By Jean-Pierre Serre

ISBN-10: 0201093847

ISBN-13: 9780201093841

This vintage publication includes an advent to structures of l-adic representations, a subject of serious value in quantity concept and algebraic geometry, as mirrored by means of the mind-blowing 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 is that a dead ringer for the Galois staff (in the corresponding l-adic illustration) is "large."

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**Example text**

G Nielsen constructed a special homeomorphism which is homotopic to f and plays the role of a “standard form” in the mapping class of f , [53, Sect. 14]. In this Chapter, we will construct a similar standard form, slightly different from Nielsen’s, and will show its essential uniqueness. 3]). 1 Definitions and Main Theorem of Chap. 1. Let A be an annulus and W Œ0; 1 S1 ! e. homeomorphism), where S 1 D R=Z. t; x/ 2 Œ0; 1 S 1; for some a; b 2 Q. A Y. M. 1007/978-3-642-22534-5 2, © Springer-Verlag Berlin Heidelberg 2011 17 18 2 Standard Form is a linear twist if f is a linear twist with respect to a certain parametrization W Œ0; 1 S 1 !

G C / W ˙g C ! ˙g C 48 2 Standard Form is isotopic to a periodic map. Then f jB W B ! B is also isotopic to a periodic map (Cf. A /; so we may assume that f j B is already periodic. s/ accordingly. Since f cyclically permutes the connected components of A . / , all the annuli contained in A . / are simultaneously non-amphidrome or amphidrome for each D 1; 2; : : : ; s. 4 (i) as the case may be, then f jA. / W A . / ! A. @ isotopic to a homeomorphism f 0jA . / W A. / ! A . / . / such that, for each annulus Aj in A.

A1 is a linear twist with respect to Similarly, if r 2, W Œ0; 1 S 1 ! A1 . f 0 /r j A2 W A2 ! f 0 /r j A2 W A2 ! A2 is a linear twist with respect to f W Œ0; 1 S 1 ! A2 ; t u and so on. This proves (i). Proof (of (ii) U NIQUENESS OF LINEARIZATION ). 2 to f r j A1 W A1 ! f 0 /r j A1 W A1 ! 1/ W A1 ! i / W Ai ! A ; @A / ! A ; @A / ! 2/ W A2 ! f j A1 / D f 0 j A1 . @ isotopic to 0Ä 1 W A2 ! A3 f 0 j A2 W A2 ! 3/ W A3 ! h1 / 1 D f 0 j A2 . i / W Ai ! h1 / D f 0 j Ai 1: W Ar ! f j A1 / W A1 ! h1 / 1 D f 0 j Ar W Ar !

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