Central values of $L$-functions and half-integral weight forms

Abstract

Abstract. We prove a relation between the Fourier coefficients of certain Hilbert modular forms of half-integral weight and central values of the corresponding Rankin L-functions. The result generalizes the classical theorem by Waldspurger. The approach is geometric and generalizes that of Gross and Hatcher.

Preimages for the Shimura map on Hilbert modular forms

“…In [Xue11], the author also follows the method of Gross for computing half-integral weight Hilbert modular forms to prove a Waldspurger's type formula, but with several restrictions such as working with prime level and odd class number of the base field, and with no focus on Hecke operators nor the Shimura correspondence.…”

An explicit Waldspurger formula for Hilbert modular forms

“…Besides extending [ST18], and thus the explicit formulas with l = 1 given in [Gro87,BSP90] when F = Q and in [Xue11] for more general F , our main result implies part of [Mao12, Theorem 1.1], where the author considers the case F = Q, odd prime level and weight 2. In that setting, the case D > 0 is covered by Mao, but not (yet) by our formula.…”

Nonvanishing of twists of -functions attached to Hilbert modular forms

“…It follows in principle from general work of Waldspurger [43] and Shimura [35], but we were not able to extract the explicit statement above: the difficult part is to construct a nonzero form g. Baruch-Mao [1] prove Conjecture 2.8 in the case F = Q (using work of Kohnen for the existence of g). Xue [46] gives a proof for general F and prime level N = p, provided L(f, . We will discuss in § 3 an algorithm that conjecturally always computes the form g associated to f ∈ S k (N) when L(f, Remark 2.10.…”

“…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.…”

“…By comparison, experiments with the distribution of twists with similar Hodge data computed without using Waldspurger's theorem are much less extensive: see for example, Watkins [45, § 6.6] and David-Fearnley-Kisilevsky [6]. Several authors have pursued Waldspurger's theorem in the setting of Hilbert modular forms, including Shimura [35], Baruch-Mao [1], Xue [46], Sirolli [38], and Hiraga-Ikeda [18]. In this paper, we develop algorithms using these formulas to compute families of central values twists of hilbert modular forms L(f, 1/2, χ D ) for Hilbert modular forms f over totally real fields F , defined in an analogous way.…”