Refined dimensions of cusp forms, and equidistribution and bias of signs

JOURNAL OF NUMBER THEORY

卷:188

Article

Martin, Kimball¹ 

[1] Univ Oklahoma, Dept Math, Norman, OK 73019 USA

关键词: Modular forms;    Root numbers;    Atkin-Lehner signs;    Equidistribution;   

DOI  :  10.1016/j.jnt.2018.01.015

来源: Elsevier

【 摘 要 】

We refine known dimension formulas for spaces of cusp forms of squarefree level, determining the dimension of subspaces generated by newforms both with prescribed global root numbers and with prescribed local signs of Atkin-Lehner operators. This yields precise results on the distribution of signs of global functional equations and sign patterns of Atkin-Lehner eigenvalues, refining and generalizing earlier results of Iwaniec, Luo and Sarnak. In particular, we exhibit a strict bias towards root number +1 and a phenomenon that sign patterns are biased in the weight but perfectly equidistributed in the level. Another consequence is lower bounds on the number of Galois orbits. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free