Relative trace formulas and subconvexity estimates for $L$-functions of Hilbert modular forms

Abstract

Abstract. We elaborate an explicit version of the relative trace formula on PGL(2) over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral equidistribution result in the level aspect for Satake parameters of holomorphic Hilbert cusp forms weighted by central L-values, and (ii) a bound of quadratic base change L-functions for Hilbert cusp forms with a subconvex exponent in the weight aspect.

Existence of Hilbert Cusp Forms with Non-vanishingL-values

“…We should note that this can be regarded as a refinement of [7,Corollary 1.2]. As for derivatives, we have a conditional result.…”

“…We write the Satake parameter of π ∈ Π * cus (l, n) at v ∈ S as diag(q νv(π)/2 v , q −νv(π)/2 v ) with ±ν v (π) belonging to the space X v = C/4πi(log q v ) −1 Z. In [7], given an even holomorphic function α(s) on X S = v∈S X v , we studied the asymptotic of the average AL * (n, α) = C l N(n) π∈Π * cus (l,n) L(1/2, π) L(1/2, π ⊗ η) L Sπ (1, π; Ad) α(ν S (π)) (1.1) with ν S (π) = {ν v (π)} v∈S and…”

“…In §2, after recalling the explicit formula of Hecke's zeta integrals for old forms ( [5]) and calculating their derivatives, we prove a formula of CT λ=0 ∂P η β,λ (Ψ l reg (n|α)) written in terms of the spectral data of cuspidal representations in Π * cus (l, n) (Proposition 8). In §3, closely following [7], we compute ∂P η β,λ (Ψ l reg (n|α)) according to the H(F ) × H(F )-double coset decomposition of GL(2, F ). By equating the two expressions of CT λ=0 ∂P η β,λ (Ψ l reg (n|α)) obtained in §2 and §3, we get a kind of relative trace formula, which is stated in Theorem 17.…”

“…The integration domain of W ηv v,2 (b) is a disjoint union of the sets D l (b) (l ∈ Z) defined in [7, 10.2]. By [7,Lemmas 10.6,10.7 and 10.8],…”

An explicit trace formula of Jacquet–Zagier type for Hilbert modular forms

“…For v ∈ Σ fin with c(π v ) = 0, we have [26] coincides with π v ( −1 0 0 1 )φ 0,v according to our notation). In the rest of the proof, [30, (2.30)], we have the formula (3.11).…”