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

Published online by Cambridge University Press:  15 December 2020

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

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.

Type
Research Article
Copyright
© The Author(s) 2020

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