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

Abstract

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.

Low-lying zeros of symmetric power $L$-functions weighted by symmetric square $L$-values

“…In §3, we introduce symmetric rth L-functions for GL 2 for any r ∈ N and prove the main theorem 1.2 on weighted density of low-lying zeros of such L-functions. The latter part of this article from Appendix A to Appendix C is independent of our main results and is devoted to the comparison of Tsuzuki and the author's trace formula [42] (called the ST trace formula in this article) with Zagier's formula [45], Mizumoto's formula [33] and Takase's formula [44]. This comparison is not so straightforward and non-trivial as the anonymous referee of [42] pointed out to the author.…”

“…We remark that our assumption l 6 can be weakened to l 4, in which the condition z ∈ [0, 1] is replaced with z ∈ [0, min(1, σ)] for any σ ∈ (0, l − 3). This condition on z and l is derived from the assumption in [42,Corollary 1.2].…”

“…Shin and Templier used Arthur's invariant trace formula by estimating orbital integrals to quantify automorphic Plancherel density theorem. Contrary to those trace formulas, the proof of Theorem 1.2 relies on the explicit Jacquet-Zagier type trace formula given by Tsuzuki and the author [42], which is a generalization of the Eichler-Selberg trace formula to a parameterized version. In our setting, we need to treat not only the main term but also the second main term of the weighted automorphic Plancherel density theorem quantitatively as in Theorem 2.6 in order to study weighted density of low-lying zeros.…”

“…This article is organized as follows. In §2, we review the explicit Jacquet-Zagier trace formula for GL 2 given by Tsuzuki and the author [42], and prove a refinement of the Plancherel density theorem weighted by symmetric square L-functions [42,Theorem 1.3] by explicating the error term. In §3, we introduce symmetric rth L-functions for GL 2 for any r ∈ N and prove the main theorem 1.2 on weighted density of low-lying zeros of such L-functions.…”

Weighted one-level density of low-lying zeros of Dirichlet $L$-functions

“…The key idea for proving the asymptotic behavior of the weighted one-level density is the use of Selberg's formula of the twisted second moment of Dirichlet L-functions with complex parameters s and s ′ [14]. Selberg's formula is a substitute of the explicit Jacquet-Zagier type trace formula by the first author and Tsuzuki [16], as we see that such a parametrized trace formula was used in [15] for analysis of the weighted one-level density for symmetric power L-functions attached to Hilbert modular forms. The computation in Proposition 2.3 is essentially the same as that in [15,Theorem 2.6].…”