Iwasawa Theory For Elliptic Curves With Complex Multiplication

Iwasawa theory for elliptic curves formulates a deep relation between the arith-metic object named compact Selmer group, and the analytic object named p-adic L-function. For the CM case, there are fairly rich results, especially Katz, Mannin-Vishik and others gave the construction of two-variable p-adic L-Functions. Yager formulated a relation between the p-adic L-function and certain Iwasawa module (local units mod-ulo elliptic units), after that Rubin completely proved the main conjecture of CM case, in which he actually used some primitive p-adic L-function. The p-adic L-function con-structed by Katz and others is not primitive, i.e. it only interpolates the special values of imprimitive Hecke L-functions in general. By the form of the main conjecture, and the proof of Rubin, we need a primitive p-adic L-function.In this thesis, we use a congruence about Eisenstein-Kronecker series discovered by Yager to construct a primitive p-adic L-function. Moreover we modify the definition of elliptic units, and get a result analogue to Yager's:The primitive p-adic L-function coincides with the characteristic element of some Iwasawa module (local units modulo modified elliptic units). We also prove that theμ-invariant of the primitive p-adic L-function is zero, so that by the main conjecture we get some important information of the arithmetic object attached with the elliptic curve:certain compact Selmer group is a finitely generated Zp-module. The thesis has eleven chapters, see below for the details:Chapter 0 is an introduction. We give some preparation and present the main results of the thesis.Chapter 1 is about notations. We give more preparation and notations in detail, together with some classical results including elliptic curves, formal groups and class field theory.In Chapter 2 we first recall the definition and some properties of Coleman power series, and prove Theorem 2.2 which is crucial in the construction of the p-adic L-function. After that we introduce a two-variable logarithmic derivative map, and study its properties. In Chapter 3 we define the group of elliptic units, and study its properties. First-ly we introduce a basic rational function, after a suitable modification we study the functional equations, and evaluated at some torsion points to get the elliptic units, we deal with the different conductors of any power of Grossencharacter in the same way, which enable us to construct the primitive p-adic L-function. At last we give the corresponding Coleman power series associated to the elliptic units.In Chapter 4 we first formulate the relation between the Eisenstein-Kronecker series and Hecke L-functions, and relate the Eisenstein-Kronecker series and the higher logarithmic derivative maps of the rational functions defined in Chapter 3. Then we introduce the p-adic period and an important congruence between the Eisenstein-Kronecker series discovered by Yager.In Chapter 5 we relate the higher logarithmic derivative maps of the elliptic units with the two-variable power series defined in Chapter 2 by the F-transform of a p-adic measure. We define two important A-homomorphisms from the groups of local units to the ring of two-variable power series, and introduce their relationship.In Chapter 6 we prove the first main result:Theorem 0.1. We use the congruence between the Eisenstein-Kronecker series to get the primitive p-adic L-Function, and study the relations with elliptic units.In Chapter 7 we prove the second main result:Theorem 0.2. We study the struc-ture of the Iwasawa module (local units modulo modified elliptic units), and prove completely that the primitive p-adic L-function coincides with the characteristic ele-ment of the Iwasawa module.Chapter 8 is devoted to study some properties of two-variable p-adic measures.In Chapter 9 we prove the third main result:Theorem 0.3. We simplify the problem to one-variable F-transform by using the tools from Chapter 8. Then we investigate the information about poles and residues to prove that theμ-invariant of the primitive p-adic L-function is zero. By the main conjecture, we get some important information about the arithmetic properties of the elliptic curve.In Chapter 10 we describe some open problems related to theμ-invariant. We first introduce the non-commutative Iwasawa theory and the mH(G)-conjecture, then using the results in Chapter 9 we relate the mH(G)-conjecture to another cyclotomicμ-conjecture, and give some ideas for the future work.