Low-lying zeros of families of elliptic curves
Low-lying zeros of families of elliptic curves
复制标题
DOI：
10.1090/s0894-0347-05-00503-5
复制
发表时间：
2004-06
影响因子：
3.9
通讯作者：
M. Young
中科院分区：
文献类型：
--
作者：
M. Young
随机矩阵模型预言，与L-函数族的零点相关的许多统计量都可以通过一个经典线性群中的大随机矩阵的特征值分布来建模（或预测）。如果一族L-函数的统计量由群G的特征值建模，那么我们说G是与该族相关联的对称群（或对称型）。在这项工作中，我们感兴趣的统计量是中心点附近的零密度（也称为1级密度）。随机矩阵模型预测，这些零点的分布应该由对称类型G（酉，辛和正交）之一的最接近1的特征值来建模。所有不同的组G在这方面都有不同的行为。因此，计算1-能级密度给出了一种理论方法来预测一个族的对称类型。1-能级密度已被研究了各种各样的L-函数族;例如参见[R]、[KSI]、[ILS]、[Mil]。在研究单能级密度时，通常假设广义黎曼假设（GRH），我们在整个工作中都是这样做的。特别是，有必要使用GRH的应用程序获得的平均解析秩从密度定理的限制。在某些情况下，GRH的使用提高了密度定理的范围，这转化为平均秩上的改进的界。除了GRH的这些重要应用之外，我们还自由地假设了GRH，即使它可以通过额外的工作被删除，因为它简化了一些非必要地方的论证。研究有理数上椭圆曲线上的L函数族的1-能级密度是特别有趣的，因为中心点的零点具有重要的算术信息（由Birch和Swinnerton Dyer猜想）。这些调查一直是这项工作的主要重点。
The random matrix model predicts that many statistics associated to zeros of a family of L-functions can be modeled (or predicted) by the distribution of eigenval ues of large random matrices in one of the classical linear groups. If the statistics of a family of L-functions are modeled by the eigenvalues of the group G, then we say that G is the symmetry group (or symmetry type) associated to the family. The statistic of interest to us in this work is the density of zeros near the central point (also known as the 1-level density). The random matrix model predicts that the distribution of these zeros should be modeled by the eigenvalues nearest 1 for one of the symmetry types G (unitary, symplectic, and orthogonal). All of the different groups G have distinct behavior in this regard. Therefore, computing the 1-level density gives a theoretical way to predict the symmetry type of a family. The 1-level density has been studied for a wide variety of families of L-functions; see [R], [KSI], [ILS], [Mil] for example. It is standard to assume the Generalized Riemann Hypothesis (GRH) to study the 1-level density and we do so throughout this work. In particular, it is necessary to use GRH for the application of obtaining a bound on the average analytic rank from a density theorem. In some cases the use of GRH improves the range of the density theorem, which translates to an improved bound on the average rank. Besides these crucial applications of GRH, we have freely assumed GRH even when it could be removed with extra work since it simplifies arguments in some non essential places. It is especially interesting to investigate the 1-level density for families of L functions attached to elliptic curves over the rationals since zeros at the central point have important arithmetic information (by the conjecture of Birch and Swinnerton Dyer). These investigations have been the main focus of this work.