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

Yueke Hu Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China Abhishek Saha School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK

Number Theory mathscidoc:2206.24014

Compositio Mathematica, 156, (11), 2368-2398, 2020.12
We improve upon the local bound in the depth aspect for sup-norms of newforms on D^×, where D is an indefinite quaternion division algebra over Q. Our sup-norm bound implies a depth-aspect subconvexity bound for L(1/2,f×θ_χ), where f is a (varying) newform on D^× of level p^n, and θ_χ is an (essentially fixed) automorphic form on GL_2 obtained as the theta lift of a Hecke character χ 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.
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@inproceedings{yueke2020sup-norms,
  title={Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity},
  author={Yueke Hu, and Abhishek Saha},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220629151626039111490},
  booktitle={Compositio Mathematica},
  volume={156},
  number={11},
  pages={2368-2398},
  year={2020},
}
Yueke Hu, and Abhishek Saha. Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity. 2020. Vol. 156. In Compositio Mathematica. pp.2368-2398. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220629151626039111490.
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