Canonical periods and congruence formulae
Canonical periods and congruence formulae
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规范周期和同余公式
DOI：
10.1215/s0012-7094-99-09811-3
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发表时间：
1999
影响因子：
2.5
通讯作者：
V. Vatsal
中科院分区：
文献类型：
--
作者：
V. Vatsal
这篇文章的目的是显示如何之间的同余的Hecke特征形的傅立叶系数引起相应的同余之间的代数部分的临界值的相关的L-功能。本研究由B发起。马祖尔在他的基本工作的爱森斯坦理想（见[Maz 77]和[Maz 79]），其中明确指出，合同的分析L-值密切相关的整体结构的某些赫克环和上同调群。[Maz 79]的结果还表明同余式在研究L-函数的非零性时是有用的。这一想法随后由Stevens [Ste 82]和Rubin-Wiles [RW 82]进一步发展。鲁宾和怀尔斯的工作，特别是使用同余研究椭圆曲线的行为塔的分圆领域。这里的一个关键成分是一个定理的华盛顿，其中指出，大致上，几乎L-值在某些家庭是非零模p。这一主题最近又被采取了，在工作的小野斯金纳[OSa]，[OSb]，詹姆斯[果酱]，和Kohnen [Koh 97]。虽然早期的历史主要关注的是分圆扭转，目前的重点是家庭的扭转二次特征标。在这里，人们想要定量估计给定模形式的二次扭曲的数量，这些模形式在s = 1处具有非零L-函数。我们继续这一趋势，在目前的工作中，使用我们的一般结果，以获得一个强非零定理的二次扭曲的模椭圆曲线的有理点的三阶。这推广了Kevin James的一个漂亮的例子，并为Goldfeld [Gol 79]的一个猜想提供了新的证据。然而，应该指出的是，即使是研究二次扭曲可以追溯到马祖尔：敦促读者看看第212-213页的[Maz 79]，特别是在脚注底部的第213页。Davenport-Heilbronn [DH 71]和华盛顿[Was 78]的定理是本文的关键，Mazur的文章中也提到了这两个定理。我们想开始讨论这篇文章的核心一致性。设f = ∑ anqn是M阶椭圆模尖形，权k ≥ 2.假设f是所有Hecke算子的同时特征形，并且a1（f）= 1。与f相关的L-函数定义为狄利克雷级数L（s，f）= ∑ ann −s，它对s的真实的部分足够大收敛，并且对s ∈ C有解析延拓。Shimura [Shi 76]的一个基本定理指出L（s，f）具有以下代数性性质：
The purpose of this article is to show how congruences between the Fourier coefficients of Hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated L-functions. This study was initiated by B. Mazur in his fundamental work on the Eisenstein ideal (see [Maz77] and [Maz79]) where it was made clear that congruences for analytic L-values were closely related to the integral structure of certain Hecke rings and cohomology groups. The results of [Maz79] also showed that congruences were useful in the study of nonvanishing of L-functions. This idea was then further developed by Stevens [Ste82] and Rubin-Wiles [RW82]. The work of Rubin and Wiles, in particular, used congruences to study the behavior of elliptic curves in towers of cyclotomic fields. A key ingredient here was a theorem of Washington, which states, roughly, that almost L-values in certain families are nonzero modulo p. This theme has recently been taken up again, in the work of Ono-Skinner [OSa], [OSb], James [Jam], and Kohnen [Koh97]. While the earlier history was primarily concerned with cyclotomic twists, the current emphasis is on families of twists by quadratic characters. Here one wants quantitative estimates for the number of quadratic twists of a given modular form, which have nonvanishing L-function at s = 1. We continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three. This generalizes a beautiful example due to Kevin James, and provides new evidence for a conjecture of Goldfeld [Gol79]. It should, however, be pointed out that even the study of quadratic twists may be traced back to Mazur: the reader is urged to look at pages 212–213 of [Maz79], and especially at the footnote at the bottom of page 213. The theorems of Davenport-Heilbronn [DH71] and Washington [Was78], which are crucial in this paper, are both mentioned in Mazur’s article. We want to begin by discussing the congruences that lie at the heart of this article. Thus let f = ∑ anq n be an elliptic modular cuspform of level M and weight k ≥ 2. Assume that f is a simultaneous eigenform for all the Hecke operators and that a1(f) = 1. The L-function associated to f is defined by the Dirichlet series L(s, f) = ∑ ann −s, which converges for the real part of s sufficiently large, and has analytic continuation to s ∈ C. A fundamental theorem of Shimura [Shi76] states that L(s, f) enjoys the following algebraicity property: