Rubinstein Michael O, Wu Kaiyu
Pure Mathematics, University of Waterloo, 200 University Avenue W, Waterloo, Ontario, Canada N2L3G1
Pure Mathematics, University of Waterloo, 200 University Avenue W, Waterloo, Ontario, Canada N2L3G1.
Philos Trans A Math Phys Eng Sci. 2015 Apr 28;239(2040). doi: 10.1098/rsta.2014.0307.
Let q be an odd prime power, and Hq,d denote the set of square-free monic polynomials D(x)∈Fq[x] of degree d. Katz and Sarnak showed that the moments, over Hq,d, of the zeta functions associated to the curves y(2)=D(x), evaluated at the central point, tend, as q→∞, to the moments of characteristic polynomials, evaluated at the central point, of matrices in USp(2⌊(d-1)/2⌋). Using techniques that were originally developed for studying moments of L-functions over number fields, Andrade and Keating conjectured an asymptotic formula for the moments for q fixed and q→∞. We provide theoretical and numerical evidence in favour of their conjecture. In some cases, we are able to work out exact formulae for the moments and use these to precisely determine the size of the remainder term in the predicted moments.
设(q)为奇素数幂，(H_{q,d})表示(\mathbb{F}q[x])中次数为(d)的无平方因子首一多项式(D(x))的集合。卡茨和萨纳克证明，对于曲线(y^2 = D(x))相关的zeta函数在中心点处的取值，在(H{q,d})上的矩，当(q \to \infty)时，趋于(\mathrm{USp}(2\lfloor(d - 1)/2\rfloor))中矩阵的特征多项式在中心点处的取值的矩。安德拉德和基廷运用最初为研究数域上(L)函数的矩而发展的技术，对固定(q)以及(q \to \infty)时的矩猜想了一个渐近公式。我们提供了支持他们猜想的理论和数值证据。在某些情况下，我们能够算出矩的精确公式，并利用这些公式精确确定预测矩中余项的大小。
