In memory of my father Sundaram Ramakrishnan (SRK) all agree at all the places. (Recall that we needed to know that they were very nearly the same to get Theorem M in the first place.) This may be of independent interest.The final application of our main result is the proof of the Tate conjecture for 4-fold products V of modular curves, asserting in particular that the order of pole at s = 2 of the L-function over any solvable (normal) number field K of the Galois module W ℓ := H 4 et (V Q , Q ℓ ) equals the rank of the group of K-rational codimension 2 Tate cycles on V Q (see Theorem 4.5.1). Moreover we show, in line with the works of Ribet ([Ri1]) and V. K. Murty ([Mu]) on the Jacobians of modular curves, that the latter number can also be computed with the Tate cycles replaced by the algebraic cycles modulo homological equivalence if the level of at least one of the curves is square-free. We refer to Chapter 5 for a precise statement.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, Université 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.
Notation and preliminaries
2.1.Let Q denote the algebraic closure of Q in C. For any subfield K of Q, let Gal(Q/K) denote the Galois group of Q over K, together with the profinite topology. For any number field F with ring of integers O F , let Σ(F ) (resp. Σ ∞ (F ), resp. Σ 0 (F )) denote the places (resp. archimedean places,
