Modularity of the Rankin-selberg L-series, and Multiplicity One for Sl(2)
نویسندگان
چکیده
Relevant objects and the strategy 9 3.2. Weak to strong lifting, and the cuspidality criterion 13 3.3. Triple product L-functions: local factors and holomorphy 15 3.4. Boundedness in vertical strips 18 3.5. Modularity in the good case 30 3.6. A descent criterion 32 3.7. Modularity in the general case 35 4. Applications 37 4.1. A multiplicity one theorem for SL(2) 37 4.2. Some new functional equations 40 4.3. Root numbers and representations of orthogonal type 42 4.4. Triple product L-functions revisited 44 4.5. The Tate conjecture for 4-fold products of modular curves 47 Bibliography 52
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Basic Rankin-selberg
We present the simplest possible example of the Rankin-Selberg method, namely for a pair of holomorphic modular forms for SL(2,Z), treated independently in 1939 by Rankin and 1940 by Selberg. (Rankin has remarked that the general idea came from his advisor and mentor, Ingham.) We also recall a proof of the analytic continuation of the relevant Eisenstein series. That is, we consider the simples...
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