Professor Greenberg intends to study a diverse set of problems related to Selmer groups, elliptic curves, and Iwasawa theory.
Selmer groups have traditionally been an important tool for studying the Mordell-Weil group of an elliptic curve over a number field.
They have also played an important role in proving special cases of the Birch and Swinnerton-Dyer conjecture which provides a relationship between arithmetic properties of an elliptic curve E and the behavior of the Hasse-Weil L-function for E. In recent years, it has become clear that certain natural generalizations of the Selmer group should provide similar conjectural relationships to the behavior of much more general kinds of L-functions.
Iwasawa theory provides a framework for studying these conjectures. In its essence, the idea is to study Selmer groups associated to a family of representations of the absolute Galois group of a number field. The formulation of these conjectures in a general setting leads to some fundamental problems. One problem is to find a simple way to measure how large the Selmer groups are.
Their size should be measured by an element in a certain ring. But it is difficult to find a way to define that element. One of the objectives of this project is to tackle that question in two contrasting settings. In one setting, the ring is rather easy to describe, but is non-commutative. In the other setting, the ring is commutative, but we know very little about its actually structure. Another objective of this project is to better understand the behavior of certain quantities which indirectly reflect the structure of Selmer groups, especially in the non-commutative setting.
Professor Greenberg hopes to show how to determine all of these quantities from just a finite number of them. One of the fundamental questions in the theory of numbers is the study of solutions of an algebraic equation. The difficulty of this question depends on the degree of the equation and the number of variables.
It has been understood since antiquity how to study this question when the degree is one or two and the number of variables is also one or two. However, the question becomes much more subtle when one considers equations of degree three, even if the number of variables is just two. A fundamental conjecture concerning this question was formulated in the s by Birch and Swinnerton-Dyer. Although considerable progress has been made since then, the conjecture remains unresolved.Ralph Greenberg born is an American mathematician who has made contributions to number theoryin particular Iwasawa theory.
He was born in Chester, Pennsylvania  and studied at the University of Pennsylvaniaearning a B. Greenberg is also well known for his many conjectures. In the s, he introduced the notion of a Selmer group for a p -adic Galois representation and generalized the "main conjectures" of Iwasawa and Barry Mazur to this setting. He has since generalized this setup to present Iwasawa theory as the theory of p -adic deformations of motives. He also provided an arithmetic theory of L -invariants generalizing his aforementioned work with Stevens.
Greenberg was an invited speaker in International Congress of MathematiciansHyderabad on the topic of " Number Theory. Inhe became a fellow of the American Mathematical Society. This article about an American mathematician is a stub. You can help Wikipedia by expanding it. From Wikipedia, the free encyclopedia. American mathematician. ChesterPennsylvania.
Institute for Advanced Study. Retrieved January 9, International Congress of Mathematicians. Retrieved Categories : births Living people 20th-century American mathematicians 21st-century American mathematicians Princeton University alumni University of Washington faculty Fellows of the American Mathematical Society Number theorists People from Chester, Pennsylvania Mathematicians from Pennsylvania University of Pennsylvania alumni American mathematician stubs.
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Wikimedia Commons. University of Pennsylvania Princeton University. University of Washington.In number theoryIwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. In the early s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties.
More recently early sRalph Greenberg has proposed an Iwasawa theory for motives. More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group. Iwasawa considered the following tower of number fields:.
If we set. This is a 2-dimensionalregular local ringand this makes it possible to describe modules over it. From this description it is possible to recover information about the p -part of the class group of K. From this beginning in the s, a substantial theory has been built up. A fundamental connection was noticed between the module theory, and the p-adic L-functions that were defined in the s by Kubota and Leopoldt.
The latter begin from the Bernoulli numbersand use interpolation to define p-adic analogues of the Dirichlet L-functions. It became clear that the theory had prospects of moving ahead finally from Kummer's century-old results on regular primes. Iwasawa formulated the main conjecture of Iwasawa theory 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.
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These proofs were modeled upon Ken Ribet 's proof of the converse to Herbrand's theorem the so-called Herbrand—Ribet theorem. Karl Rubin found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin's Euler systemsdescribed in Lang and Washingtonand later proved other generalizations of the main conjecture for imaginary quadratic fields.
The Galois group of the infinite tower, the starting field, and the sort of arithmetic module studied can all be varied. In each case, there is a main conjecture linking the tower to a p -adic L-function. From Wikipedia, the free encyclopedia.
Study of objects of arithmetic interest over infinite towers of number fields. L -functions in number theory. Analytic class number formula Riemann—von Mangoldt formula Weil conjectures. Main conjecture of Iwasawa theory Selmer group Euler system.MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
What papers should we read to start? What basic knowledge do we need to understand the question? What is this area really about?
And what are people researching on it?Gla 250 4matic 2021
Iwasawa theory has its origins in the following counterintuitive insight of Iwasawa: instead of trying to describe the structure of any particular Galois module, it is often easier to describe every Galois module in an infinite tower of fields at once.
This ring is much easier to understand; it's a complete 2-dimensional regular local ring that is non-canonically isomorphic to a power series ring, and there is a strong structure theorem for modules over this ring. Using this structure theorem, Iwasawa proved many theorems about the class numbers of cyclotomic fields. This was Iwasawa's "main conjecture" and is now a theorem. It implies the Herbrand-Ribet theorem and essentially every classical result relating cyclotomic fields and zeta values.
There's a lot of current research in this area. To start reading, I recommend Washington's book on cyclotomic fields. Chapter 13 is fun and is a good use of some of the main techniques of Iwasawa theory. You don't need anything but the basic background in chapters to read sections of Chapter 13, which contain the types of theorem I was referring to in the first two paragraphs of this answer.
If you know some algebraic number theory, you should be fine to read this book.
If you're comfortable with class field theory, and have read the first few sections of Chapter 13 in Washington, then Coates and Sujatha's recent book, Cyclotomic Fields and Zeta Valuesis a pleasure to read. Volume 56, by Ralph Greenberg. Also check the continuation of Greenberg's paper. After the above, I highly recommend: Rational points of abelian varieties with values in towers of number fields.
Check out Otmar Venjakob thesis, and his papers, to have an idea of non commutative Iwasawa theory. Also Ralph Greenberg has been working on a book on Iwasawa theory, and I think you can find some chapters of it in his webpage. Part 2 of Kato's lectures was never published as I am sure everyone is aware, however, Kato's paper with Fukaya- "A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory" is a replacement and of course an extension of part 2.
Kato's lectures on the generalized Iwasawa conjecture adn how it fits into the general picture of zeta values, p-adic periods etc. An other overview is given in Kato's ICM lecture.An angel came down to me
Pottharst wrote a short note "What is Iwasawa theory about in my opinion? You can look up Iwasawa's old papers like Analogy between number and function fields which is pretty much the motivation for the subject. Also Iwasawa's book on p-adic L-functions is a hard but good book to read. I collected a list of references for Iwasawa theory hereincluding links to various surveys which hasn't been mentioned so far. Sign up to join this community. The best answers are voted up and rise to the top.
How do we study Iwasawa theory?Wagner lacrosse schedule 2021
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Detroit Lions at Baltimore Ravens (-2. Danny Woodhead and Buck Allen should also create havoc in third-down situations, as their pass-catching prowess should give the Lions fits. But the game will be won between the tackles. Look for Collins to cross the century mark as the Ravens control the time of possession and win a low-scoring matchup.
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