Weighted Distribution of Low-lying Zeros of GL(2) -functions

Abstract

We show that if the zeros of an automorphic L-function are weighted by the central value of the L-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the L-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted L-values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

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

“…A density function not arising in random matrix theory is also seen for families of L-functions attached to elliptic curves in Miller [32]. Therefore, our Theorem 1.2 gives us a new example of the phenomenon that central L-values effect to change of density of low-lying zeros, as seen in Knightly and Reno [26] for GL 2 and Kowalski, Saha and Tsimerman [27], Dickson [9] for GSp 4 . From these observations, it might be meaningful to suggest the following, which is not a rigorous form.…”

“…Recently, Knightly and Reno [26] studied density of low-lying zeros of the standard Lfunctions L(s, f ) attached to holomorphic elliptic cusp forms f weighted by central values L(1/2, f ), and found the change of the symmetry type of the density from orthogonal to symplectic. Their study was originated by a change of symmetry type by central L-values in the case of Siegel modular forms of degree 2 in Kowalski, Saha and Tsimerman [27] and Dickson [9] as an evidence of Böcherer's conjecture which was not proved at that time (Now this conjecture is known to be true by Furusawa and Morimoto [12,Theorem 2]).…”

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

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

“…Motivated by the work of Knightly and Reno [9] in 2019, the one-level density for a family of L-functions weighted by L-values has been studied in the context of random matrix theory. Knightly and Reno [9] discovered the phenomenon that the symmetry type of a family of automorphic L-functions attached to elliptic modular forms changes from orthogonal to symplectic due to the weight factors of central L-values, by comparing [9] with the usual one-level density [4].…”

“…Motivated by the work of Knightly and Reno [9] in 2019, the one-level density for a family of L-functions weighted by L-values has been studied in the context of random matrix theory. Knightly and Reno [9] discovered the phenomenon that the symmetry type of a family of automorphic L-functions attached to elliptic modular forms changes from orthogonal to symplectic due to the weight factors of central L-values, by comparing [9] with the usual one-level density [4]. Their work was inspired by Kowalski, Saha and Tsimerman [10], who treated the one-level density for a family of spinor L-functions attached to Siegel modular forms of degree 2 weighted by Bessel periods, identical to central L-values by [2].…”