Download Iwasawa theory of elliptic curves with complex by Ehud De Shalit PDF

By Ehud De Shalit

Within the final fifteen years the Iwasawa concept has been utilized with impressive good fortune to elliptic curves with complicated multiplication. a transparent but basic exposition of this thought is gifted during this book.

Following a bankruptcy on formal teams and native devices, the p-adic L services of Manin-Vishik and Katz are built and studied. within the 3rd bankruptcy their relation to classification box idea is mentioned, and the functions to the conjecture of Birch and Swinnerton-Dyer are handled in bankruptcy four. complete proofs of 2 theorems of Coates-Wiles and of Greenberg also are provided during this bankruptcy which can, furthermore, be used as an creation to the newer paintings of Rubin.

The publication is basically self-contained and assumes familiarity purely with primary fabric from algebraic quantity concept and the speculation of elliptic curves. a few effects are new and others are awarded with new proofs.

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1-2, its dimension must be finite. In other words, the structure of V* as a right vector-space over K is defined by the formula (8) (av,v*)v = (v, v* a)v. Conversely, if V and V* are dual groups, and V* has a structure of right vector-space over K, (8) may be used in order to define V as a left vectorspace over K. Thus we may still identify V with the dual of V* when their structures as vector-spaces over K are taken into account. If L is any closed subgroup of V, the subgroup L* of V* associated with L by duality consists of the elements v* of V* such that (v, v*)v = 1 for all VEL; in view of (8), this implies that, if L is a left module for some subring of K, L* is a right module for the same subring, and conversely.

V d } of Kover Qp such that R = I Zp Vi. By (3), we have now, for 1 ~ i~d, v~ 1, aiEZp: and therefore: (4) n (l+p2 vi)pV-l d ai =1+pV+I i= I d I aivi (pv+2R). avivi i with av;EZp for 1 ~ i ~d, and then 1 + Xv + I = (1 + xv) n(1 + p2 Vi) i It is now clear that we have (5) 1+ Xl = n(1 + P2Vi)b i 4· i p v-I avi • Lattices and duality over local fields 34 II where the bi are given, for 1 ~ i ~ d, by +00 b i = L pv-l avi · v= 1 This shows that, as a multiplicative Zp-module, the group 1 + p2 R is generated by the d elements 1 + p2 Vi; as it is an open subgroup of the compact group 1 + P, hence of finite index in 1 + P, and as 1 + P, as a Zp-module, is generated by the elements 1 + p2 Vi and by a full set of representatives of the classes modulo 1 + p2 R in 1 + P, this implies that 1 +P is finitely generated.

The concept of lattice, as developed for p-fields in §§ 1-2, cannot be applied to R-fields. The appropriate concept is here as follows: DEFINITION 3. By an R-lattice in a vector-space V of finite dimension over an R-field, we understand a discrete subgroup L of V such that VIL is compact. We have to recall here some elementary facts about discrete subgroups. Let G be a topological group, r a discrete subgroup of G, and

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