On sup-norm bounds part I: ramified Maaß newforms over number fields

Abstract

We prove new upper bounds for the sup-norm of Hecke Maaß newforms on GL(2) over a number field. Our newforms are more general than those considered in a recent paper by Blomer, Harcos, Maga, and Milicevic: we do not require square free level. Furthermore, we allow for non-trivial central character. Over the rationals we cover the best bounds as obtained by Saha.

Explicit Burgess-like subconvex bounds for GL2 × GL1

“…For ǫ > 0 small, we first take M > 2 (say M = 3), then take 1) , we deduce from (3.6), (3.7), (3.8) and (3.9) and conclude by…”

“…(1) Our bound of the local factors at ramified places is via bounding the absolute value of the relevant matrix coefficients, which does not seem to give the true size by comparison with an existing computation in some special cases due to (2) As an alternative approach, one may apply the sup-norm techniques, such as those in [1,5,33],…”

“…) is holomorphic unless ξ is trivial on A (1) . If ξ = 1 and τ = 0, ζ(s, E(iτ, Φ)) admits two simple poles at s = 1 ± iτ with residue…”

“…Let ξ run over characters of F × \A (1) , extended to a Hecke character by triviality on R + according to a fixed section s F : R + → A × , such that for p < ∞…”

Hybrid bounds for the sup-norm of automorphic forms in higher rank

“…This was shown for instance in [AU95] for squarefree N , where it was used to compare the Arakelov and the Poincaré metrics on X 0 (N ). A great amount of work was dedicated to achieving subconvex bounds in more and more general settings, for example in [BH10], [Tem10], [TH13], [Sah17], [Ass17]. See the introduction of [HS20] for a more complete set of references with the corresponding bounds.…”