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Author: Barry Mazur
Publisher: Amer Mathematical Society
ISBN: 9780821835128
Category : Mathematics
Languages : en
Pages : 96
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Book Description
Since their introduction by Kolyvagin, Euler systems have been used in several important applications in arithmetic algebraic geometry. For a $p$-adic Galois module $T$, Kolyvagin's machinery is designed to provide an upper bound for the size of the Selmer group associated to the Cartier dual $T^*$. Given an Euler system, Kolyvagin produces a collection of cohomology classes which he calls ``derivative'' classes. It is these derivative classes which are used to bound the dual Selmer group. The starting point of the present memoir is the observation that Kolyvagin's systems of derivative classes satisfy stronger interrelations than have previously been recognized. We call a system of cohomology classes satisfying these stronger interrelations a Kolyvagin system. We show that the extra interrelations give Kolyvagin systems an interesting rigid structure which in many ways resembles (an enriched version of) the ``leading term'' of an $L$-function. By making use of the extra rigidity we also prove that Kolyvagin systems exist for many interesting representations for which no Euler system is known, and further that there are Kolyvagin systems for these representations which give rise to exact formulas for the size of the dual Selmer group, rather than just upper bounds.
Author: Barry Mazur
Publisher: Amer Mathematical Society
ISBN: 9780821835128
Category : Mathematics
Languages : en
Pages : 96
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Book Description
Since their introduction by Kolyvagin, Euler systems have been used in several important applications in arithmetic algebraic geometry. For a $p$-adic Galois module $T$, Kolyvagin's machinery is designed to provide an upper bound for the size of the Selmer group associated to the Cartier dual $T^*$. Given an Euler system, Kolyvagin produces a collection of cohomology classes which he calls ``derivative'' classes. It is these derivative classes which are used to bound the dual Selmer group. The starting point of the present memoir is the observation that Kolyvagin's systems of derivative classes satisfy stronger interrelations than have previously been recognized. We call a system of cohomology classes satisfying these stronger interrelations a Kolyvagin system. We show that the extra interrelations give Kolyvagin systems an interesting rigid structure which in many ways resembles (an enriched version of) the ``leading term'' of an $L$-function. By making use of the extra rigidity we also prove that Kolyvagin systems exist for many interesting representations for which no Euler system is known, and further that there are Kolyvagin systems for these representations which give rise to exact formulas for the size of the dual Selmer group, rather than just upper bounds.
Author:
Publisher: American Mathematical Soc.
ISBN: 9780821865217
Category :
Languages : en
Pages : 96
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Book Description
Author: Karl Rubin
Publisher: Princeton University Press
ISBN: 1400865204
Category : Mathematics
Languages : en
Pages : 241
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Book Description
One of the most exciting new subjects in Algebraic Number Theory and Arithmetic Algebraic Geometry is the theory of Euler systems. Euler systems are special collections of cohomology classes attached to p-adic Galois representations. Introduced by Victor Kolyvagin in the late 1980s in order to bound Selmer groups attached to p-adic representations, Euler systems have since been used to solve several key problems. These include certain cases of the Birch and Swinnerton-Dyer Conjecture and the Main Conjecture of Iwasawa Theory. Because Selmer groups play a central role in Arithmetic Algebraic Geometry, Euler systems should be a powerful tool in the future development of the field. Here, in the first book to appear on the subject, Karl Rubin presents a self-contained development of the theory of Euler systems. Rubin first reviews and develops the necessary facts from Galois cohomology. He then introduces Euler systems, states the main theorems, and develops examples and applications. The remainder of the book is devoted to the proofs of the main theorems as well as some further speculations. The book assumes a solid background in algebraic Number Theory, and is suitable as an advanced graduate text. As a research monograph it will also prove useful to number theorists and researchers in Arithmetic Algebraic Geometry.
Author: David Burns
Publisher: American Mathematical Soc.
ISBN: 0821834800
Category : Education
Languages : en
Pages : 234
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Book Description
Stark's conjectures on the behavior of USDLUSD-functions were formulated in the 1970s. Since then, these conjectures and their generalizations have been actively investigated. This has led to significant progress in algebraic number theory. The current volume, based on the conference held at Johns Hopkins University (Baltimore, MD), represents the state-of-the-art research in this area. The first four survey papers provide an introduction to a majority of the recent work related to themes currently under exploration in the area, such as non-abelian and USDpUSD-adic aspects of the conjectures, abelian refinements, etc. Among others, some important contributors to the volume include Harold M. Stark, John Tate, and interested in number theory.
Author: Cristian Popescu
Publisher: American Mathematical Soc.
ISBN: 0821886983
Category : Mathematics
Languages : en
Pages : 517
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Book Description
Author: Samuel Rufus Williams
Publisher:
ISBN:
Category : Curves, Elliptic
Languages : en
Pages : 156
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Book Description
Author: Thanasis Bouganis
Publisher: Springer
ISBN: 3642552455
Category : Mathematics
Languages : en
Pages : 487
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Book Description
This is the fifth conference in a bi-annual series, following conferences in Besancon, Limoges, Irsee and Toronto. The meeting aims to bring together different strands of research in and closely related to the area of Iwasawa theory. During the week before the conference in a kind of summer school a series of preparatory lectures for young mathematicians was provided as an introduction to Iwasawa theory. Iwasawa theory is a modern and powerful branch of number theory and can be traced back to the Japanese mathematician Kenkichi Iwasawa, who introduced the systematic study of Z_p-extensions and p-adic L-functions, concentrating on the case of ideal class groups. Later this would be generalized to elliptic curves. Over the last few decades considerable progress has been made in automorphic Iwasawa theory, e.g. the proof of the Main Conjecture for GL(2) by Kato and Skinner & Urban. Techniques such as Hida’s theory of p-adic modular forms and big Galois representations play a crucial part. Also a noncommutative Iwasawa theory of arbitrary p-adic Lie extensions has been developed. This volume aims to present a snapshot of the state of art of Iwasawa theory as of 2012. In particular it offers an introduction to Iwasawa theory (based on a preparatory course by Chris Wuthrich) and a survey of the proof of Skinner & Urban (based on a lecture course by Xin Wan).
Author: Stefano Pigola
Publisher: American Mathematical Soc.
ISBN: 0821836390
Category : Mathematics
Languages : en
Pages : 118
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Book Description
Aims to introduce the reader to various forms of the maximum principle, starting from its classical formulation up to generalizations of the Omori-Yau maximum principle at infinity obtained by the authors.
Author: Robert Oliver
Publisher: American Mathematical Soc.
ISBN: 0821838288
Category : Mathematics
Languages : en
Pages : 116
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Book Description
We prove here the Martino-Priddy conjecture at the prime $2$: the $2$-completions of the classifying spaces of two finite groups $G$ and $G'$ are homotopy equivalent if and only if there is an isomorphism between their Sylow $2$-subgroups which preserves fusion. This is a consequence of a technical algebraic result, which says that for a finite group $G$, the second higher derived functor of the inverse limit vanishes for a certain functor $\mathcal{Z}_G$ on the $2$-subgroup orbit category of $G$. The proof of this result uses the classification theorem for finite simple groups.
Author: Fabrizio Andreatta
Publisher: American Mathematical Soc.
ISBN: 0821836099
Category : Mathematics
Languages : en
Pages : 114
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Book Description
We study Hilbert modular forms in characteristic $p$ and over $p$-adic rings. In the characteristic $p$ theory we describe the kernel and image of the $q$-expansion map and prove the existence of filtration for Hilbert modular forms; we define operators $U$, $V$ and $\Theta_\chi$ and study the variation of the filtration under these operators. Our methods are geometric - comparing holomorphic Hilbert modular forms with rational functions on a moduli scheme with level-$p$ structure, whose poles are supported on the non-ordinary locus.In the $p$-adic theory we study congruences between Hilbert modular forms. This applies to the study of congruences between special values of zeta functions of totally real fields. It also allows us to define $p$-adic Hilbert modular forms 'a la Serre' as $p$-adic uniform limit of classical modular forms, and compare them with $p$-adic modular forms 'a la Katz' that are regular functions on a certain formal moduli scheme. We show that the two notions agree for cusp forms and for a suitable class of weights containing all the classical ones. We extend the operators $V$ and $\Theta_\chi$ to the $p$-adic setting.
