个人简介 | |
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Dr. Daniel C. Mayer Austrian Science Fund, Austria |
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标题: Deep Transfers of p-Class Tower Groups | |
摘要:
Given a prime number p and an algebraic number field F, the Galois group G of
the maximal unramied 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 unramied 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 identied by the principalization behavior of
p-classes of unramied abelian p-extensions of F with low degree (the prime p) in the rst
Hilbert p-class eld of F, that is, an unramied 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.
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简介:
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 rened the
cohomological classication of dihedral elds by Nicole Moser with dierential 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 scientic 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 scientic 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 unramied 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.
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