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Our Mission:

1. To advance European science
to the forefront
of international research
and to stabilize this position.

2. To strengthen cooperation
with international research centers:
  • Harish-Chandra Research Institute,
    Allahabad, India
  • Washington and Lee University,
    Lexington, Virginia, USA
  • Université Mohammed Premier,
    Oujda, Morocco
  • Australian National University,
    Canberra, Capital Territory
  • Aichi University of Education,
    Nagoya, Japan
  • University of Calgary,
    Alberta, Canada



International Conferences:

January 21 - 25, 2019:
Asia-Australia Algebra Conference 2019
Western Sydney University, Parramatta City campus
Sydney, New South Wales, Australia


April 25 - 27, 2019:
Conference on Algebra, Number Theory and Their Applications
Université Mohammed Premier, Faculté des Sciences
Oujda, Region Oriental, Morocco
Daniel C. Mayer's 3 Lectures:
1. Proving the Conjecture of Scholz
2. Differential principal factors
3. Pure Metacyclic Fields


July 1 - 5, 2019:
31st Journées Arithmétiques
Istanbul University, Faculty of Science
JA 2019, Istanbul, Turkey



Brussels - Town Hall

Open Problem and Conjecture concerning
Real Quadratic Number Fields with 3-Class Group of Type (3,3):

  1. ASSUMPTIONS. Let K = Q(d1/2) be a real quadratic field
    with 3-class group Cl3(K) ∼ C(3)×C(3),
    that is, with abelian type invariants (3,3).

    Denote by N1,N2,N3,N4 the four unramified cyclic cubic extensions of K
    within the Hilbert 3-class field F31(K) of K,
    which exist according to the Artin reciprocity law of class field theory.

    Let Ji : Cl3(K) → Cl3(Ni) be the natural extension homomorphisms of 3-classes, for 1 ≤ i ≤ 4.
    Their kernels are called the capitulation kernels of K in N1,N2,N3,N4.

  2. CONJECTURE. The situation with four total capitulation kernels,
    ker(Ji) = Cl3(K) , for 1 ≤ i ≤ 4,
    cannot occur for identical 3-class groups Cl3(Ni) ∼ C(3)×C(3), for 1 ≤ i ≤ 4.
    (In terms of the Artin transfer pattern:
    The structure (τ(K),κ(K)) ∼ ((11,11,11,11),(0000)) is forbidden.)

    HINT. To discourage unnecessary waste of time, we point out the following:
    For each 1 ≤ i ≤ 4, let Li be one of the three isomorphic totally real cubic subfields of Ni.
    Since we necessarily would have Cl3(Li) ∼ C(3), for 1 ≤ i ≤ 4,
    the class number relation would not be violated by the problematic situation:
    h3(Ni) = 9 = (1/9) × 9 × 32 = (Q/32) × h3(K) × h3(Li)2, for 1 ≤ i ≤ 4,
    where Q = 1 denotes the coinciding index of subfield units for all extensions N1,N2,N3,N4.

  3. REMARK. The simplest situation with four total capitulation kernels,
    which actually occurs, is the following:
    a single distinguished 3-class group Cl3(N1) ∼ C(9)×C(9) and Cl3(Ni) ∼ C(3)×C(3), for 2 ≤ i ≤ 4
    (In terms of the Artin transfer pattern:
    The admissible structure is (τ(K),κ(K)) ∼ ((22,11,11,11),(0000)).)
    The situation occurs indeed, for instance, in the case d = 62501.


    Since we have Cl3(L1) ∼ C(9) and Cl3(Li) ∼ C(3), for 2 ≤ i ≤ 4,
    the class number relation for the first extension in the actual situation is:
    h3(N1) = 81 = (1/9) × 9 × 92 = (Q/32) × h3(K) × h3(L1)2.


Daniel C. Mayer


Principal Investigator and
Project Leader of several

International Scientific Research Lines:






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Web master's e-mail address:
contact@algebra.at
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Fame For Austria 2018
Prime Number 2017
IPAD and IPOD 2016
29ièmes Journées Arithmétiques 2015
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Research Frontier 2013
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