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

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.

A geometric approach to the sup-norm problem for automorphic forms: the case of newforms on $GL_2(\mathbb F_q(T))$ with squarefree level

“…An exponent below 5/24 < 1/4 appears, for division algebras, in the recent work of Hu and Saha [2019] and it is likely possible to obtain a similar statement, in particular one with exponent < 1/4, over GL 2 (for highly ramified level, with the conductor of the central character not too large), by the same method. Doing this well by our method would require further geometric ideas.…”

Theta functions, fourth moments of eigenforms, and the sup-norm problem II

“…Remark 1.4. In the depth aspect, where N = p n is a prime power, Hu and Saha [HS20] establish for an indefinite quaternion algebra the strong bound ϕ ∞ ≪ λϕ,p,dB,ε N 5/24+ε . The local depth aspect and the global squarefree aspect, that we address in this paper, are arguably disparate.…”

A geometric approach to the sup‐norm problem for automorphic forms: the case of newforms on GL2(Fq(T)) with squarefree level

“…Combining both amplification and estimates for Whittaker coefficients, Saha [17] obtained a bound of in the limit where becomes more powerful while the conductor of the central character divides . Hu and Saha [10] obtained a bound of by the amplification method for forms on division algebras, with level a high power of a small prime, where the conductor of the central character is not too large. It is likely possible to obtain a similar statement for modular forms, with the same‐level condition, by the same method.…”