John H. Coates, Kenneth A. Ribet, Ralph Greenberg, Karl's Arithmetic Theory of Elliptic Curves: Lectures given at the PDF

By John H. Coates, Kenneth A. Ribet, Ralph Greenberg, Karl Rubin (auth.), Carlo Viola (eds.)

ISBN-10: 3540481605

ISBN-13: 9783540481607

ISBN-10: 3540665463

ISBN-13: 9783540665465

This quantity comprises the accelerated models of the lectures given by means of the authors on the C.I.M.E. educational convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers gathered listed here are wide surveys of the present study within the mathematics of elliptic curves, and likewise include numerous new effects which can't be came upon in other places within the literature. because of readability and magnificence of exposition, and to the historical past fabric explicitly incorporated within the textual content or quoted within the references, the amount is easily fitted to study scholars in addition to to senior mathematicians.

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Extra info for Arithmetic Theory of Elliptic Curves: Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, July 12–19, 1997

Example text

Let Ho~ denote the maximal unramified extension of Fv which is contained in F~,w, and put M ~ = Hoo(#p~). Put G1 = G(Foo,w/M~), G2 = G ( M ~ / H ~ ) , G3 = G(H~/Fv). 8, we choose a basis of Tp(E) whose first element is a basis of Tp(Ev,po~). Then the representation p of Z ~ on Tp(E) has the form Iol : ~ a(o-)~ ' where z] : Zw --4 Z p is the character giving the action of 57w on A Tp(Ev,p~), Tp(Ev,p~). and c : Zw --+ Z p is the character giving the action of Zw on Now we first remark t h a t each of G1, G2 and G3 is the direct product of Z v with a finite abelian group of order prime to p, and is topologically generated by a single element.

8. Assume that (i) p ~> 5, (ii) S ( E / F ) is finite, and (iii) E has good ordinary reduction at all primes v of F dividing p. Then H i ( E , S(E/Foo)) is finite, and its order divides # ( H 3 ( Z , Ep~)). # ( C o k e r ( r (115) Proof. From (42), we have the exact sequence 0 ) S(E/Foo) ----+ HI(GT(Foo),Epr Im(AT(F~)) ) 0. (116) Taking 2:-cohomology, and recalling (95), we obtain the exact sequence H I ( G T ( F ~ ) , E p ~ ) n -~ Im(AT(F~)) E -+ H I ( ~ , 8(E/Foo)) ~ H 3 ( Z , Ep~), (117) where 6 is the obvious induced map.

Then H i ( E , S(E/Foo)) is finite, and its order divides # ( H 3 ( Z , Ep~)). # ( C o k e r ( r (115) Proof. From (42), we have the exact sequence 0 ) S(E/Foo) ----+ HI(GT(Foo),Epr Im(AT(F~)) ) 0. (116) Taking 2:-cohomology, and recalling (95), we obtain the exact sequence H I ( G T ( F ~ ) , E p ~ ) n -~ Im(AT(F~)) E -+ H I ( ~ , 8(E/Foo)) ~ H 3 ( Z , Ep~), (117) where 6 is the obvious induced map. 8 is now plain from (115), and its proof is complete. 9. Assume that (i) p ~> 5, (ii) S ( E / F ) is finite, (iii) E has good ordinary reduction at all primes v of F dividing p, and (iv) p is unramified in F.

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Arithmetic Theory of Elliptic Curves: Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, July 12–19, 1997 by John H. Coates, Kenneth A. Ribet, Ralph Greenberg, Karl Rubin (auth.), Carlo Viola (eds.)


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