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Galois Theory and Modular Forms : Developments in Mathematics - Ki-Ichiro Hashimoto

Galois Theory and Modular Forms

Developments in Mathematics

By: Ki-Ichiro Hashimoto (Editor), Katsuya Miyake (Editor), Hiroaki Nakamura (Editor)


Published: 30th November 2003
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This volume is an outgrowth of the research project "The Inverse Ga- lois Problem and its Application to Number Theory" which was carried out in three academic years from 1999 to 2001 with the support of the Grant-in-Aid for Scientific Research (B) (1) No. 11440013. In September, 2001, an international conference "Galois Theory and Modular Forms" was held at Tokyo Metropolitan University after some preparatory work- shops and symposia in previous years. The title of this book came from that of the conference, and the authors were participants of those meet- All of the articles here were critically refereed by experts. Some of ings. these articles give well prepared surveys on branches of research areas, and many articles aim to bear the latest research results accompanied with carefully written expository introductions. When we started our re~earch project, we picked up three areas to investigate under the key word "Galois groups"; namely, "generic poly- nomials" to be applied to number theory, "Galois coverings of algebraic curves" to study new type of representations of absolute Galois groups, and explicitly described "Shimura varieties" to understand well the Ga- lois structures of some interesting polynomials including Brumer's sextic for the alternating group of degree 5. The topics of the articles in this volume are widely spread as a result. At a first glance, some readers may think this book somewhat unfocussed.

Prefacep. ix
Arithmetic geometryp. 1
The arithmetic of Weierstrass points on modular curves X[subscript 0](p)p. 3
Semistable abelian varieties with small division fieldsp. 13
Q-curves with rational j-invariants and jacobian surfaces of GL[subscript 2]-typep. 39
Points defined over cyclic quartic extensions on an elliptic curve and generalized Kummer surfacesp. 65
The absolute anabelian geometry of hyperbolic curvesp. 77
Galois groups and Galois extensionsp. 123
Regular Galois realizations of PSL[subscript 2](p[superscript 2]) over Q(T)p. 125
Middle convolution and Galois realizationsp. 143
On the essential dimension of p-groupsp. 159
Explicit constructions of generic polynomials for some elementary groupsp. 173
On dihedral extensions and Frobenius extensionsp. 195
On the non-existence of certain Galois extensionsp. 221
Frobenius modules and Galois groupsp. 233
Algebraic number theoryp. 269
On quadratic number fields each having an unramified extension which properly contains the Hilbert class field of its genus fieldp. 271
Distribution of units of an algebraic number fieldp. 287
On capitulation problem for 3-manifoldsp. 305
On the Iwasawa [mu]-invariant of the cyclotomic Z[subscript p]-extension of certain quartic fieldsp. 315
Modular forms and arithmetic functionsp. 327
Quasimodular solutions of a differential equation of hypergeometric typep. 329
Special values of the standard zeta functionsp. 337
p-adic properties of values of the modular j-functionp. 357
Thompson series and Ramanujan's identitiesp. 367
Generalized Rademacher functions and some congruence propertiesp. 375
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9781402076893
ISBN-10: 1402076894
Series: Developments in Mathematics : Book 11
Audience: General
Format: Hardcover
Language: English
Number Of Pages: 394
Published: 30th November 2003
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 3.18
Weight (kg): 1.65