Integral presentations of the shifted convolution problem and subconvexity estimates for GL_n-automorphic L-functions

2019/03/18 by Van Order, Jeanine
#FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1903.07284

Abstract

Fix n ≥ 2 an integer, and F be a totally real number field. We reduce the shifted convolution problem for L-function coefficients of GL_n(\bfA_F)-automorphic forms to the better-understood setting of GL₂(\bfA_F). The key idea behind this reduction is to use the classical projection operator \mathbb Pⁿ₁ φ together with properties of its Fourier-Whittaker expansion. This allows us to derive novel integral presentations for the shifted convolution problem as Fourier-Whittaker coefficients of certain L²-automorphic forms on the mirabolic subgroup P₂(\bfA_F) of GL₂(\bfA_F) or its two-fold metaplectic cover P₂(\bfA_F). We then construct liftings of these mirabolic forms to GL₂(\bfA_F) and its two-fold metaplectic cover G(\bfA_F) to justify expanding the underlying forms into linear combinations of Poincaré series. Decomposing each of the Poincaré series spectrally then allows us to derive completely new bounds for the shifted convolution problem in dimensions n ≥ 3. As an application, we derive a uniform subconvexity bound for GL_n(\bfA_F)-automorphic L-functions twisted by Hecke characters. This uniform level-aspect subconvexity estimate appears to the the first of its kind for dimensions n ≥ 3.

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