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

This quantity comprises the extended models of the lectures given through the authors on the C.I.M.E. educational convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers accumulated listed here are vast surveys of the present examine within the mathematics of elliptic curves, and likewise include a number of new effects which can't be came upon somewhere else within the literature. due to readability and magnificence of exposition, and to the historical past fabric explicitly incorporated within the textual content or quoted within the references, the quantity is easily suited for learn scholars in addition to to senior mathematicians.

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**Example text**

And thus it is clear that the Galois group of F~,w over Hcr is isomorphic to Zp. If E does not have split multiplicative reduction at v, there exists a finite extension L of F~ such that E has split multiplicative reduction over L. But then our previous argument shows that the profinite degree of LFoo,~ over LHoo is divisible by p ~ , whence the same must be true for the profinite degree of F~,w over H ~ since L is of finite degree over Fv. 3. 4. Let p be a prime >1 5, and let v be a place o f F such that ordv(jE) >/ 0 and v does not divide p.

We are very grateful to Greenberg for pointing out to us that one can establish a first result in this direction using recent work of Hachimori and Matsuno [15]. Let K ~ denote the cyclotomic Zp-extension of K. Put r -- G ( K ~ / K ) , and let A(F) denote the Iwasawa algebra of F, We recall that S ( E / K ~ ) denotes the Selmer group of E over K ~ , and C ( E / K ~ ) denotes the Pontrjagin dual of S(E/Koo). 5. Let p be a prime number such that (i) p /> 5, (ii) 5: = G ( F ~ / F ) is a pro-p-group, (iii) E has good ordinary reduction at all primes v of F dividing p, and (iv) C(E/Koo) is a torsion A(r)-module and has p-invariant equal to O.

Denotes the kernel of multiplication by pn on A. Passing to the inductive limit as n -+ co, we obtain the surjection Hi(Fv, Ev,po~) , > Hi(Fv,Ev(~)) for all i/> 1. Hence the final assertion of the lemma will follow if we can show that Hi(Fv, E,,p-) -- 0 (76) for all i /> 2. This is automatic from cohomolog~cal dimension when i ~> 3. For i -- 2, we use the well known fact that E~,p. is its own orthogonal complement in the Weil pairing of Ep. x Ep. into #p.. Hence, by Tate local duality, H2(Fv, Ev,por is dual to H~ and this latter group is zero since only finitely many elements of Ev,p~ are rational over Fv.