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Download e-book for iPad: Abelian l-adic representations and elliptic curves by Jean-Pierre Serre

By Jean-Pierre Serre

ISBN-10: 0201093847

ISBN-13: 9780201093841

This vintage e-book includes an creation to structures of l-adic representations, a subject matter of significant value in quantity conception and algebraic geometry, as mirrored via the excellent 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 with out advanced multiplication, the most results of that's that clone of the Galois team (in the corresponding l-adic illustration) is "large."

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N -1 and fi = i = 0, .. ,n (as power series). it, i Zi = Zi, Proof. 11). The conclusion of the proposition is equivalent to the assertion that F must be the identity map of C. By definition F is a complex analytic automorphism of C, that is, a projective transformation. 13) F 0 -1 IBmin(ro,ro). 15) for some nonzero complex number a. 15) we have lim wfo w-+O Since fo is of the form w+ (W) = 1. 16) implies a = 1. Thus F is the identity map of C. 4. 5. 4. 26 1. 17) is called a canonical sphere with tubes of type (1,0).

Though in this work we do not need that proof, the idea and the method used in that proof might be useful for further studies, especially of the maps E and E- 1 • Also, that proof involves some interesting formal calculus and is related to formal groups. So here we give the heuristic idea of that proof and leave the rigorization to the reader as an exercise. L. 00 3=30 F(z) = J (E:-1(a)) . ~l E·30 (a) Z3+1, gtz)dZ. 1. FORMAL POWER SERIES AND EXPONENTIALS OF DERIVATIONS 43 g (exp (Eio1(a):) F-l(y») j y y=F(",) = g(p-l(y + Eio1(a)))\y=F(",) = = exp (Eio1(a): ) y g(F-1(y»j y=F(",) = e1/(:C)g(:c).

I " (z). ml···mn ,A,ao ml···m .. t,{. )) = eA(' )a: d. 4). 2. From the above proposition we see that heuristically the composition (f~') )-' C~{. )) is generated by the "formal infinitesimal conformal transformations" This composition is a formal analogue of the right-hand side of the sewing equation or the "formal transition map" of a sphere with tubes sewn from two canonical spheres with tubes (see Chapters 1). This is the reason why we construct the "formal conformal transformations" from the "formal infinitesimal conformal transformations" in this way.

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Abelian l-adic representations and elliptic curves by Jean-Pierre Serre


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