H. Boerner, Darstellungen von Gruppen. 2. überarb. Auflage. (Die Grundlehren der mathematischen Wissenschaften, Band 74). XIV + 317 S. m. 14 Abb. Berlin/Heidelberg/New York 1967. Springer‐Verlag. Preis geb. DM 58,–

Space groups and groups of prime-power order I

Inspired by a construction by Arnaud Beauville of a surface of general type with K 2 = 8, p g = 0, the second author defined Beauville surfaces as the surfaces which are rigid, i.e., without nontrivial deformations, and which admit an unramified covering which is isomorphic to a product of curves of genus at least 2.

Beauville surfaces without real structures

Sets of elements that pairwise generate a linear group

In this paper we study the influence of the partial cover and avoidance property on the subgroups of some relevant families of subgroups in a finite group.

Local embeddings of some families of subgroups of finite groups

Introduction. Historically, two approaches have been followed to study the classical Fermat equation x r + y r = z r. The first, based on cyclotomic fields, leads to questions about abelian extensions and class numbers of K = Q(ζ r) and values of the Dedekind zeta-function ζ K (s) at s = 0. Many open questions remain, such as Vandiver's conjecture that r does not divide the class number of Q(ζ r) +. The second approach is based on modular forms and the study of 2-dimensional representations of Gal(¯ Q/Q). Even though 2-dimensional representations are more subtle than abelian ones, it is by this route that Fermat's last theorem was finally proved (cf. [Fre], [Se2], [Ri2], [W3], and [TW]; or [DDT] for a general overview). This article examines the equation x p + y q = z r. (1) Certain 2-dimensional representations of Gal(¯ K/K), where K is the real subfield of a cyclotomic field, emerge naturally in the study of equation (1), giving rise to a blend of the cyclotomic and modular approaches. The special values ζ K (−1), which 413 414 HENRI DARMON in certain cases are related to the class numbers of totally definite quaternion algebras over K, appear as obstructions to proving that (1) has no solutions. The condition that r is a regular prime also plays a key role in the analysis leading to one of our main results about the equation x p + y p = z r (see Theorem 3.22). One is interested in primitive solutions (a, b, c) to equation (1), that is, those satisfying gcd(a, b, c) = 1. (Such a condition is natural in light of the abc conjecture, for example; see also [Da2].) A solution is called nontrivial if abc = 0. It is assumed from now on that the exponents p, q, and r are prime and that p is odd. Let (a, b, c) be a nontrivial primitive solution to equation (1). One wishes to show that it does not exist. The program for obtaining the desired contradiction, following the argument initiated by Frey and brought to a successful conclusion by Wiles in the case of x p + y p = z p , can be divided into four steps. Step 1 (Frey, Serre). Associate to (a, b, c) a mod p Galois representation ρ : Gal(¯ K/K) −→ GL 2 (F) having \" …

http://swc.math.arizona.edu/aws/1998/98Darmon.pdf

H. Boerner, Darstellungen von Gruppen. 2. überarb. Auflage. (Die Grundlehren der mathematischen Wissenschaften, Band 74). XIV + 317 S. m. 14 Abb. Berlin/Heidelberg/New York 1967. Springer‐Verlag. Preis geb. DM 58,–

Citations

Space groups and groups of prime-power order I

Beauville surfaces without real structures

Sets of elements that pairwise generate a linear group

Local embeddings of some families of subgroups of finite groups

Rigid local systems, Hilbert modular forms, and Fermat's last theorem

Related Papers (2)

Endliche Gruppen I

Atlas of finite groups : maximal subgroups and ordinary characters for simple groups