Random Matrix Theory and the Fourier Coefficients of Half-Integral-Weight Forms

Abstract

Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of L-functions at the center of the critical strip are used to motivate a series of conjectures concerning the value distribution of the Fourier coefficients of halfintegral- weight modular forms related to these L-functions. Our conjectures may be viewed as being analogous to the Sato–Tate conjecture for integral-weight modular forms. Numerical evidence is presented in support of them.

Keywords:

• 11M, 15A52
• L-functions
• elliptic curve
• random matrix theory
• half-integral weight form