Arithmetic Theory of Elliptic Curves: Lectures given at the by J. Coates

By J. Coates

This quantity comprises the accelerated 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 huge surveys of the present study within the mathematics of elliptic curves, and likewise include numerous new effects which can't be discovered somewhere else within the literature. due to readability and style of exposition, and to the heritage fabric explicitly integrated within the textual content or quoted within the references, the quantity is definitely fitted to learn scholars in addition to to senior mathematicians.

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Hence G K . , v 11. ) = IE(F,),I, which in turn equals (B,/(y, - l)BuI. Hence I ker(r,)l = c$)' when E has additive reduction a t v. (It is known that c, 5 4 when E has additive reduction at v. ) Now assume that E has split, multiplicative reduction at v. (g$)) = -ordU(jE), where denotes the Tate period for E at v. Now a,, is clearly surjective. Hence ( ker(r,, ) 1 = [ ker(a,,) 1 . I ker(b,,) 1. 5, we have I ker(a,,)I = ~,@(f,,),l. 2, just the boundedness of J ker(a,,)l (and of I ker(b,,)J) suffices.

For if go is a topological generator of A x H , then the torsion subgroup of X/(go - $(go))X is isomorphic to the kernel of go -$(go) acting on At/X 2 W. ) But this in turn is isomorphic to W/(go - $(go))W, whose dual is easily identified with HorncMq(8, (I),($,/H,)(+)). We have attempted to give a rather self-contained "Iwasawa-theoretic" approach to studying the above local Galois cohomology group. 2. But using results of Poitou and Tate is often Let T easier and more effective. We will illustrate this.

Iwasawa found examples of Zp-extensions F,/F where p(F,/F) > 0. In his examples there are infinitely many primes of F which decompose completely in F,/F. In these lectures, we will concentrate on the "~yclotomic &,-extensionv of F which is defined as the unique subfield F, of F(pp) with r = Gal(F,/F) % Z,. Here pPm denotes the ppower roots of unity. It ~ t ( ~ ~ ) ~ ~ I 53 is easy to show that all nonarchimedean primes of F are finitely decomposed in F,/F. More precisely, if v is any such prime of F, then the corresponding decomposition subgroup r ( v ) of I' is of finite index.

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