An introduction to the theory of elliptic curves the discrete logarithm problem fix a group g and an element g 2 g. The iwasawa main conjectures for elliptic curves provide a scope to study birch and swinnertondyer conjecture. Iwasawa theory, elliptic curves, complex multiplication, cm, birch. An introduction to iwasawa theory for elliptic curves. We study this subject by first proving that the pprimary subgroup of the classical selmer group for an elliptic curve. Iwasawa theory of modular elliptic curves of analytic rank at. Anticyclotomic iwasawa theory of cm elliptic curves adebisi agboola and benjamin howard with an appendix by karl rubin abstract. These are a preliminary set ot notes for the authors lectures for the 2018 arizona winter school on iwasawa theory. Iwasawa theory of modular elliptic curves of analytic rank at most 1 john coates department of pure mathematics and mathematical statistics, university of cambridge, 16 mill lane, cambridge cb2 1sb. In this seminar we will focus on the classical case of zpextensions. A pair of main conjectures article in journal of number theory 27 march 2009 with 29 reads how we measure reads. Pdf iwasawa theory for elliptic curves at supersingular. 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. The main conjecture of iwasawa theory for elliptic curves.
Noncommutative iwasawa theory of elliptic curves at. Anticyclotomic iwasawa theory of cm elliptic curves. The original motivations for considering the selmer groups of elliptic curves along a zpextension are however quite different from what has been suggested above. Elliptic curves, modular forms and iwasawa theory in. On the main conjectures of iwasawa theory for certain elliptic curves with complex multiplication. Vatsal, on the iwasawa invariants of elliptic curves, in preparation. Following a chapter on formal groups and local units, the padic l functions of maninvishik and katz are constructed and studied. On the main conjectures of iwasawa theory for certain. Dyer conjecture for elliptic curves of conductor 6 5000 with rank 6 1 by a computational application of euler system results of kato and kolyvagin combined with explicit descent. Iwasawa theory of elliptic curves with complex multiplication, by. Kwokwing tsois homepage iwasawa theory of cm elliptic curves.
Given an elliptic curve e, understand how the mordellweil group ef varies as f varies. The main unifying theme is iwasawa theory, a field that john coates himself has done much to create. Lectures on the iwasawa theory of elliptic curves 3 some notational preliminaries. Rational points of abelian varieties with values in towers of number fields. Seminar on iwasawa theory of elliptic curves gabor wiese sommersemester 2008 abstract iwasawa theory studies arithmetic objects in certain padic towers of number. Noncommutative iwasawa theory of elliptic curves at primes. The iwasawa main conjecture for supersingular elliptic curves is conjecture 1. In the last fifteen years the iwasawa theory has been applied with remarkable success to elliptic curves with complex multiplication.
Iwasawa theory for elliptic curves at supersingular primes. Iwasawa theory of elliptic curves and bsd in rank zero. The main motivation for the present paper is to develop algorithms using iwasawa theory, in order to enable veri cation of the. It has been generalized to the study of the relation between zeta values and more general arithmetic objects rational points and selmer groups of elliptic curves, galois representations, galois cohomology groups. The smallest integer m satisfying h gm is called the logarithm or index of h with respect to g, and is denoted. Namely, we define a new selmer group, and show that it is of. Iwasawa main conjecture for supersingular elliptic curves. In the early 1970s, barry mazur considered generalizations of iwasawa theory to abelian varieties. Introduction the study of the main conjectures of iwasawa theory, which relates the padic lfunctions to the characteristic power series of certain iwasawa modules, has been very in. The problem given an elliptic curve e, understand how the mordellweil group ef varies as f varies. Lectures on the iwasawa theory of elliptic curves christopher skinner abstract. Then we formulate the iwasawa main conjecture as that the characteristic ideal is generated by pollacks padic lfunction. By constructing padic lfunctions at primes of additive reduction, we formulate a main conjecture linking this lfunction with a certain selmer group for e over the zpextension.
Iwasawa theory for elliptic curves at unstable primes. When the complex lfunction of e vanishes to even order, ru. The mordellweil theorem says the abelian group of rational points on an elliptic curve over q is finitely generated. Sujatha to parimala on the occasion of her sixtieth birthday 1.
Noncommutative iwasawa theory for elliptic curves with multiplicative reduction. Iwasawa theory of elliptic curves with complex multiplication anna seigal 2nd may 2014 contents 1 introduction 2 1. 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. Iwasawa theory elliptic curves with complex multiplication. We present the rst few sections of greenbergs article \introduction to iwasawa theory for elliptic curves. In particular, we give analogous definitions of the plus and minus coleman maps for normalised new forms of arbitrary weights and relate pollack s padic lfunctions to the plus and minus selmer groups. Iwasawa theory studies arithmetic objects in certain padic towers of number fields.
Ralph greenberg university of washington download pdf abstract. Noncommutative iwasawa theory for elliptic curves with. Iwasawa theory of elliptic curves the philosophy 2. Classical iwasawa theory describes the relation between zeta values and ideal class groups. Student iwasawa theory seminar karl schaefer and eric stubley winter 2018 the goal of this seminar is to prepare for the 2018 arizona winter school by learning some iwasawa theory. We then prove theorems of mazur, schneider, and perrinriou on the basis of this description. It began as a galois module theory of ideal class groups, initiated by kenkichi iwasawa, as part of the theory of cyclotomic fields. An elliptic curve ek is one of the simplest nontrivial examples of an abelian variety. The above papers are somehow the foundations of classical iwasawa theory.
This formulation is similar to mazurs at good ordinary primes. Student iwasawa theory seminar university of chicago. In mathematics, the main conjecture of iwasawa theory is a deep relationship between padic lfunctions and ideal class groups of cyclotomic fields, proved by kenkichi iwasawa for primes satisfying the kummervandiver conjecture and proved for all primes by mazur and wiles. Introduction a fundamental problem in algebraic number theory concerns the study of the absolute galois group of the. 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. 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. Iwasawa theory of modular elliptic curves of analytic rank. Iwasawa theory and generalizations 339 let e be an elliptic curve over a number. In this paper we examine the iwasawa theory of modular elliptic curves e defined over q without semistable reduction at p.
Iwasawa theory of elliptic curves with complex multiplication. This article is dedicated to the memory of kenkichi iwasawa, who passed away on october 26th, 1998. Computing tateshafarevich groups of elliptic curves using. Algorithms for the arithmetic of elliptic curves using. Several of the contributions in this volume were presented at the conference elliptic curves, modular forms and iwasawa theory, held in honour of the 70 th birthday of john coates in cambridge, march 2527, 2015. We study the iwasawa theory of a cm elliptic curve e in the anticyclotomic zpextension of the cm. Pdf download iwasawa theory 2012 free unquote books. Check out otmar venjakob thesis, and his papers, to have an idea of non commutative iwasawa theory. The characteristic power series c divides the padic lfunction l pe,k. The dual selmer group x e is torsion over q and the characteristic ideal of x e is generated by l e as. 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 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. In number theory, iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. We give a new formulation in iwasawa theory for elliptic curves at good supersingular primes.
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