Prof. Daniel C. Mayer
Prof. Daniel C. Mayer
Austrian Science Fund, Austria
Title: Deep Transfers of p-Class Tower Groups
Given a prime number p and an algebraic number fi eld F, the Galois group G of the maximal unrami ed pro-p extension of F is called the p-tower group of F. In a previous paper, we have proved that the abelian type invariants of p-class groups of unrami ed abelian extensions with degree p of F, and the capitulation of p-classes of F in these extensions, which correspond to kernels of shallow transfers from G to its maximal subgroups of index p and their abelian quotient invariants, determine a nite batch of candidates for the second derived quotient G/G'' of G, usually consisting of isoclinic but non-isomorphic p-groups. In the present lecture, we describe our most recent success in proving that individual members in such a batch of isoclinic p-groups can be identi ed by the principalization behavior of p-classes of unrami ed abelian p-extensions of F with low degree (the prime p) in the rst Hilbert p-class eld of F, that is, an unrami ed abelian p-extension with high degree (a power of p), which corresponds to kernels of deep transfers from maximal subgroups of the p-tower group G to its commutator subgroup G', according to the Artin reciprocity law.
Daniel C. Mayer, born 1956, has received his Ph.D. for mathematics, physics and chemistry in 1983 from the University of Graz, Austria, and his M.Sc. for psychology, pedagogics and didactics in 1995 from the University of Regensburg, Germany. Since 1982, he is university teacher for algebra, arithmetic and analysis. His primary research interests are algebraic number theory, class eld theory and group theory. In 1990, he visited Hugh C. Williams at the University of Manitoba in Winnipeg, Canada, where he re ned the cohomological classi cation of dihedral elds by Nicole Moser with di erential principal fac- torizations and capitulation kernels, thereby proving a conjecture of Arnold Scholz. Further he developed the multiplicity theory for cubic and dihedral discriminants, based on quadratic ring class groups, awarded by an Erwin Schrodinger Grant from the Austrian Science Fund. A generalization followed 1991 for pure metacyclic discriminants in cooperation with Pierre Barrucand at Paris. Since 1992, he serves the scienti c community as reviewer and referee for various international mathematical journals. His joint research with Abdelmalek Azizi and collaborators in Morocco started in 2002 and concerns capitulation problems of ideal classes in number eld extensions. Since 2012, he is the principal investigator and project leader of an international scienti c cooperation between Austria, Morocco, Australia, USA and Japan, supported by the Austrian Science Fund. In these last ve years, his group theoretic innovations, in particular the investigation of descendant trees of p-groups, have led to considerable progress in the theory of maximal unrami ed pro-p extensions of algebraic number elds, which are known as towers of Hilbert p-class elds, for an assigned prime number p. In particular, as a striking world record, he provided rigorous proofs for the rst non-metabelian p-tower groups with soluble length three for p = 3 and p = 5, partially in cooperation with Michael R. Bush at Washington and Lee University in Lexington, Virginia, USA.