Chapter 1 modules up to pseudoisomorphism let abe a commutative noetherian integrally closed domain. E we where on the righthand side, he is the order of the tateshafarevich group xe of e which is an elliptic curve analogue of the ideal class. Elliptic curves and iwasawa theory uci mathematics. Lectures on the iwasawa theory of elliptic curves christopher skinner abstract. Broadly, i study the arithmetic of elliptic curves.
Later this would be generalized to elliptic curves. Iwasawa theory for elliptic curves and bsdp algant. Then we consider the iwasawa theory of class groups of abelian extensions of f. Among all these, elliptic curves provide an example in which the elementary request of.
For simplicity, we shall take our motive always to be an elliptic curve e over q. Given an elliptic curve e, understand how the mordellweil group ef varies as f varies. In this seminar we will focus on the classical case of zpextensions. This article is dedicated to the memory of kenkichi iwasawa, who passed away on october 26th, 1998. Speed through some background on elliptic curves chapter 1. In 1970s, mazur formulated the iwasawa theory for elliptic curves at good ordinary primes. Introduction an important problem in number theory is to study.
Letuscheckthisinthecase a 1 a 3 a 2 0 andchark6 2,3. In this paper we complete rubins partial verication of the con jecture for a large class of elliptic curves with complex. Algorithms for the arithmetic of elliptic curves using. In fact, the growth pattern is very similar to the classical case. We present the first few sections of greenbergs article introduction to iwasawa. The smallest integer m satisfying h gm is called the logarithm or index of h with respect to g, and is denoted. Introduction the iwasawa main conjectures for elliptic curves provide a scope to study birch and swinnertondyer conjecture. A gentle introduction to elliptic curve cryptography. These are a preliminary set ot notes for the authors lectures for the 2018 arizona winter school on iwasawa theory. Iwasawa theory of elliptic curves with complex multiplication anna seigal 2nd may 2014 contents 1 introduction 2 1. Then we will move on to elliptic curves, treating the contents of greenbergs introduction to. We study the iwasawa theory of a cm elliptic curve e in the anticyclotomic zpextension of the cm. Iwasawa theory of elliptic curves and bsd in rank zero, pdf.
Our four lectures will give a very brief introduction to the noncommutative theory. On the anticyclotomic iwasawa theory of rational elliptic curves at eisenstein primes 3 our most complete results towards conjecturesaandbare proved under the additional hypothesis that sel the z pcorank of sel p1ek is 1. Then we will move on to elliptic curves, treating the contents of greenbergs introduction to iwasawa theory for elliptic curves. A gentle introduction to elliptic curve cryptography je rey l. Noncommutative iwasawa theory of elliptic curves at. An excellent ve page introduction to this material is the introduction to shari s notes on iwasawa theory, available at iwasawa. The galois group is known explicitly from the isomorphism. Sujatha to parimala on the occasion of her sixtieth birthday 1. Let eq be an elliptic curve, and pa prime where ehas good reduction, and assume that e admits a rational pisogeny. However there is more than one case of the elliptic curve version.
Some references john coates and ramdorai sujatha, galois cohomology of elliptic curves, tata institute of fundamental research lectures on mathematics, vol. 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 curve eover an abelian extension f of the imaginary quadratic eld k. We study this subject by first proving that the pprimary subgroup of the classical selmer group for an elliptic curve. In order to explain the iwasawa theory of elliptic curves we. The dual selmer group x e is torsion over q and the characteristic ideal of x e is generated by l e as. May 09, 2015 iwasawa theory for elliptic curves at supersingular primes. Introduction to elliptic curves part 1 of 8 youtube. Euler systems, iwasawa theory, and selmer groups introduction. Introduction to elliptic curves to be able to consider the set of points of a curve cknot only over kbut over all extensionsofk.
Iwasawa theory for elliptic curves with complex multiplication. Voting systems, mass murder, and the enigma machine, pdf. It is a branch of mathematics, at the intersection of number theory, algebra, arithmetic algebraic geometry and. The iwasawa main conjecture for supersingular elliptic curves is conjecture 1. Pdf arithmetic theory of elliptic curves pp 51144 cite as. Let kq be a nite extension and k 1k a z pextension with layers k n. 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. Let eq be an elliptic curve and pan odd supersingular prime for e. Introduction eighty years have passed since mordell proved that the mordellweil group of rational points on an elliptic curve e is. For another example of iwasawa main conjecture, take e an elliptic curve over q and. Noncommutative iwasawa theory of elliptic curves at primes. Guo, on a generalization of tate dualities with application to iwasawa theory, compositio math. In 10 and 17, asymptotic formulas for the size of xeq np1 have been established where q n runs through the cyclotomic z pextension of q.
Iwasawa theory for elliptic curves over imaginary quadratic. We then prove theorems of mazur, schneider, and perrinriou on the basis of this description. Moreover, iwasawa theory is a comparatively technical subject. It is also the story of alice and bob, their shady friends, their numerous and crafty enemies, and. Greenberg, iwasawa theory for motives, lms lecture notes series 153 1991, 211233. Noncommutative iwasawa theory of elliptic curves at primes of multiplicative reduction by chernyang lee the school of mathematical sciences, the university of nottingham, nottingham ng72rd. Additional lecture use of sage for computational iwasawa theory of elliptic curves. Roe, elliptic operators, topology and asymptotic methods, cambridge univ. Iwasawa theory of elliptic curves at supersingular primes. When the complex lfunction of e vanishes to even order, rubins proof the two variable main conjecture of iwasawa theory implies that.
We have tensored the usual iwasawa algebras with qp. This textbook covers the basic properties of elliptic curves and modular forms, with emphasis on certain connections with number theory. Pdf download elliptic curves number theory and cryptography discrete. Iwasawa theory and generalizations international congress of. Iwasawa theory was introduced into the study of the arithmetic of elliptic curves by mazur in the 1970s. Then we discuss some recent progresses in the proof of the corresponding iwasawa main conjectures formulated by kato conjecture 4. Recently, pollack and rubin informed the author that the above main conjecture is true for cm elliptic curves 19.
Nichifor, iwasawa theory for elliptic curves with cyclic isogenies, university of washington, ph. Classical iwasawa theory is concerned with the case f1 q p1 p1 the group of all ppower roots of unity when g z p is abelian. The main motivation for the present paper is to develop algorithms using iwasawa theory, in order toenable veri. By way of the most interesting example, let e be an elliptic curve over f with ordinary reduction at p, and let ep. Iwasawa theory for elliptic curves at supersingular primes 3 for the more precise statement of the theorem, see sect. Iwasawa theory and generalizations 339 let e be an elliptic curve over a number. Iwasawa theory for elliptic curves over imaginary quadratic fields par massimo bertolini.
An introduction to iwasawa theory for elliptic curves, part ii thursday, september 1 math 215 3. We will focus here ontheonevariablemainconjectureandcontinuetorestricttothesimplestsetup of. Contents 1 modules up to pseudoisomorphism 1 2 iwasawa modules 7 3 z pextensions 14 4 iwasawa theory of elliptic curves 21 0. Iwasawa main conjecture for supersingular elliptic curves. The ancient congruent number problem is the central motivating example for most of the book. This paper is devoted to the study of the amodule structure of the above groups, when e is an elliptic curve defined over the rationals and k. Iwasawa theory 2012 state of the art and recent advances. 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 characteristic power series c divides the padic lfunction l pe,k. Given an elliptic curve e over qand a prime p there are two parts to such a program. Number theory seminar edray goins purdue university an introduction to iwasawa theory for elliptic curves, part ii thursday, september 1 math 215 3. An introduction to the theory of elliptic curves the discrete logarithm problem fix a group g and an element g 2 g. Anticyclotomic iwasawa theory of cm elliptic curves arxiv.
Jk divcpc where divc is the group of divisors of degree 0. In the past, the plusminus iwasawa theory was mostly limited to elliptic. Iwasawa theory for elliptic curves at supersingular primes core. We will construct the statement of the main conjecture. Kato, padic hodge theory and values of zeta functions of modular forms. Topics in number theory introduction to iwasawa theory david burns giving a onelecture introduction to iwasawa theory is an unpossibly dif. The main conjecture of iwasawa theory for elliptic curves. Introduction the theory of diophantine equation is about.
Iwasawa theory of elliptic curves with complex multiplication. Introduction to iwasawa theory yi ouyang department of mathematical sciences tsinghua university. Euler systems, iwasawa theory, and selmer groups 317 qiqcdp is a finite product of fields, and hence hq. This is an introduction to iwasawa theory and its generalizations. We present the rst few sections of greenbergs article \ introduction to iwasawa theory for elliptic curves. Introduction a fundamental problem in algebraic number theory concerns the study of the absolute galois group of the. This is an overview of the theory of elliptic curves, discussing the mordellweil theorem, how t. Iwasawa theory of elliptic curves with complex multiplication, by. We then prove theorems of mazur, schneider, and perrin. This results in a more transparent definition of the muinv.
Vatsal, on the iwasawa invariants of elliptic curves, in preparation. Cv and bibliography karl rubin home uci mathematics. Brauers theorems and the meromorphicity of lfunctions, pdf. Introduction the iwasawa theory for an elliptic curve over a z pextension of a number. Let il be a number field, and let e be an elliptic curve defined over il. 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. Anticyclotomic iwasawa theory of cm elliptic curves adebisi agboola and benjamin howard with an appendix by karl rubin abstract. We will include here the ps or pdf files for various papers and expository articles as. 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. The theory of diophantine equation is about finding integral solution to algebraic equations.
Selmer group of the elliptic curve e over a padic lie extension f. We will impose the following hypothesis on the splitting type of the prime pin k 1. Each talk will be 1 hour long, followed by 15 minutes for questions and chatting about the talk. Denote by the iwasawa algebra z pgalk 1k and let n galk 1k n. Topics in number theory introduction to iwasawa theory.
Washington, introduction to cyclotomic fields, grad. An introduction to the theory of elliptic curves pdf 104p covered topics are. This article is intended to be a fairly selfcontained introduction to some of the. Seminar on iwasawa theory of elliptic curves gabor wiese sommersemester 2008 abstract iwasawa theory studies arithmetic objects in certain padic towers of number.
The intention is to give an overview of some topics in. This requires the introductionofafewmoreconcepts,mostimportantlyastructuretheoremforatypeof. E we where on the righthand side, he is the order of the tateshafarevich group xe of e which is an elliptic curve analogue of the ideal class group and is conjectured to be. Introduction to iwasawa theory for elliptic curves university of. Iwasawa theory for elliptic curves at supersingular primes. Just as for the classical case of iwasawa theory, this case 1.
However, it is still not known how to extend these iwasawa theoretic arguments to the prime p 2, whereas our elementary arguments work well for p 2. Assumption 6 on e,k,p given at the end of this introduction theorem 1 below, a weak form of the main conjecture of iwasawa theory for elliptic curves in the ordinary and anticyclotomic setting. The proof of this result relies on iwasawa theoretic techniques. We also obtain an asymptotic formula for the padic order of the tateshafarevich group x ef. Chapter 1 modules up to pseudoisomorphism let abe a commutative noetherian integrally closed. Introduction kolyvagin discovered the method of euler system, and used it to analyze ideal class groups of certain cyclotomic fields and selmer groups of elliptic curves. The above groups are equipped with a natural structure of discrete amodules.
Kurihara, on the tate shafarevich groups over cyclotomic. 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. Kobayashi, iwasawa theory for elliptic curves at supersingular primes, invent. Iwasawa theory of elliptic curves the philosophy 2. For the remainder of this introduction, e will denote an elliptic curve defined over. In this paper we give an overview of some aspects of iwasawa theory for modular forms. Introduction to iwasawa theory yi ouyang department of mathematical sciences tsinghua university beijing, china 84.
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