Lower-Order Biases in the Second Moment of Dirichlet Coefficients in Families of L -Functions
Lower-Order Biases in the Second Moment of Dirichlet Coefficients in Families of L -Functions
复制标题
L 函数族狄利克雷系数二阶矩的低阶偏差
DOI：
10.1080/10586458.2021.1980453
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发表时间：
2023
影响因子：
0.5
通讯作者：
Winsor, Karl
中科院分区：
文献类型：
--
作者：
Asada, Megumi;Chen, Ryan C.;Fourakis, Eva;Kim, Yujin Hong;Kwon, Andrew;Lichtman, Jared Duker;Mackall, Blake;Miller, Steven J.;Winsor, Eric;Winsor, Karl
设是一个非平凡的单参数椭圆曲线族，其中.考虑狄利克雷系数的k时刻。罗森和西尔弗曼证明了长尾的猜想，第一个时刻与家族的排名有关，米歇尔证明了IFJ(T)不是常数，那么第二个时刻等于。上同调论证表明，低阶项的大小为1。在每种情况下，我们都可以用封闭形式分析，二阶矩展开式中不平均为零的最大低阶项平均为负，尽管数值表明这可能对中等阶数的家庭无效。我们在几大类族上证明了这一偏差猜想，包括具有秩族、复乘族和常数j(T)不变族。我们还研究了Dirichlet特征标族、GL上的全纯形式以及它们的对称幂和Rankin-Selberg卷积的相似偏差猜想。我们在大类族中识别所有低阶项，揭示控制这些项的算术对象。这些低阶项中的负偏向意味着超额秩猜想和中心点附近的零点行为。
Letbe a nontrivial one-parameter family of elliptic curves over, with. Consider thekthmomentsof the Dirichlet coefficients. Rosen and Silverman proved Nagao’s conjecture relating the first moment to the family’s rank over, and Michel proved ifj(T) is not constant then the second moment equals. Cohomological arguments show the lower order terms are of sizesand 1. In every case, we can analyze in closed form, the largest lower order term in the second moment expansion that does not average to zero is on average negative, though numerics suggest this may fail for families of moderate rank. We prove this Bias Conjecture for several large classes of families, including families with rank, complex multiplication, and constantj(T)-invariant. We also study the analogous Bias Conjecture for families of Dirichlet characters, holomorphic forms on GL, and their symmetric powers and Rankin-Selberg convolutions. We identify all lower order terms in large classes of families, shedding light on the arithmetic objects controlling these terms. The negative bias in these lower order terms has implications toward the excess rank conjecture and the behavior of zeros near the central point.