No abstract

In this paper we consider the sup-norm problem in the context of analytic Eisenstein series for GL(2) over number fields. We prove a hybrid bound which is sharper than the corresponding bound for Maaß forms. Our results generalise those of Huang and Xu where the case of Eisenstein series of square-free levels over the base field Q had been considered. theory of Eisenstein series is used to give lower bounds for ζ(1 + it). This last method has vast generalizations. See for example [8]. Our case is reverse. We exploit that the theory of Hecke L-functions over number fields is well developed and use this to produce an improved amplifier. This leads to the remarkable fact that (as in [23]) we get sup-norm bounds which are in some aspects better than the state of the art results for cusp forms. It was already observed in [13] that one can improve the sup-norm bound in the spectral aspect if one has a better understanding of the Hecke eigenvalues. This is exactly the ingredient we gain from the classically well developed GL 1 theory. More precisely, we use highly non-trivial zero free regions for Hecke L-functions to derive an asymptotic expansion for generalized divisor sums. This will serve as a lower bound for the amplifier.Note that recently in [16] it was shown that the quality of the sup-norm bound that we achieve for Eisenstein series holds for SL 2 (Z)-Maaß forms on average.Before we can give a precise statement of our theorems we have to introduce some notation. We assume that the reader is familiar with the results and the notation in [1] and only focus on the aspects where Eisenstein series differ from cusp forms.

In this paper we tackle a question raised by N. Templier and A. Saha concerning the size of Whittaker new vectors appearing in infinite dimensional representations of GL 2 over non-archimedean fields. We derive precise bounds for such functions in all possible situations. Our main tool is the p-adic method of stationary phase.

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.

We prove Sarnak's density conjecture for the principal congruence subgroup of SLn(Z) of squarefree level and discuss various arithmetic applications. The ingredients include new bounds for local Whittaker functions and Kloosterman sums.