Integral moments of automorphic L-functions

Abstract

Abstract. This paper exposes the underlying mechanism for obtaining second integral moments of GL 2 automorphic L-functions over an arbitrary number field. Here, moments for GL 2 are presented in a form enabling application of the structure of adele groups and their representation theory. To the best of our knowledge, this is the first formulation of integral moments in adele-group-theoretic terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers Q, we recover the classical results.

Moments for L-Functions for GL r×GL r-1

“…As in [Diaconu-Garrett 2009], a direct computation shows that the spectral expansion of the GL 2 Poincaré series with constant term removed is…”

“…By elliptic regularity, solutions f to this differential equation are in the local Sobolev space with index 2ν − 1 2 − ε, and by Sobolev's lemma are locally at least C 2ν−1−2ε ⊂ C 2ν−2 . That is, by increasing ν solutions are made as differentiable as desired, and their Fourier transforms will have corresponding decay, giving convergence of the Poincaré series (for suitable s, β), as in [Diaconu-Garrett 2009].…”

“…The spectral expansion of the GL 2 Poincaré series P β s,ν formed with this archimedean data ϕ β s,ν is special case of the computation in [Diaconu-Garrett 2009], recalled above, but in fact gives a much simpler outcome. For example, the cuspidal components are directly computed by unwinding, integrating by parts, and applying the characterization of ϕ β s,ν by the differential equation:…”

“…For example, see [Hardy-Littlewood 1918], [Ingham 1926], [Heath-Brown 1975], [Sarnak 1985], [Good 1983[Good ,1986, [Motohashi 1997], [Jutila 1997], [PetridisSarnak 2001], [Bruggeman-Motohashi 2001,2003, [CFKRS 2005], [Diaconu-Goldfeld-Hoffstein 2005], [Diaconu-Goldfeld 2006a,2006b], [Diaconu-Garrett 2009]. Second integral moments of level-one holomorphic elliptic modular forms were first treated in [Good 1983[Good ,1986, the latter using an idea that is a precursor of part of the present approach.…”

“…Also, [Li 2009] obtains a t-aspect subconvexity bound for standard L-functions for GL 3 (Q) for Gelbart-Jacquet lifts. For context, we review the [Diaconu-Goldfeld 2005] treatment of spherical waveforms f for GL 2 (Q). In that case, the sum of moments is a single term Z A GL2(Q)\GL2(A) P(g, z, w) |f (g)| 2 dg = 1 2πi Re (s)= 1 2 L(z + s, f ) · L(s, f ) · Γ(s, z, w, f ∞ ) ds where Γ(s, z, w, f ∞ ) is a ratios of products of gammas, with arguments depending upon the archimedean data of f .…”