We give an exact formula of the average of adjoint L-functions of holomorphic Hilbert cusp forms with a fixed weight and a square-free level, which is a generalization of Zagier's formula known for the case of elliptic cusp forms on SL 2 (Z). As an application, we prove that the Satake parameters of Hilbert cusp forms with a fixed weight and with growing square-free levels are equidistributed in an ensemble constructed by values of the adjoint L-functions.
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.
Given a modular embedding j : ∆\H/H ∩ K → Γ\G/K associated with an equivariant embedding (H, H/H ∩ K) → (G, G/K) of symmetric domains with actions of a semisimple Lie group G and a reductive subgroup H, both defined over Q compatibly, together with some other conditions. Starting from a certain harmonic left H-invariant "spherical" current on G/K with sigularity along HK/K, we can define a Poincaré series. Apply ∂∂ operator to the analytic continutation of this with respect to the parameter of eigenvalues of the "Laplacian". Then as an analogue of the Kronecker limit formula, we can construct a Green current for the cycle defined by j. This is a continuation of the previous paper [26], and here we treat the case of higher-codimensional cycles with compact Γ\G/K.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.