Introduction to iwasawa theory for elliptic curves pdf

Mcconnel, iwasawa theory of modular elliptic curves of analytic rank at most 1, j. Several of the contributions in this volume were presented at the conference elliptic curves, modular forms and iwasawa theory, held in honour of the 70th birthday of john coates in cambridge, march 2527, 2015. Introduction the iwasawa main conjectures for elliptic curves provide a scope to study birch and swinnertondyer conjecture. Fragments of the gl 2 iwasawa theory of elliptic curves without complex multiplication. The main conjecture of iwasawa theory for elliptic curves. An introduction to the theory of elliptic curves outline introduction elliptic curves the geometry of elliptic curves the algebra of elliptic curves what does ek look like. An introduction to iwasawa theory for elliptic curves. Elliptic curves, the geometry of elliptic curves, the algebra of elliptic curves, elliptic curves over finite fields, the elliptic curve discrete logarithm problem, height functions, canonical heights on elliptic curves, factorization using elliptic curves, lseries. We study the iwasawa theory of a cm elliptic curve e in the anticyclotomic zpextension of the cm.

Elliptic curves, the geometry of elliptic curves, the algebra of elliptic curves, elliptic curves over finite fields, the elliptic curve discrete logarithm problem, height functions, canonical heights on elliptic curves, factorization using elliptic curves, lseries, birchswinnertondyer. Iwasawa theory of elliptic curves with complex multiplication. Then we will move on to elliptic curves, treating the contents of greenbergs introduction to iwasawa theory for elliptic curves. Introduction let eq be an elliptic curve of squarefree level n. Similar formulae appear in khares work 21 on establishing isomorphisms between deformation. The problem given an elliptic curve e, understand how the mordellweil group ef varies as f varies. In the early 1970s, barry mazur considered generalizations of iwasawa theory to abelian varieties. Iwasawa theory of elliptic curves at supersingular primes over z pextensions of number fields adrian iovita and robert pollack june 21, 2005 abstract in this paper, we make a. We illustrate our general theory by concrete examples of such modules arising from the iwasawa theory of elliptic curves without.

From classical to noncommutative iwasawa theory an introduction to the gl 2 main conjecture otmar venjakob this paper, which is an extended version of my talk the gl 2 main conjecture for elliptic curves without complex multiplication given on the 4ecm, aims to give a survey on recent developments in noncommutative iwasawa theory. The main di erence between the present paper and previous works such as 15 is the development of the plusminus iwasawa theory for a cm elliptic. Introduction to elliptic curves part 1 of 8 duration. In number theory, iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. We study this subject by first proving that the pprimary subgroup of the classical selmer group for an elliptic curve with good, ordinary reduction at a prime p has a very simple and elegant description which involves just the galois module of ppower torsion points. This will then allow us to derive iwasawas theorem on the behaviour of the ppart of the class number in azpextension of a number. Iwasawa theory 2012 state of the art and recent advances.

Introduction in this paper we prove the iwasawagreenberg main conjecture for a large class of elliptic curves and modular forms. Iwasawa theory of elliptic curves with complex multiplication anna seigal 2nd may 2014 contents. The main motivation for the present paper is to develop algorithms using iwasawa theory, in order toenable veri. Using this theory, it is possible to describe the growth of the size of the ppart of selmer groups of abelian varieties in z ptowers. We then prove theorems of mazur, schneider, and perrinriou on the basis of this description. We present the rst few sections of greenbergs article \introduction to iwasawa theory for elliptic curves. An introduction to the theory of elliptic curves pdf 104p covered topics are. Introduction to iwasawa theory yi ouyang department of mathematical sciences tsinghua university.

Whilst much is known about the iwasawa theory of efor primes of ordinary reduction, the same is unfortunately not true of iwasawa theory at supersingular primes, for in this case the iwasawa modules that one naturally considers are not torsion, and. A clear yet general exposition of this theory is presented in this book. We present the first few sections of greenbergs article introduction to iwasawa. Chapter 1 modules up to pseudoisomorphism let abe a commutative noetherian integrally closed domain. It began as a galois module theory of ideal class groups, initiated by kenkichi iwasawa, as part of the theory of cyclotomic fields. The topics that we will discuss have their origin in mazurs. Iwasawa theory of elliptic curves and bsd in rank zero. The original motivations for considering the selmer groups of elliptic curves along a zpextension are however quite different from what has been suggested above. However, it is still not known how to extend these iwasawatheoretic arguments to the prime p 2. Springer new york berlin heidelberg hong kong london milan paris tokyo. This will then allow us to derive iwasawa s theorem on the behaviour of the ppart of the class number in azpextension of a number. The intention is to give an overview of some topics in.

The discrete logarithm problem fix a group g and an element g 2 g. Following a chapter on formal groups and local units, the padic l functions of maninvishik and katz are constructed and studied. Speed through some background on elliptic curves chapter 1. Later this would be generalized to elliptic curves. Topics in iwasawa theory ralph greenberg december 15, 2006 1 ideal class groups. Elliptic curves over finite fields the elliptic curve discrete logarithm problem reduction modulo p, lifting, and height functions canonical heights on elliptic curves. An introduction to the theory of elliptic curves pdf 104p. This article is dedicated to the memory of kenkichi iwasawa, who passed away on october 26th, 1998. Pdf download iwasawa theory 2012 free unquote books. Jk divcpc where divc is the group of divisors of degree 0. We study this subject by first proving that the pprimary subgroup of the classical selmer group for an elliptic curve.

The mordellweil theorem says the abelian group of rational points on an elliptic curve over q is finitely generated. Iwasawa theory elliptic curves with complex multiplication. On the algebraic side, assuming that e has good ordinary reduction at p, it is. Noncommutative iwasawa theory of elliptic curves at primes of multiplicative reduction. Given an elliptic curve e over qand a prime p, the goal or main conjecture of this program is to relate a padic lfunction attached to e with a selmer group, which contains information. In fact, the growth pattern is very similar to the classical case.

Iwasawa theory of elliptic curves with complex multiplication, by. Iwasawa theory of elliptic curves at supersingular primes. Greenberg, kummer theory for abelian varieties over local fields, invent. Criticalslope padic lfunctions by david hansen june 15, 2016. In general, the explicit determination of h f, let alone the structure of cl f as a. Iwasawa theory of elliptic curves and bsd in rank zero jordan schettler classical theory for number fields theory for elliptic curves application to a special case of bsd three concrete examples connection between growth formula and x it turns out that x is a. It has been an object of intense study since the nineteenth. It is an historical introduction to the basic ideas of this subject going back to the first papers of iwasawa, various versions of the main conjecture, etc introduction to iwasawa theory for elliptic curves. Introduction iwasawa theory gained a place in the study of elliptic curves in the 1970. Also, a more appropriate name for our theorem would be \a conjecture analogous to the main conjecture of iwasawa theory because the plusminus iwasawa theory is a relatively new eld.

Introduction to elliptic curves to be able to consider the set of points of a curve cknot only over kbut over all extensionsofk. Kwokwing tsois homepage iwasawa theory of cm elliptic. These are a preliminary set ot notes for the authors lectures for the 2018 arizona winter school on iwasawa theory. Noncommutative iwasawa theory of elliptic curves at. An elliptic curve ef has complex multiplication if. Algorithms for the arithmetic of elliptic curves using. Roe, elliptic operators, topology and asymptotic methods, cambridge univ. Anticyclotomic iwasawa theory of cm elliptic curves adebisi agboola and benjamin howard with an appendix by karl rubin abstract.

The study of elliptic curves over this tower has been undertaken by a number of authors, notably in hv from the algebraic point of view of descent, and iwasawa theory and in dov from the analytic point of view of lfunctions and root numbers. Second lecture control theorem for curves with good ordinary reduction. In the last fifteen years the iwasawa theory has been applied with remarkable success to elliptic curves with complex multiplication. The iwasawa main conjecture for elliptic curves at odd. Here i will mostly follow the approach taken in the iwasawa theory of elliptic. Recall basics about the arithmetic of elliptic curves. Lectures on the iwasawa theory of elliptic curves christopher skinner abstract. Then we will move on to elliptic curves, treating the contents of greenbergs introduction to.