Review of Abelian l-adic Representations and Elliptic Curves
نویسنده
چکیده
Addison-Wesley has just reissued Serre’s 1968 treatise on l-adic representations in their Advanced Book Classics series. This circumstance presents a welcome excuse for writing about the subject, and for placing Serre’s book in a historical perspective. The theory of l-adic representations is an outgrowth of the study of abelian varieties in positive characteristic, which was initiated by Hasse and Deuring (see, e.g., [3], [1]) and continued in Weil’s 1948 treatise [12]. Over the complex field C, an abelian variety A of dimension g may be viewed as an (algebrizable) complex torus W/L, where L ≈ Z is a lattice in the Cvector space W of dimension g. The classical study of A relies heavily on the lattice L, which is intrinsically the first homology group H1(A,Z). However, the quotients L/nL (for n ≥ 1) have a purely algebraic definition. Indeed, over C the quotient L/nL is canonically the group
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