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

This quantity includes the extended types 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 amassed listed here are vast surveys of the present study within the mathematics of elliptic curves, and in addition include numerous new effects which can't be chanced on in different places within the literature. because of readability and magnificence of exposition, and to the heritage fabric explicitly integrated within the textual content or quoted within the references, the amount is easily suited for examine 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**

**Example text**

Now Bdiv has finite Zp-corank. It is clear that Thus, H1(rn,B) has order bounded by [B:Bdiv], which is independent of If we use the fact that B is finite, then ker(h,) has the same order as H"(rn,B), namely IE(Fn)pI. n. 2. Coker(h,) = 0. Proof. The sequence H1(F,, E[pw]) -+ H1(F,, ~ [ p , ] ) ~ --+ H 2 ( r n ,B) is exact, where B = H o(F,, E[pW]) again. But r, % H, is a free pro-p group. Hence H 2 ( r n ,B) = 0. Thus, h, is surjective as claimed. Let v be any prime of F. We will let v, denote any prime of F, lying over v.

Should be in Ax, where T L= (1 + T)-l - 1. The analogue of this statement is true for fE(T). 14. Assume that E is an elliptic curve defined over F with good, ordinary reduction or multiplicative reduction at all primes of F lying over p. Assume that SelE(F,), is A-cotorsion. Then the characteristic ideal of XE(Fm) is fied by the involution L of A induced by ~ ( y = ) y-' for all y E r. A proof of this result can be found in [Gr2] using the Duality Theorems of Poitou and Tate. There it is dealt with in a much more general context-that of Selmer groups attached to "ordinary" padic representations.

Now if one considers the A-module Y = A/( fi (T)ai),where f i (T) is irreducible in A, then Y/TY is infinite if and only if fi(T) is an associate of T. Therefore, if F is an imaginary quadratic field in which p splits and if F, is the cyclotomic Bpextension of F, then TI f (T), where f (T) is a generator of the characteristic ideal of X . One can prove that T2 I( f (T). (This is an interesting exercise. It is easy to show that X/TX has Zp-rank 1. One must then show that X/T2X also has Zp-rank 1.