Galois Theory and Modular Forms

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.

I. Arithmetic geometry.- The arithmetic of Weierstrass points on modular curves X0(p).- Semistable abelian varieties with small division fields.- Q-curves with rational j-invariants and jacobian surfaces of GL2-type.- Points defined over cyclic quartic extensions on an elliptic curve and generalized Kummer surfaces.- The absolute anabelian geometry of hyperbolic curves.- II. Galois groups and Galois extensions.- Regular Galois realizations of PSL2(p2) over ?(T).- Middle convolution and Galois realizations.- On the essential dimension of p-groups.- Explicit constructions of generic polynomials for some elementary groups.- On dihedral extensions and Frobenius extensions.- On the non-existence of certain Galois extensions.- Frobenius modules and Galois groups.- III. Algebraic number theory.- On quadratic number fields each having an unramified extension which properly contains the Hilbert class field of its genus field.- Distribution of units of an algebraic number field.- On capitulation problem for 3-manifolds.- On the Iwasawa ?-invariant of the cyclotomic ?p-extension of certain quartic fields.- IV. Modular forms and arithmetic functions.- Quasimodular solutions of a differential equation of hypergeometric type.- Special values of the standard zeta functions.- p-adic properties of values of the modular j-function.- Thompson series and Ramanujan’s identities.- Generalized Rademacher functions and some congruence properties.