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Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  15 December 2020

Yueke Hu  and
Abhishek Saha
Yueke Hu
Affiliation:
Yau Mathematical Sciences Center, Tsinghua University, Beijing100084, Chinayhumath@mail.tsinghua.edu.cn
Abhishek Saha
Affiliation:
School of Mathematical Sciences, Queen Mary University of London, LondonE1 4NS, UKabhishek.saha@qmul.ac.uk
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Abstract

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

Keywords

automorphic formnewformMaass formsubconvexitysup-normamplificationL-functionlevel aspectdepth aspectmatrix coefficientquaternion algebra

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F12: Automorphic forms, one variable 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F72: Spectral theory; Selberg trace formula 11F85: $p$-adic theory, local fields 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 11 , November 2020 , pp. 2368 - 2398
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Copyright
© The Author(s) 2020

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