Subconvexity for twisted 𝐿-functions on 𝐺𝐿₃ over the Gaussian number field

Abstract

Let q P Zris be prime and χ be the primitive quadratic Hecke character modulo q. Let π be a self-dual Hecke automorphic cusp form for SL 3 pZrisq and f be a Hecke cusp form for Γ 0 pqq Ă SL 2 pZrisq. Consider the twisted L-functions Lps, π b f b χq and Lps, π b χq on GL 3ˆG L 2 and GL 3 . We prove the subconvexity boundsfor any ε ą 0.2010 Mathematics Subject Classification. 11M41.

Moments of Central $L$-values for Maass Forms over Imaginary Quadratic Fields

“…The asymptotics in (1) are analogous to those in [Qi3,Lemma 5.1 (1)]. See also [IK,Proposition 5.4], [Blo,Lemma 1], and [Qi1,Lemma 3.7]. To derive (4.17) and (4.18), we choose U " a Cptq, say, and shift the integral contour in (4.14) and (4.15) from Repvq " ε further down to Repvq " ´A; the main term is the residue from the pole at v " 0 while the error term is from the Stirling formula.…”

Subconvexity for $L$-Functions on $\mathrm{GL}_3$ over Number Fields

“…t ˘. (5.12) Properties of Vpy; νq and Gpu, νq are contained in the following lemma (see [Blo, Lemma 1] and [Qi3,Lemma 3.7]).…”

“…Finally, when the phase is of the form λ f pxq, we record here a generalization of the stationary phase estimate in [Sog, Theorem 1.1.1] (X " 1 in [Sog]). See [Qi3,§2.4].…”

“…The main difficulty in the analysis over C is the lack of suitable stationary phase results in two dimensions. Considerably more efforts are needed particularly for the Hankel transform and the Mellin transform; the former was studied in the author's previous work [Qi3]. We would rather not to discuss the technical details-It is more worthwhile and interesting to present here the features of certain trigonometric-hyperbolic functions arising in the phases.…”

On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over number fields

“…We quote from Lemma 4.1 in [Qi4] the following uniform estimate for J it pzq, tJ it pzq Î p|t| `1q 3 min 1, 1{|z| ( . (6.5) Further, it is clear from [Wat,3.13 (1)] (note that |Γp1 `itq| 2 " πt{ sinhpπtq) that t pJ it pzq ´Rit pzqq Î |z| 2 , |z| ď 1.…”