Rankin-Selberg L-functions in the level aspect

Abstract

In this paper we calculate the asymptotics of various moments of the central values of Rankin-Selberg convolution L-functions of large level, thus generalizing the results and methods of W. Duke, J. Friedlander, and H. Iwaniec and of the authors. Consequences include convexity-breaking bounds, nonvanishing of a positive proportion of central values, and linear independence results for certain Hecke operators.

The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II

“…More precisely, Bernstein and Reznikov considered the subconvexity problem in the spectral parameter aspect and established a subconvex bound for the central value L π φ ⊗ π φ ⊗ π φ , 1 2 of the triple product L-function associated to three Hecke-Maass cusp forms (over Q), two of which (φ and φ , say) are fixed and the remaining one (φ) has large Laplacian eigenvalue. In a different direction, A. Venkatesh considered the subconvexity problem in the level aspect for automorphic L-functions over a general number field F .Amongst other cases, he obtained subconvex bounds for the standard L-function L(π, s), the Rankin-Selberg L-function L(π ⊗ π , s), and the central value of the triple product Lfunction L π ⊗ π ⊗ π , 1 2 , where π and π are some fixed cuspidal automorphic representations of GL 2 (A F ) and π is a cuspidal automorphic representation of GL 2 (A F ) with large conductor and trivial central character: in particular, these bounds generalize, to any number field, the subconvex bounds obtained in [DFI93,DFI94a,KMV02]. The proof of these new subconvexity cases build on soft but powerful methods which are very different from the ones developed so far: indeed these methods are geometric in nature, and rely on the expression of the central values of automorphic L-functions in terms of periods of automorphic forms; then, the subconvexity problem becomes tantamount to bounding non-trivially these periods.…”

“…• The first one is via the δ-symbol method 3 which has been used for instance in [DFI93,DFI94a], in [KMV02], and-in a different form-in [J99, H03a, H03b]; the method builds on a formula for the Kronecker symbol δ m−n=h in terms of additive characters of small moduli (i.e., of size ∼ √ q instead of q). Plugging this formula into the shifted convolution sum separates the variable m from n; then a double application of Voronoi's summation formula for g yields a linear combination of Kloosterman sums in a short range.…”

“…If g is induced from a holomorphic form of weight l, then by Appendix A.3 of [KMV02] (see also [DI90]),…”

“…This is in sharp contrast with the former approaches (like the one presented here) where the relationship between central values and automorphic period is exploited, but only after the subconvex bound has been obtained (via the method of amplified moments for instance), to deduce an equidistribution result from subconvexity. Although these alternative approaches are very different from the ones coming from analytic number theory, an astute reader will nevertheless identify some common patterns on each side: this is especially striking when one compares [V05] with say [DFI94a] or [KMV02]. In fact, it seems plausible that one can pursue the analogy further so as, for example, to incorporate some features of the present paper within the framework developed in [V05]: this would yield to a generalization of Theorem 1 (which is not covered by the results of [V05]) to GL 2 -automorphic forms on a general number field.…”

Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for GL m (ℤ)

“…Ivić [1] generated the original Voronoi formula to non-cuspidal forms as given by multiple divisor functions. For cuspidal representations of GL 2 (Z), a Voronoi-type summation formula was proved by Sarnak [2] for holomorphic cusp forms and by Kowalski-Michel-Vanderkam [3] for Maass forms. Voronoi's summation formula for Maass forms for SL 3 (Z) was proved by Miller-Schmid [4] and Goldfeld-Li [5].…”