Download Arithmetic theory of elliptic curves: lectures given at the by J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola PDF

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

This quantity comprises the multiplied models of the lectures given by means of the authors on the C. I. M. E. educational convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers gathered listed below are huge surveys of the present study within the mathematics of elliptic curves, and likewise comprise numerous new effects which can't be stumbled on in other places 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 amount is easily suited for learn scholars in addition to to senior mathematicians.

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3 by studying the structure of X as a module for +,[[A x HI] = A[A], where A = B,[[H]] E Z,[[T]], with = h - 1. The results are due to Iwasawa. 66 Iwasawa theory for elliptic curves Ralph Greenberg > MZ. For any n 0, let Hn = H P ~Let . M, = The commutator subgroup of Gal(L,/M,) is (hpn - 1 ) X and so, if L, is the maximal abelian extension of Mn contained in L,, then Gal(L,/M,) 2 Hn x (x/(hpn - 1)X). But L, is the maximal abelian pro-p extension of M, and, by local class field theory, this Galois group is isomorphic to Z,[Mn:Qpl+l x W,, where W, denotes the group of ppower roots of unity contained in M,.

3-20 and 35 (1971), 731-737. -P. Serre, La distribution d'Euler-Poincari d'un groupe profini, to appear. -P. Serre, J. Tate, Good reduction of abelian varieties, Ann. of Math. 88 (1968), 492-517. -[32] - J. Silverman, The arithmetic of elliptic curves, Graduate Texts in Math. 106 (1986), Springer. [33] K. Wingberg, On Poincare' groups, J. London Math. Soc. 33 (1986), 271-278. Iwasawa Theory for Elliptic Curves Ralph Greenberg University of Washington 1. Introduction The topics that we will discuss have their origin in Mazur's synthesis of the theory of elliptic curves and Iwasawa's theory of Pp-extensions in the early 1970s.

We let denote the reduction of E at v. Then we have where y,, is a topological generator of Gal((F,),/(F,),,). Now E(f,), is finite and the kernel and cokernel of y,, - 1 have the same order, namely This is the order of ker(d,). 4. follows. IP( 1 Let COdenote the finite set of nonarchimedean primes of F which either lie over p or where E has bad reduction. Eo and v, is a prime of F, lying over v, then ker(r,,) = 0. 4 show that 1 ker(r,,)) is bounded as n varies. The number of primes v, of F,, lying over any nonarchimedean prime v is also bounded.

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