Fredholm理论与检测L函数族中心点消失的最优检验函数
Fredholm Theory and Optimal Test Functions for Detecting Central Point Vanishing Over Families of L-functions
摘要：
黎曼ζ函数是研究最多的L函数;它的零点给出了关于素数的信息。我们可以将L函数与大量对象关联起来，一般来说，这些L函数的零给出了关于这些对象的信息。对于任意的L函数，在中心点处消失的阶是特别重要的。例如，Birch和Swinnerton-Dyer猜想指出，在椭圆曲线L函数的中心点消失的阶是该椭圆曲线的Mordell-Weil群的秩。卡茨-萨纳克密度猜想指出，这种阶消失和其他行为可以用从经典紧群中提取的随机矩阵很好地模拟。特别地，该猜想指出，仅使用给定的权函数和选择的检验函数，φ，可以有界的平均阶消失在一个家庭的L函数。当检验函数被适当地限制时，该猜想对于许多族是已知的。很自然地，我们会问，对于每个系列和对phi的每一组自然限制，哪一个检验函数是最好的。我们的主要结果是一个减少，否则无限维优化到有限维最优化问题的所有家庭和所有的限制。我们显式地解决了许多这些最优化问题，并计算改进的界限，我们得到的平均秩。虽然我们没有验证这些新的、更宽松的限制的密度猜想，但通过这个项目，我们能够精确地量化这些努力在平均秩方面的好处。最后，我们能够证明，这个界限严格提高，因为我们增加支持。
英文摘要：
The Riemann Zeta-Function is the most studied L-function; it's zeroes give information about the prime numbers. We can associate L-functions to a wide array of objects, and in general, the zeroes of these L-functions give information about those objects. For arbitrary L-functions, the order of vanishing at the central point is of particular important. For example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing at the central point of an elliptic curve L-function is the rank of the Mordell-Weil group of that elliptic curve. The Katz-Sarnak Density Conjecture states that this order vanishing and other behavior are well-modeled by random matrices drawn from the classical compact groups. In particular, the conjecture states that an average order vanishing over a family of L-functions can be bounded using only a given weight function and a chosen test function, phi. The conjecture is known for many families when the test functions are suitably restricted. It is natural to ask which test function is best for each family and for each set of natural restrictions on phi. Our main result is a reduction of an otherwise infinite dimensional optimization to a finite-dimensional optimization problem for all families and all sets of restrictions. We explicitly solve many of these optimization problems and compute the improved bound we obtain on average rank. While we do not verify the density conjecture for these new, looser restrictions, with this project, we are able to precisely quantify the benefits of such efforts with respect to average rank. Finally, we are able to show that this bound strictly improves as we increase support.
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