Correction: "Modularity of the Rankin-Selberg $L$-series, and multiplicity one for SL(2)"

By Dinakar Ramakrishnan. 2000

271 citations recorded by the source corpus.

Let f, g be primitive cusp forms, holomorphic or otherwise, on the upper half-plane H of levels N,M respectively, with (unitarily normalized) L-functions L(s, f) = [equation] and L(s, g) = [equation]. When p does not divide N (resp. M), the inverse roots αp, βp (resp. α′p, β′p ) are nonzero with sum ap (resp. bp). For every p prime to NM, set Lp(s, f × g) = [(1 − αpα′pp−s)(1 − αpβ′pp−s)(1 − βpα′pp−s)(1 − βpβ′pp−s)]^−1. Let L∗(s, f × g) denote the (incomplete Euler) product of Lp(s, f × g) over all p not dividing NM. This is closely related to the convolution L-series [sum over n≥1] a[sub]n b[sub] n n^−s, whose miraculous properties were first studied by Rankin and Selberg.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/94836eb22acf1360632727e67cbbeb48307803f1

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Citations in the corpus

A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations (2000, methodology): “We have also found a non-trivial extension of this result (in characteristic zero) to Maass forms using an array of results on automorphic L functions, and this is the subject matter of another paper [ Ra2 ].”
On the Exceptional Zeros of Rankin–Selberg L-Functions (2001, methodology): “[ Ra2000 ]) that their Rankin-Selberg product π π� of”
On the complex moments of symmetric power $L$-functions at $s = 1$ (2004, background): “Proof. Our proof is by recursion on k. The case k = 1 is classical and the case k = 2 (from which the casek = 1 follows) was proved in a stronger form by Ramakrishnan [24, 25 ].”
On modular signs (2009, background): “The argument is short and simple, but it depends crucially on a very deep result: Ramakrishnan’s proof [19] the Rankin–Selberg convolution L-function is the L-function of some modular form on GL(4).”
Weak subconvexity without a Ramanujan hypothesis (2018, methodology): “ functoriality conjectures, one expects that π1 × π2 corresponds to an automorphic representation of GL(m1m2) (not necessarily cuspidal), but this remains unknown, apart from the work of Ramakrishnan [26] in the case m1 = m2 = 2 and the work of Kim and Shahidi [17] in the case m1 = 2 and m2 = 3. Pro”
Cohomology of congruence subgroups of SL(4,Z) II (2002, unknown): “Since the Rankin–Selberg product (π1,π2) → π is functorial at all places [13], in particular at ∞, we must have σ∞ = σ1,∞ ⊗ σ2,∞,”
SIEGEL MODULAR FORMS OF GENUS 2 ATTACHED TO ELLIPTIC CURVES (2006, background): “To elaborate, one knows multiplicity one for SL(2) ( [ 35 ]), which implies that two cusp forms ·;· 0 on GL(2) are twist-equivalent if their adjoint L-functions are the same (outside a flnite number of factors).”