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

Abstract

For a totally real number field F and its adèle ring A F , let π vary in the set of irreducible cuspidal automorphic representations of PGL 2 (A F ) corresponding to primitive Hilbert modular forms of a fixed weight. Then, we determine the symmetry type of the one-level density of low-lying zeros of the symmetric power L-functions L(s, Sym r (π)) weighted by special values of symmetric square L-functions L( z+1 2 , Sym 2 (π)) at z ∈ [0, 1] in the level aspect. If 0 < z 1, our weighted density in the level aspect has the same symmetry type as Ricotta and Royer's density of low-lying zeros of symmetric power L-functions for F = Q with harmonic weight. Hence our result is regarded as a zinterpolation of Ricotta and Royer's result. If z = 0, density of low-lying zeros weighted by central values is a different type only when r = 2, and it does not appear in random matrix theory as Katz and Sarnak predicted. Moreover, we propose a conjecture on weighted density of low-lying zeros of L-functions by special L-values.In the latter part, Appendices A, B and C are dedicated to the comparison among several generalizations of Zagier's parameterized trace formula. We prove that the explicit Jacquet-Zagier type trace formula (the ST trace formula) by Tsuzuki and the author recovers all of Zagier's, Takase's and Mizumoto's formulas by specializing several data. Such comparison is not so straightforward and includes non-trivial analytic evaluations.

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

“…Here, d is a fundamental discriminant, χ d is the primitive quadratic Dirichlet character corresponding to the quadratic field Q( √ d) by class field theory, and ∆ is the delta function (a cusp form of weight 12 and level 1). The first author's result [15] on the change of the symmetry type provides new evidence of Fazzari's conjecture.…”

“…(see [9,Proposition 3.1] and [15,Corollary 2.9]). The first term on the right-hand side is then evaluated as…”

“…We notice by comparing the weighted one-level density [10] and the usual one-level density [7,8] that the symmetry type of the family of those spinor Lfunctions changes from orthogonal to symplectic. Recently, the first author [15] observed a new phenomenon occurring in a family of symmetric square L-functions attached to Hilbert modular forms. He found that its symmetry type changes from symplectic to a new type of density function which does not occur in random matrix theory as Katz and Sarnak predicted.…”

“…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].…”

“…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]. Both computations include the derivation at s = 1/2 for deducing the main term and the second main term.…”