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 centre of the critical strip are used to motivate a series of conjectures concerning the value-distribution of the Fourier coefficients of half-integral 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.

Derivatives of random matrix characteristic polynomials with applications to elliptic curves

“…The T 3/4 has been predicted previously by Sarnak using different arguments, but random matrix theory adds more detailed information in the form of the power on the logarithm. For numerical evidence supporting the conjecture, see [9] and [10].…”

“…(The conjecture was originally stated in [9] with c E > 0, but in fact numerics have revealed that the constant can be zero [10] for certain families. An explanation of such a case with c E = 0 was given by Delaunay [14].)…”

Computational methods and experiments in analytic number theory

“…The constant α E depends on c −1/2 and the real period of E, but also on some extra subtle arithmetic information that seems to be related to Delaunay's heuristics for TateShafarevich groups [De] and is not yet fully understood. Numerical evidence in favour of this conjecture is presented in [CKRS2]. One can skirt these delicate issues, the power on the logarithm and the constant α E , as follows.…”

“…We see the values fluctuating about zero, most of the time agreeing to within about two percent. The convergence in X is predicted from secondary terms to be logarithmically slow and one gets a better fit by including more terms [CKRS2]. …”

“…We end this paper with a plot that substantiates this conjecture. Figure 13 compares, for one hundred elliptic curves E, the predicted value of R p to the actual value R p (X), with X = 10 8 and the set of d's restricted to certain residue classes depending on E as described in [CKRS2]. The L-values were computed in this special case by exploiting their connection to the coefficients of certain weight three halves modular forms and using a table of Rodriguez-Villegas and Tornaria [RT].…”