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

2019/05/15 by Hu, Yueke, Saha, Abhishek
#11F66 #11F67 #11F72 #11F85 #22E50 #FOS: Mathematics #Number Theory (math.NT) #Primary 11F70 #Secondary 11F12

paper · doi:10.48550/arxiv.1905.06295

Abstract

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 pⁿ, and θ_χ is an (essentially fixed) automorphic form on GL₂ 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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