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Abelian L'Adic Representations and Elliptic Curves (Advanced Book Classics): Jean Pierre Serre

The "l-adic representations" considered in this book are the algebraic analogue of the locally constant sheaves (or "local coefficients" of Topology). A typical example is given by the l^n_th division pints of abelian varieties (cf Chap. I., 1.2); the correspnding l-adic spaces, first introduced by Weil [40] are one of our main tools in the study of these varieties. Even the case of dimension 1 presents non trivial problems; some of them will be studied in Chap. IV .

The general notion of an l-adic representation was first defined by Taniyama [35] (see also the review of this paper given by Weil in Math. Rev., 20, 1959, rev. 1667). He showed how one can relate L-adic representations relative to different prime numbers l via the properties of the Frobenius elements (see below) . In the same paper, Taniyama also stdied some abelian representations which are closely related to complex multiplication (cf. Weil [41], [42] and Shimura-Taniyama [34]). These abelian representations, together with some applications to elliptic curves, are the subject matter of this book.

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