On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction I
On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction I
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具有超奇异约简 I 的椭圆曲线分圆域上的 Tate Shafarevich 群
DOI：
10.1007/s002220100206
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发表时间：
2002
影响因子：
3.1
通讯作者：
M. Kurihara
中科院分区：
文献类型：
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作者：
M. Kurihara
令E 为在Q 上定义的椭圆曲线。我们固定素数p，并假设K∞/Q 是分圆Zp 扩展。 PutΛ=Zp[[Gal(K∞/Q)]]。 Rubin（CM 情况）和 Kato（非 CM 情况）[24][14] 提出的一个显着定理，是 Mazur 的猜想，指出如果 E 在 p 处有常约约，则 E 的 Tate Shafarevich 群的 p 一次分量在 K∞ 上的 Pontrjagin 对偶是一个挠率 Λ 模。通过这个事实以及 Mazur 的控制定理 [17] 和可追溯到 Iwasawa 的挠率 Λ 模的一般理论，我们知道 Kn 上 Tate Shafarevich 群的渐近行为为 n→∞，其中 Kn 表示 K∞ 的子域，使得 [Kn: Q]= pn。也就是说，如果 Kn 上的 E 的 Tate Shafarevich 群的 p 主分量是有限的，并且如果我们用 pen 表示阶数，我们知道存在 λ、με Z≥ 0 且 νε Z，使得对于所有足够大的 n，en= λn+ µpn+ ν。当然，这类似于 Iwasawa 著名的 Zp 扩展中中间域的类数公式 [12]。但如果 E 没有潜在的普通约简，我们对 Tate Shafarevich 群的渐近行为几乎一无所知。我们本文的目的是研究 E 在 p 处具有超奇异约简的情况下的渐近行为。（有关此问题的更多详细信息，请参见 [3] 第 4 章和 [6] § 5。）在第一部分中，我们考虑最简单的情况。假设 E 在 p 处有超奇异约简。令 L (E, s) 为 E 的 L 函数。我们的主要假设是 p 不能整除 L (E, 1)/ΩE，其中 ΩE 是 Néron 周期。如果伯奇和斯温纳顿-戴尔猜想为真，这将是
Let E be an elliptic curve defined over Q. We fix a prime number p, and suppose that K∞/Qis the cyclotomicZp-extension. PutΛ= Zp [[Gal (K∞/Q)]]. A remarkable theorem by Rubin (CM case) and Kato (non CM case)[24][14], which was a conjecture of Mazur, states that the Pontrjagin dual of the p-primary component of the Tate Shafarevich group of E over K∞ is a torsion Λ-module if E has ordinary reduction at p. By this fact together with Mazur’s control theorem [17] and the general theory of torsion Λ-modules which goes back to Iwasawa, we know the asymptotic behaviour of the Tate Shafarevich groups over Kn as n→∞ where Kn denotes the subfield of K∞ such that [Kn: Q]= pn. Namely, if the p-primary components of the Tate Shafarevich groups of E over Kn are finite and if we denote the order by pen, we know that there exist λ, µ∈ Z≥ 0 and ν∈ Z such that en= λn+ µpn+ ν for all sufficiently large n. This is, of course, an analogue of Iwasawa’s famous formula for the class numbers of the intermediate fields in a Zp-extension [12]. But if E does not have potentially ordinary reduction, we know almost nothing about the asymptotic behaviour of the Tate Shafarevich groups1. Our aim in this paper is to study the asymptotic behaviour in the case that E has supersingular reduction at p.(See [3] Chap. 4 and [6] § 5 for more details on this problem.)In this part I, we consider the simplest situation. Suppose that E has supersingular reduction at p. Let L (E, s) be the L-function of E. Our main assumption is that p does not divide L (E, 1)/ΩE where ΩE is the Néron period. If the Birch and Swinnerton-Dyer conjecture is true, this would