By Walter J Savitch

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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,).