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

Abstract

We provide a few natural applications of the analytic newvectors, initiated in \cite{JN} arXiv:1911.01880, to some analytic questions in automorphic forms for ${\mathrm{PGL}}_n(\mathbb{Z})$ with $n\ge 2$, 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.

arXiv comment34 pages, Nearly accepted version

References in corpus (13)

• The subconvexity problem for $\GL_{2}$
• Uniform approximate functional equation for principal L-functions
• Low-lying zeros of families of elliptic curves
• Applications of the Kuznetsov formula on GL(3)
• On the absolute convergence of the spectral side of the Arthur trace formula for GL(n)
• Upper bounds on L-functions at the edge of the critical strip
• Weighted Sato-Tate Vertical Distribution of the Satake Parameter of Maass Forms on PGL(N)
• Density theorems for GL(n)
• Counting cusp forms by analytic conductor
• An orthogonality relation for GL(4,R)
• Analytic newvectors for ${\mathrm{GL}}_n(\mathbb{R})$
• The Whittaker period formula on Metaplectic SL(2)
• Rankin-Selberg gamma factors of level zero representations of $G{L}_n$

Cited by in corpus (8)

• An unconditional $\mathrm{GL}(n)$ large sieve
• The density hypothesis for horizontal families of lattices
• Analytic newvectors for ${\mathrm{GL}}_n(\mathbb{R})$
• Zeros of Rankin-Selberg $L$-functions in families
• Optimal Diophantine Exponents for $\mathrm{SL}(n)$
• The second moment of $\mathrm{GL}(n)\times \mathrm{GL}(n)$ Rankin--Selberg $L$-functions
• A Density Theorem for $\mathrm{Sp}(4)$
• An Asymptotic Orthogonality Relation for $\mathrm{GL}(n,\mathbb{R})$