A Bessel Delta Method and Exponential Sums for Gl(2)

Abstract

<jats:title>Abstract</jats:title> <jats:p>In this paper, we introduce a simple Bessel $\delta $-method to the theory of exponential sums for $\textrm{GL}_2$. Some results of Jutila on exponential sums are generalized in a less technical manner to holomorphic newforms of arbitrary level and nebentypus. In particular, this gives a short proof for the Weyl-type subconvex bound in the $t$-aspect for the associated $L$-functions.</jats:p>

Beyond the Weyl barrier for $\mathrm{GL}(2)$ exponential sums

“…We start with the following result from [AHLQ,§4], which is a consequence of applications of the Voronoï summation formula along with the Bessel δ-identity.…”

“…After applying Poisson summation with modulus p 1 p 2 to the n-sum, in view of the discussions in [AHLQ,§ §5.2,5.4], we arrive at (our notation here is slightly different)…”

“…In view of (2.13), the condition K 2 {T ą N ε in [AHLQ,Lemma 5.4 (2)] may be weakened into K 2 {N ą N ε , so we only require…”

Sub-convexity bound for $GL(3) \times GL(2)$ $L$-functions: Hybrid level aspect

“…After initial success of the second author [27], the method has been extended, simplified and generalised by several researchers, e.g. see [14], [2], [3], [4], [18], [19], [29], [30] and [23].…”

Hybrid subconvexity bounds for twists of $\rm GL(3)$ $L$-functions

“…In the last two decades, People extend the results on GL(2) and GL(3) L-functions to different aspects, either in q-aspect or in t-aspect. (see [Agg20], [AHLQ20], [BM15], [Li11],[Mun15], [Mun18], [LS], [RY15] and so on).…”