Galois Theory and Modular Forms

1. Front Matter

   Pages i-xi

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2. Arithmetic geometry

   1. Front Matter

      Pages 1-1

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   2. The Arithmetic of Weierstrass Points on Modular Curves X ₀ (p)

      • Scott Ahlgren

      Pages 3-12

   3. Semistable Abelian Varieties with Small Division Fields

      • Armand Brumer, Kenneth Kramer

      Pages 13-37

   4. Q-curves with Rational j-invariants and Jacobian Surfaces of GL₂-type

      • Ki-ichiro Hashimoto

      Pages 39-64

   5. Points Defined over Cyclic Quartic Extensions on an Elliptic Curve and Generalized Kummer Surfaces

      • Masato Kuwata

      Pages 65-76

   6. The Absolute Anabelian Geometry of Hyperbolic Curves

      • Shinichi Mochizuki

      Pages 77-122

3. Galois groups and Galois extensions

   1. Front Matter

      Pages 123-123

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   2. Regular Galois Realizations of PSL₂(p ²) Over ℚ(T)

      • Takehito Shiina

      Pages 125-142

   3. Middle Convolution and Galois Realizations

      • Michael Dettweiler

      Pages 143-158

   4. On the Essential Dimension of p-Groups

      • Arne Ledet

      Pages 159-172

   5. Explicit Constructions of Generic Polynomials for Some Elementary Groups

      • Yūichi Rikuna

      Pages 173-194

   6. On Dihedral Extensions and Frobenius Extensions

      • Masafumi Imaoka, Yasuhiro Kishi

      Pages 195-220

   7. On the Non-Existence of Certain Galois Extensions

      • Hyunsuk Moon

      Pages 221-231

   8. Frobenius Modules and Galois Groups

      • B. Heinrich Matzat

      Pages 233-267

4. Algebraic number theory

   1. Front Matter

      Pages 269-269

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   2. On Quadratic Number Fields Each Having an Unramified Extension Which Properly Contains the Hilbert Class Field of its Genus Field

      • Ken Yamamura

      Pages 271-286

   3. Distribution of units of an algebraic number field

      • Yoshiyuki Kitaoka

      Pages 287-303

   4. On Capitulation Problem for 3-Manifolds

      • Masanori Morishita

      Pages 305-313

   5. On the Iwasawa μ-Invariant of the Cyclotomic ℤ _p -Extension of Certain Quartic Fields

      • Tsuyoshi Itoh

      Pages 315-325

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.

• Abelian variety
• algebra
• algebraic curve
• Dimension
• field
• Galois group
• Galois theory
• manifold
• modular form
• number theory
• polynomial

• Waseda University Tokyo, Tokyo, Japan

  Ki-ichiro Hashimoto

• Tokyo Metropolitan University, Tokyo, Japan

  Katsuya Miyake

• Okayama University, Okayama, Japan

  Hiroaki Nakamura

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