By J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola

ISBN-10: 3540665463

ISBN-13: 9783540665465

This quantity comprises the elevated models of the lectures given via the authors on the C. I. M. E. educational convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers gathered listed below are large surveys of the present study within the mathematics of elliptic curves, and likewise comprise numerous new effects which can't be came across somewhere else within the literature. because of readability and magnificence of exposition, and to the history fabric explicitly integrated within the textual content or quoted within the references, the amount is easily fitted to learn scholars in addition to to senior mathematicians.

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**Extra resources for Arithmetic Theory of Elliptic Curves**

**Sample text**

Hence it follows that Im(XK)/Im(nK) is a finite group. On the other hand, GK has pcohomological dimension 1 because of the hypothesis that Gal(K/F,) contains an infinite pro-p subgroup. ) Thus if C is a divisible, pprimary GK-module, then the exact sequence 0 + C[p] + C 4 C + 0 induces the cohomology exact se, + H ~ ( K C[p]). , The last group is zero and quence H1(K, C ) 4 H ~ ( K C) hence H1(K, C ) is divisible. Applying this to C = C,, we see that Im(XK) is divisible and so Im(nK) = 1 m ( X ~ ) .

Suppose that F has only one prime lying over p and that this prime is totally ramified in F,/F. ) Then one can indeed recover Gal(Ln/Fn) from the A-module X . We have - The isomorphism is induced from the restriction map X + Gal(Ln/Fn). Here is a brief sketch of the proof: Gal(F,/Fn) is topologically generated by ypn; one verifies that (ypn - l ) X is the commutator subgroup of Gal(L,/Fn); and one proves that the maximal abelian extension of Fn contained in L, is precisely F,Ln. ) Then one notices that Gal(Ln/Fn) F Gal(F,L,/F,).

It depends on the Tate period q~ for E, which is defined by the equation j(qE) = jE, solving this equation for q~ E F c . , the image of q under the Kummer homomorphism M /(MX)pn -+ H1(M, ppn). 6 follows by just taking a direct limit. Still assuming that E has split, multiplicative reduction at v, the statement that ker(r,,) is finite is equivalent t o the assertion that ker(d,J Im(n,,). In this case, we show that I ker(r,,)l is bounded as n varies. For let = '~CF"(QE)I(F,)~E Gal((F,),/F,). Then we have '~C(F,,),,, ( Q E ) ~ ( F ~=) aen , .

### Arithmetic Theory of Elliptic Curves by J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola

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