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Topic: Iwasawa theory

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Wolf Prize

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June 2004

	Iwasawa theory - Wikipedia, the free encyclopedia
		In number theory, Iwasawa theory is a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa as part of the theory of cyclotomic fields.
		Iwasawa's starting observation was that there are towers of fields in algebraic number theory, having Galois group isomorphic with the additive group of p-adic integers.
		The main conjecture of Iwasawa theory was formulated as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined.
	en.wikipedia.org /wiki/Iwasawa_theory (496 words)

	Kenkichi Iwasawa - Wikipedia, the free encyclopedia
		Kenkichi Iwasawa (岩澤健吉 Iwasawa Kenkichi, September 11, 1917 - October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory.
		Iwasawa was born in Shinshuku-mura, a town near Kiryu, in Gunma Prefecture.
		However, this same year Iwasawa became sick with pleurisy, and was unable to return to his position at the university until April 1947.
	en.wikipedia.org /wiki/Kenkichi_Iwasawa (262 words)

	Iwasawa theory (Site not responding. Last check: 2007-10-08)
		Iwasawa's starting observation was that there are of fields in algebraic number theory having Galois group isomorphic with the additive group of p-adic integers.
		That group usually written Γ in theory and with multiplicative notation can be as a subgroup of Galois groups of field extensions (which are by their nature pro-finite groups).
		The main conjecture of Iwasawa theory was formulated as an assertion that methods of defining p-adic L-functions (by module by interpolation) should coincide as far as was well-defined.
	www.freeglossary.com /Iwasawa_theory (613 words)

	Iwasawa (Site not responding. Last check: 2007-10-08)
		Artin was at the Institute during Iwasawa's two years there and he was one of the main factors in changing the direction of Iwasawa's research to algebraic number theory.
		It was Iwasawa's intention to return to Japan in 1952 after his visit to the Institute for Advanced Study but when he received the offer of a post of assistant professor at the Massachusetts Institute of Technology he decided to accept it.
		In the late 1960s Iwasawa made a conjecture for algebraic number fields which, in some sense, was the analogue of the relationship which Weil had found between the zeta function and the divisor class group of an algebraic function field.
	www-gap.dcs.st-and.ac.uk /~history/Mathematicians/Iwasawa.html (1213 words)

	Iwasawa (Site not responding. Last check: 2007-10-08)
		Takagi had retired in 1936, the year before Iwasawa began his studies, but his students Iyanaga and Suetuna were bringing to the university many ideas which they had developed during studies with the leading experts in Europe.
		Iwasawa graduated in 1940 and remained at Tokyo University to undertake graduate studies.
		Iwasawa's results are related to Hibert's fifth problem which asks whether any locally Euclidean topological groups is necessarily a Lie group.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Iwasawa.html (1213 words)

	Interview with a mathematician
		Iwasawa: I had an interest in number theory from the time when I was a student.
		Iwasawa: In about 1950, cohomology theory was brought to class field theory by Nakayama and Hochschild.
		Iwasawa said that he found his notebook for his lectures in Princeton, which were given the same year that he moved there.
	myhome.iolfree.ie /~alexandros/articles/iwasawa.htm (2985 words)

	Open Questions and Recent Developments in Iwasawa Theory -- in honor of Ralph Greenberg's 60th Birthday
		Iwasawa theory of elliptic curves and modular forms (cyclotomic and anticyclotomic)
		Algebraic p-adic L-functions and main conjectures in non-commutative Iwasawa theory
		Iwasawa theory and several elements in Galois cohomology
	math.bu.edu /people/rpollack/greenberg.html (184 words)

	Princeton - PWB 111698 - Obituaries (Site not responding. Last check: 2007-10-08)
		A leading researcher in algebra and number theory, Iwasawa was recognized particularly for his work on the theory of algebraic number fields, which became known as "Iwasawa Theory." This earned him several prizes, including the American Mathematical Society's 1962 Cole Prize.
		Born in Japan, Iwasawa graduated from the University of Tokyo in 1940, where he earned his DrSci degree in 1945 and was appointed an assistant professor in 1949.
		Iwasawa was a member of the American Mathematical Society and Mathematical Society of Japan, and a fellow of the American Academy of Arts and Sciences.
	www.princeton.edu /pr/pwb/98/1116/obits.htm (395 words)

	Iwasawa (Site not responding. Last check: 2007-10-08)
		In his 1949 paper Iwasawa gives what is now known as the 'Iwasawa decomposition' of a real semisimple Lie group.
		A proof of the Riemann-Roch theorem is given, and the theory of Riemann surfaces and their topology is studied.
		Iwasawa himself produced a series of deep papers throughout the 1960s which pushed his ideas much further.
	www-history.mcs.st-and.ac.uk /history/Mathematicians/Iwasawa.html (1213 words)

	Kenkichi Iwasawa (Site not responding. Last check: 2007-10-08)
		'''Kenkichi Iwasawa''' (&23721;&28580; &20581;&21513;) (September 11 1917 - October 26 1998) was a Japanese mathematician who is known for his influence on algebraic number theory.
		He spent the next two years at Institute for Advanced Study at Princeton, and in Spring of 1952 was offered a job at the Massachusetts Institute of Technology, where he worked until 1967.
		Iwasawa, Kenkichi Iwasawa, Kenkichi Iwasawa, Kenkichi Iwasawa, Kenkichi Iwasawa, Kenkichi ja:&23721;&28580;&20581;&21513;
	kenkichi-iwasawa.area51.ipupdater.com (245 words)

	Iwasawa theory - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-08)
		Hilbert Modular Forms and Iwasawa Theory (Oxford Mathematical Monographs)
		Iwasawa Theory Elliptic Curves with Complex Multiplication : P-Adic L Functions (Perspectives in Mathematics, Vol 3)
		Arithmetic Geometry: Conference on Arithmetic Geometry With an Emphasis on Iwasawa Theory March 15-18, 1993 Arizona State University (Contemporary Mathematics)
	encyclopedia.worldsearch.com /iwasawa_theory.htm (480 words)

	Topics in Algebraic Number Theory
		Algebraic number theory has a long and distinguished history and remains one of the most significant areas of research in mathematics.
		The analysis of problems in number theory, even those of a seemingly concrete and explicit nature, may well however involve the interplay of results and techniques from may different branches of pure mathematics.
		The topics to be discussed have been chosen both because they have been of pivotal significance to recent developments and also because they illustrate well the wide variety of techniques and the nature of the problems which arise in much of the fundamental research which is being conducted today.
	www.mth.kcl.ac.uk /events/short_courses/ANT_Sep_2002.html (659 words)

	Algebraic Number Theory Archive (Site not responding. Last check: 2007-10-08)
		ANT-0342: 28 Mar 2002, On an Archimedean analogue of Tate's conjecture, by Dipendra Prasad and C.S.Rajan.
		ANT-0296: 8 Jun 2001, On the Iwasawa theory of p-adic Lie extensions, by Otmar Venjakob.
		ANT-0295: 8 Jun 2001, On the structure theory of the Iwasawa algebra of a p-adic Lie group, by Otmar Venjakob.
	front.math.ucdavis.edu /ANT (12251 words)

	Coates (Site not responding. Last check: 2007-10-08)
		He worked on Iwasawa's theory and wrote a number of articles with Andrew Wiles published around 1977-78 including Kummer's criterion for Hurwitz number, Explicit reciprocity laws and On p-adic L-functions and elliptic units.
		During the 1980s Coates's work was concerned with elliptic curves, Iwasawa theory and p-adic L-functions, all work closely related to the direction that would eventually yield the proof of Fermat's Last Theorem.
		Coates's insights into the Iwasawa theory of the symmetric square of an elliptic curve were instrumental in the recent proof by Wiles of the Shimura-Taniyama conjecture for semistable elliptic curves over Q.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Coates.html (760 words)

	LMS Proceedings Abstract, paper PLMS 1368 (Site not responding. Last check: 2007-10-08)
		If G is a pro-p, p-adic, Lie group containing no element of order p and if $\Lambda (G)$ denotes the Iwasawa algebra of G then we propose a number of invariants associated to finitely generated $\Lambda (G)$-modules, all given by various forms of Euler characteristic.
		The first turns out to be none other than the rank, and this gives a particularly convenient way of calculating the rank of Iwasawa modules.
		We explore some properties and give applications to the Iwasawa theory of elliptic curves.
	www.lms.ac.uk /publications/proceedings/abstracts/p1368a.html (117 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The origins of Iwasawa theory can be traced back to the famous class number formula of Dirichlet, which states that the number of inequivalent binary quadratic forms of a given fundamental discriminant $D$ is determined by the value $L(1,\chi_D)$ of the associated quadratic L-function.
		In modern terms, the class number formula is a very special case of a {\em Main Conjecture in Iwasawa theory}, which relates analytic information (as given by L-values) to algebraic information given by cohomology (the so-called Selmer group, which reduces in the classical case to quadratic forms and genus theory).
		We illustrate the theory by describing the proof of the $p$-adic Artin conjecture.
	garsia.math.yorku.ca /seminar/Previous/F1998/F1998/Vatsal1.html (204 words)

	AMS Prize - Frank Nelson Cole Prize in Number Theory
		This prize (and the Frank Nelson Cole Prize in Algebra) was founded in honor of Professor Frank Nelson Cole on the occasion of his retirement as Secretary of the American Mathematical Society after twenty-five years of service and as Editor-in-Chief of the Bulletin for twenty-one years.
		The prize is for a notable paper in number theory published during the preceding six years.
		Fourth award, 1951: To Paul Erdös for his many papers in the theory of numbers, and in particular for his paper, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem, Proceedings of the National Academy of Sciences, volume 35 (1949), pp.
	e-math.ams.org /prizes/cole-prize-number-theory.html (805 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The Iwasawa module MF is the Galois group of the maximal abelian `-extension of F1 that is unramified a* *way from `; it is a profinite 0F-module.
		Stable homotopy theory In this section we recall some facts from stable homotopy theory that are ne* *eded for the rest of the paper.
		Suppose that E is a homology theory, or equivalently, that E is a spectrum w* ith associated homology theory* EX = ss(E ^X).
	hopf.math.purdue.edu /Dwyer-Mitchell/spectrum.txt (17100 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The first is concerned with Iwasawa theory and L-functions.
		These approaches have their origins in the theory of arithmetic Galois module structure, and they lead to new ways of studying the Iwasawa theory of abelian varieties and p-adic representations.
		This is a subject that blends two of the oldest branches of mathematics---number theory and geometry---and which has blossomed to the point where it has solved problems that have stood for centuries.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9700937.txt (198 words)

	Class field theory (Site not responding. Last check: 2007-10-08)
		Class field theory is a major branch of algebraic number theory, including most of the central results that were proved in the period about 1900-1950.
		The class field theory project included the 'higher reciprocity laws' (cubic reciprocity and so on), but is not limited to that one, classical line of generalisation.
		In the 1930s and subsequently the use of infinite extensions and the theory of Krull of their Galois groups was found increasingly useful.
	www.omniknow.com /common/wiki.php?in=en&term=Class_field_theory (1011 words)

	Atlas: Toward equivariant Iwasawa theory by Jürgen Ritter (Site not responding. Last check: 2007-10-08)
		Let K/k be a Galois extension of totally real number fields, with k/Q finite and with K finite over the cyclotomic l-extension of k, where l is an odd prime number.
		The talk formulates a nonabelian " equivariant main conjecture " of Iwasawa theory and comments on the present state of a verification.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calz-61.
	atlas-conferences.com /cgi-bin/abstract/calz-61 (95 words)

	Cyclotomic Fields (L24) (Site not responding. Last check: 2007-10-08)
		This is the central result of the theory of cyclotomic fields, giving a precise link between the values of the Riemann zeta function at the odd negative integers and the arithmetic of cyclotomic fields.
		It is also of great importance for current research in Iwasawa theory because it remains the basic model of what one tries to establish for a vast array of generalizations in arithmetic geometry involving the arithmetic of motives over
		Background knowledge will be kept to a minimum throughout the course (in particular, no prior knowledge of Iwasawa theory or of cyclotomic fields will be assumed), although some basic knowledge about local fields will be needed.
	www.maths.cam.ac.uk /postgrad/courses/damtp/node35.html (175 words)

	EMS Prizes
		Although he has worked widely in ergodic theory, his recent proof of the quantum unique ergodicity conjecture for arithmetic hyperbolic surfaces breaks fertile new ground, with great promise for future applications to number theory.
		In arithmetic geometry, Iwasawa theory is the only general technique known for studying the mysterious relations between exact arithmetic formulae and special values of L-functions, as typified by the conjecture of Birch and Swinnerton-Dyer.
		With Hachimori he discovered the first examples of arithmetic Iwasawa modules which are completely faithful, as well as proving a remarkable asymptotic upper bound for the rank of the Mordell Weil group of elliptic curves in certain towers of number fields over Q whose Galois group is a p-adic Lie group of dimension 2.
	www.math.kth.se /4ecm/prizes.ecm.html (1412 words)

	ABC-Dir: Theory
		Research areas: recursion theory, model theory, set theory and foundations, proof theory, and in applications to algebra,...
		Algebraic number theory: Iwasawa theory, the arithmetic of elliptic curves, Galois theory; questions in computational number theory.
		Includes new theory of physics based on the theory of limits; the limits to physical measurement.
	www.abc-directory.com /view/theory (281 words)

	On the Iwasawa theory of p-adic Lie Extensions, by Otmar Venjakob (Site not responding. Last check: 2007-10-08)
		On the Iwasawa theory of p-adic Lie Extensions, by Otmar Venjakob
		In this paper the new techniques and results concerning the structure theory of modules over non-commutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions K of number fields k "up to pseudo-isomorphism".
		In particular, a close relationship is revealed between the Selmer group of abelian varieties, the Galois group of the maximal abelian unramified p-extension of K as well as the Galois group of the maximal abelian outside S unramified p-extension where S is a finite set of certain places of k.
	front.math.ucdavis.edu /ANT/0296 (127 words)

	Open Directory - Science: Math: Number Theory: Events: Past Events (Site not responding. Last check: 2007-10-08)
		Colloquium on Number Theory - In honor of the 60th birthday of Professors Kálmán Gyõry and András Sárközy.
		Number Theory Conference in Honor of Harold Stark - On the occasion of his 65th birthday.
		Theory of Motives, Homotopy Theory of Varieties, and Dessins d'Enfants - AIM Research Conference Center (ARCC), Palo Alto, CA, USA; 12--15 March 2004.
	www.dmoz.org /Science/Math/Number_Theory/Events/Past_Events (3702 words)

	London Number Theory Seminar
		The London Number Theory Seminar is held weekly during term times.
		Abstract: There have been several important developments in non-commutative Iwasawa theory over the last few years.
		In one case the Riemann Hypothesis of Deligne is the crucial ingredient, in the other the spectral theory of automorphic forms appears naturally.
	www.mth.kcl.ac.uk /events/numbtheo.html (502 words)

	Citebase - Characteristic elements in noncommutative Iwasawa theory
		In this article we construct characteristic elements for a certain class of Iwasawa modules in noncommutative Iwasawa theory.
		Fukaya and K. Kato, A formulation of conjectures on p-adic zeta functions in noncommutative Iwasawa theory, preprint (2003).
		Howson, Iwasawa theory of Elliptic Curves for p-adic Lie extensions, Ph.D. thesis, University of Cambridge, July 1998.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0311446 (783 words)

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