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# Modularity of the Rankin-Selberg {$L$}-series, and multiplicity one for {${\rm SL}(2)$}

Ramakrishnan, Dinakar (2000) Modularity of the Rankin-Selberg {$L$}-series, and multiplicity one for {${\rm SL}(2)$}. Annals of Mathematics, 152 (1). pp. 45-111. ISSN 0003-486X. https://resolver.caltech.edu/CaltechAUTHORS:RAMaom00

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## Abstract

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.

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Additional Information:In memory of my father Sundaram Ramakrishnan (SRK) (Received October 26, 1998) We would like to express our gratitude to Ilya Piatetski-Shapiro for his continued interest in this project, and for kindly writing down, with J. Cogdell, the form of the converse theorem for GL(4) which we need ([CoPS]). Thanks are also due to T. Ikeda for writing down his calculations of the archimedean factors of the triple product L-functions ([Ik2]), to S. Rallis for useful remarks on these L-functions, to F. Shahidi for explaining his approach to the same via Langlands’s theory of Eisenstein series and for commenting on an earlier version, to my colleague T. Wolff for helpful conversations on an analytic lemma we use in Section 3.4, and to many others, including H. Jacquet, R. P. Langlands, J. Rogawski and P. Sarnak, who have shown encouragement and interest. Special thanks must go to J. Cogdell for reading the earlier and the revised versions thoroughly and making crucial remarks. Part of the technical typing of this paper was done by Cherie Galvez, whom we thank. Finally, we would like to express our appreciation to the following: the National Science Foundation for support through the grants DMS-9501151 and DMS-9801328, Universit´e Paris-sud, Orsay, where we spent a fruitful month during September 1996, the DePrima Mathematics House in Sea Ranch, CA, for inviting us to visit and work there during August 1996 and 1998, MATSCIENCE, India, for hospitality in February 98, and — last, but not the least — the MSRI, Berkeley, for (twice) providing the right climate to work in; this project was started (in 1994) and essentially ended there.
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