Subconvex bounds on GL(3) via degeneration to frequency zero

Abstract

For a fixed cusp form π on GL 3 (Z) and a varying Dirichlet character χ of prime conductor q, we prove that the subconvex bound2 ) ≪ q 3/4−δ holds for any δ < 1/36. This improves upon the earlier bounds δ < 1/1612 and δ < 1/308 obtained by Munshi using his GL 2 variant of the δ-method. The method developed here is more direct. We first express χ as the degenerate zero-frequency contribution of a carefully chosen summation formula à la Poisson. After an elementary "amplification" step exploiting the multiplicativity of χ, we then apply a sequence of standard manipulations (reciprocity, Voronoi, Cauchy-Schwarz and the Weil bound) to bound the contributions of the nonzero frequencies and of the dual side of that formula. Contents 1. Introduction 2. Preliminaries 3. Division of the proof 4. Estimates for F 5. Estimates for O Appendix A. Correlations of Kloosterman sums Appendix B. Comparison with Munshi's approach References

Bounds for twists of $\rm GL(3)$ $L$-functions

“…Again, the approach that Munshi took does not require the non-negativity of certain L-functions, which removes the self-duality assumption on the forms π and χ in Blomer's work. Recently Holowinsky and Nelson [5] discovered a new look at Munshi's delta method, which removes the use of Petersson trace formula in [16] altogether as well as improves the exponent of saving to any δ ă 1{36.…”

“…We make such assumption so as to control the error term of the stationary phase analysis in our approach. For the case |t| ă M ε , the bound (1) follows from the work [5], since there their bound L p1{2 `it, π b χq ! t,π,ε M 3{4´1{36`ε is of polynomially dependence in t.…”

“…For subconvexity bounds on GLp3q in other aspects, see [2,3,15,21]. Our approach is a variant of the methods introduced in the works [16] and [5]. In Section 2, we will give a brief outline of our approach for the simpler case L p1{2 `it, πq, to guide the readers through.…”

“…Our approach is inspired by the work [16] and is a further variant to the recent work [5]. We now give a brief introduction to the approach in [16].…”

Sub-convexity problem for Rankin-Selberg $L$-functions

“…He also came up with the "GL(2) circle method" to establish a sub-convexity bound for GL(3) cusp forms twisted by Dirichlet character [26]. Holowinsky and Nelson have simplified the latter result's proof considerably in [12]. Munshi also used the GL(2) circle method to re-establish a Burgess type bound for character twists of GL(2) automorphic forms (including the Eisenstein series, which recovers Burgess's original bound for Dirichlet L-function) in [28].…”