。只有[Si 2]中证明的最后一句是有缺陷的：在最后一个等价链的左手侧的“i（x）∈ Jtors或τp = 0”应该用“τp（i（x））∈ Jtors”代替。因此，如果用“τpx ∈ Jtors”代替“x ∈ Jtors”，则语句1为真。[Si 2]中的定理0.4只起一个注释的作用：该论文的主要结果与Hecke对应下点的图像高度有关，不受校正的影响。西尔弗曼向我解释说，他归因于定理0.4在[Si 2]，以“马祖尔，未发表”，因为马祖尔勾勒了一个声明和证明，他口头上，因此他认为马祖尔应该得到信贷的想法，而他接受的责任，轻微的错误，在其书面。定理2.设J在Q上同构于椭圆曲线E×F的乘积。则存在无穷多个非挠、非尖点、非CM点x ∈ X（Q），使得存在无穷多个素数p不整除N，且秩ZTp（x）= p。2000年数学学科分类。第14 G35
We correct a theorem in the literature describing the rank of the span of the images of a point on a modular curve under Hecke correspondences. Let X be a modular curve over Q associated to one of the congruence subgroups Γ0(N), Γ1(N), or Γ(N). Assume that X has genus at least 2. Identify X with its image in the jacobian J under the map taking x to the class of x −∞, where ∞ ∈ X(Q) denotes the usual cusp. Let Jtors denote the torsion subgroup of J(Q). For any prime p not dividing N , the Hecke correspondence Tp on X induces an endomorphism τp of J . Finally, let ZTp(x) denote the Z-span in J(Q) of the p + 1 points of X(Q) obtained by applying Tp to x. The main result of this note is Theorem 2, which contradicts the following. Statement 1 (Theorem 0.4 in [Si2]). Let x ∈ X(Q) be a noncuspidal, non-CM point. Then for p sufficiently large, rank ZTp(x) = { p, if x ∈ Jtors. p + 1, otherwise. It is only the last sentence of the proof in [Si2] that is flawed: the “i(x) ∈ Jtors or τp = 0” on the left hand side of the last chain of equivalences should be replaced by “τp(i(x)) ∈ Jtors”. Therefore Statement 1 becomes true if “x ∈ Jtors” is replaced by “τpx ∈ Jtors”. Theorem 0.4 in [Si2] plays the role only of a remark: the main results of that paper, which are concerned with the heights of the images of a point under a Hecke correspondence, are unaffected by the correction. Silverman explained to me that he attributed Theorem 0.4 in [Si2] to “Mazur, unpublished” because Mazur sketched a statement and proof to him verbally; therefore he feels that Mazur should get credit for the idea, while he accepts responsibility for the minor error in its write-up. Theorem 2. Suppose that J is isogenous over Q to a product of elliptic curves E×F . Then there exist infinitely many nontorsion, noncuspidal, non-CM points x ∈ X(Q) such that there exist infinitely many primes p not dividing N for which rank ZTp(x) = p. Received October 23, 2001. 2000 Mathematics Subject Classification. Primary 14G35.