Applications of analytic newvectors for $\mathrm{GL}(n)$

Abstract

We provide a few natural applications of the analytic newvectors, initiated in [24], to some analytic questions in automorphic forms for PGL r (Z) with r ≥ 2, in the archimedean analytic conductor aspect. We prove an orthogonality result of the Fourier coefficients, a density estimate of the non-tempered forms, an equidistribution result of the Satake parameters with respect to the Sato-Tate measure, and a second moment estimate of the central L-values as strong as Lindelöf on average. We also prove the random matrix prediction about the distribution of the low-lying zeros of automorphic L-function in the analytic conductor aspect. The new ideas of the proofs include the use of analytic newvectors to construct an approximate projector on the automorphic spectrum with bounded conductors and a soft local (both at finite and infinite places) analysis of the geometric side of the Kuznetsov trace formula.

Conductor zeta function for the GL(2) universal family

“…Use in the trace formula. A definition of archimedean conductor as suggested above lends itself quite naturally to use in the Arthur-Selberg trace formula (or other types of trace formulae, such as that of Kuznetsov), as has been powerfully demonstrated in the recent thesis of Jana [19] and subsequent outgrowths. In this usage, one takes as an archimedean test function a smoothened characteristic function of K 1 (X, τ 0 ), see [20,Section 8].…”

Zeros of Rankin-Selberg $L$-functions in families

“…We mention three results that improve on (2.5) in an average sense, each having complementary strengths. Jana [36,Theorem 6] extended the GLH-on-average bound of Deshouillers and Iwaniec to the family of cuspidal automorphic representations of GL n (A Q ) of arithmetic conductor 1. Blomer [4, Corollary 5] proved the corresponding result for the family of cuspidal automorphic representations of GL n (A Q ) of a large given arithmetic conductor q, trivial central character, and whose archimedean components are principal series representations such that the associated Laplace eigenvalue is bounded.…”

The second moment of $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions

“…We point out on the naive similarities between the spectral weight here and the one which is used in e.g. [15,Theorem 1]. However, the invariance property which is needed here is a bit stronger than the invariance used in [15, Theorem 1]: We only needed invariance at points near the identity in GL n (R) in [15], where as here we have to gain an invariance which is uniform for all elements in GL n−1 (R).…”

“…[15,Theorem 1]. However, the invariance property which is needed here is a bit stronger than the invariance used in [15, Theorem 1]: We only needed invariance at points near the identity in GL n (R) in [15], where as here we have to gain an invariance which is uniform for all elements in GL n−1 (R). The method of using the approximate invariance of the newvectors is similar as in [27] for GL (2) where in the non-archimedean aspect the exact invariance is used.…”