By J. Coates

This quantity includes the improved types of the lectures given via the authors on the C. I. M. E. tutorial convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers accrued listed below are large surveys of the present examine within the mathematics of elliptic curves, and in addition comprise numerous new effects which can't be stumbled on in different places within the literature. due to readability and style of exposition, and to the history fabric explicitly incorporated within the textual content or quoted within the references, the amount is definitely suited for learn scholars in addition to to senior mathematicians.

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**Example text**

Also, the value of the number of roots of fE(T) of the form C - 1, where is a ppower root of unity, if we assume in addition the finiteness of UIE(Fn), for all n. 12 would imply that this number is equal to the Z,-rank of X~(Foo)/enx~(Fm for) n >> 0. In section 4 we will introduce some theorems due to B. Perrin-Riou and to P. Schneider which give a precise relationship between SelE(F), and the behavior of ~ E ( T at ) T = 0. These theorems are important because they allow one to study the Birch and Swinnerton-Dyer conjecture by using the so-called "Main Conjecture" which states that one can choose the generator ~ E ( T SO ) that it satisfies a certain interpolation property.

Under the above assumptions, one has 2 T(E,F ) . This result is due to P. Schneider. He conjectures that equality should hold here. 3, where one assumes just that E has potentially ordinary or potentially multiplicative reduction at all primes of F lying over p. : (The ring A/enA is for n 2 0. ) One uses the fact that there is a pseudo-isomorphism from XE(F,) to A' @ Y, where T = rankA(xE(F,)), which is the A-corank of SelE(F,),, and Y is the A-torsion submodule of XE(F,). However, it 'is reasonable to make the following conjecture.

To study ker(g,), we focus on each factor in PE(F,) by considering where q is any prime of F, lying above v,. , F, = K,. Thus, ker(rUn) = 0. For nonarchimedean v, we consider separately v 1 p and v 1 p. 3. Suppose v is a nonarchimedean prime not dividing p. Then ker(rvn) is finite and has bounded order as n varies. If E has good reduction at v, then ker(rvn)= 0 for all n. Proof. Let Bv = H"(K, E[pm]), where K = (F,),. is unramified and finitely decomposed in F,/F, K is the unramified Z,extension of Fv (in fact, the only +,-extension of F,).