By Jean-Pierre Serre

This vintage booklet includes an creation to structures of l-adic representations, an issue of serious value in quantity concept and algebraic geometry, as mirrored through the fabulous contemporary advancements at the Taniyama-Weil conjecture and Fermat's final Theorem. The preliminary chapters are dedicated to the Abelian case (complex multiplication), the place one unearths a pleasant correspondence among the l-adic representations and the linear representations of a few algebraic teams (now referred to as Taniyama groups). The final bankruptcy handles the case of elliptic curves without advanced multiplication, the most results of that's that identical to the Galois team (in the corresponding l-adic illustration) is "large."

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**Example text**

He probably observed that two pseudo-cycles, ^ S, intersect in a pseudo-cycle; but there for the same imbedding K no indication that he considered the question of the topological invariance of the resulting ring. This was proved first in a paper 1935. e. without using also presented a definition of a cupAlexander imbeddings K<^S). under product. It was not satisfactory since it did not correspond intrinsic definitions of the NORMAN 36 E. STEENROD duality to the intersection of pseudo-cycles. It deviated from this by a numerical factor.

Of Math. (2), 28: 342-354. The a complex on a manifold and related questions. Proc. , 13: 614-622, 805-807. [39] On the functional independence of ratios of theta functions. Proc. Nat. Acad. Sci. , 13: 657-659. [38] residual set of Nat. Acad. Sci. 1928 [40] Transcendental theory ; Singular correspondences between algebraic curves ; Hyperelliptic surfaces and Abelian varieties. Chap. 15-17, p. 310-395, vol. 1, of Selected Topics in Algebraic Geometry; Report of the Com- mittee on Rational Transformations of the National Research Council.

Two oriented (/-cells f:ar->X and g:r->X are called equivalent if there exists a 1-1 bary centric map h:or-*T preserving orientation such that gh =/. He took the free group generated by the oriented singular #-cells and reduced by the relations which identified the negative of an oriented singular cell with the oppositely cell. The resulting group was the singular chain boundary operator d:Cq (X)->Cq_ l (X) was defined by specifying its values on the generating cells in the obvious manner. oriented singular group The Cq (X).