On the Waldspurger formula and the metaplectic Ramanujan conjecture over number fields

Abstract

In this paper, by inputting the Bessel identities over the complex field in previous work of the authors, the Waldspurger formula of Baruch and Mao is extended from totally real fields to arbitrary number fields. This is applied to give a non-trivial bound towards the Ramanujan conjecture for automorphic forms on the metaplectic group Ă SL 2 for the first time in the generality of arbitrary number fields.

A Voronoï--Oppenheim summation formula for number fields

“…The kernel formula for unitary representations of PGL 2 pRq first appears in the book of Cogdell and Piatetski-Shapiro [CPS], and has been generalized to GL 2 pRq and GL 2 pCq in [Qi5,§ §17,18]. For its applications to establishing the Kuznetsov formula and the Waldspurger formula, we refer the reader to [CPS,Qi1,BM1,CQ2,BM2,CQ1]. For the unitary case, the kernel formula is actually valid for all W φ papxqq in the Kirillov model.…”

On the Fourier Transform of Regularized Bessel Functions on Complex Numbers and Beyond Endoscopy Over Number Fields

“…Recently, by a rather indirect method that combines asymptotic analysis and differential equations, the formulae in (1.1)-(1.4) have been extended successfully over C in [Qi1,Qi2,Qi6], so that the Waldspurger formula in [BM3] and the Beyond Endoscopy in [Ven] are now valid over arbitrary number fields as in [CQ1] and [Qi6]. 1 1.1.…”

Eisenstein series and the cubic moment for PGL(2)

“…The above "subconvex" estimates improve upon the respective "trivial" or "convexity" bounds of C(χ) 1/2+ε and C(χ) 1/4+ε , and also upon earlier nontrivial subconvex bounds (see [7,62,61,37] and references). Via period formulas as in [51,15,3] (see also [59,31,56,30,32,32,34]) these estimates lead to improved bounds for the Fourier coefficients of half-integral weight modular forms over number fields (cf. [7,Cor 1]), hence to improved estimates for representation numbers of ternary quadratic forms over number fields.…”