Computation of central value of quadratic twists of modular L-functions

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

“…For d > 0, the work of Baruch, Mao, Rodriguez-Villegas, and Tornaria [1] [21] gives the relevant weight 3/2 form for Q prime and d's satisfying χ d (Q) = a Q . One has the same relation as above, but with a different proportionality constant.…”

On elliptic curves and random matrix theory

“…This was indicated to us 2 by H. Darmon and G. Tornaría, and a similar idea can be used in the case of even parity [27]. The idea is that the Hilbert class field of Q √ −dl contains that of Q √ −d , and if we construct a point in the former, then tracing by the genus character χ l gives a point in Q √ −d .…”

Nonvanishing of twists of -functions attached to Hilbert modular forms

“…If we follow this construction with L(f, 1/2) = 0, then the corresponding form has g = 0: see Gross [15] and Böcherer-Schulze-Pillot [3] for the case F = Q and Xue [46] and Sirolli [38] for general F . On the other hand, one expects a nonzero g in Conjecture 2.8; see Mao-Rodriguez-Villegas-Tornaría [24] for an extension of this algorithm to compute g in this case.…”

“…For how many fundamental discriminants D with |D| X does the twisted L-function L(f, s, χ D ) vanish at the center of the critical strip? In a collection of papers [24,29,30], a number of variants of this problem were considered: the weight of f was allowed to vary, the level of f was allowed to be composite, and so on.…”

“…The central values L(f, 1 2 , χ D ) encode interesting arithmetic information about the form f , and a number of explicit investigations have been carried out examining the family of these values [3,15,24,27,28].…”