JournalsowrVol. 3 , No. 1pp. 317–378

The Arithmetic of Fields

  • Wulf-Dieter Geyer

    Universität Erlangen-Nünberg, Germany
  • Moshe Jarden

    Tel Aviv University, Israel
  • Florian Pop

    University of Pennsylvania, Philadelphia, United States
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Abstract

The fifth conference with the title {\bf The Arithmetic of Fields}, organized by Wulf-Dieter Geyer (Erlangen), Moshe Jarden (Tel Aviv), and Florian Pop (Philadelphia), was held February 5--11th, 2006. In contrast to the fourth conference held in February 3--9th, 2002, this conference was a 'full' one, namely as many participants were invited as the Institute could host. Due to support from the European Union, more young people were invited in the last few weeks prior to the conference, so that the total number of participants reached 54. The participants came from 13countries: Germany(20), USA~(10), Israel~(7), France~(7), Denmark~(2), Austria~(1), Brazil~(1), Canada~(1), Hungary~(1), Japan~(1), Romania~(1), Russia~(1), and South Africa~(1). Among the participants there were 9 graduate students and 8 young researchers. Six women attended the conference.

The organisers asked four people before the conference to give surveys of one hour on recent progress made by other colleagues in Field Arithmetic.

Tam\'as Szamuely (Budapest) reported on the solution by J\'anos Koll\'ar of a Problem due to Ax from 1968: Every PAC field of characteristic 00 is C1C_1. The case where the characteristic is positive remains open.

Alexandra Shlapentokh (Greenville) described the progress Bjorn Poonen made on Hilbert's Tenth Problem: There exists a recursive set T1T_1 of prime numbers of natural density 00 and a set T2T_2 of prime numbers of natural density 11 such that T1T2T_1\subseteq T_2 and for each set SS with T1ST2T_1\subseteq S\subseteq T_2 Hilbert's Tenth Problem for O\bbQ,SO_{\bbQ,S} has a negative solution. Here O\bbQ,SO_{\bbQ,S} is the ring of all rational numbers whose denominators are divided only by primes in SS. Whether Hilbert's Tenth Problem for \bbQ\bbQ has a negative solution is still open.

Alexander Prestel (Konstanz) presented a theorem of Jochen Koenigsmann: If a pp-Sylow extension PP of a field KK is Henselian and PP is neither separably closed nor real closed, then KK itself is Henselian.

Pierre D\`ebes (Lille) surveyed Fried's problem on Modular Towers. He mentioned that the main conjecture is close to completion in the case of 44 branch points (Bayley-Fried). He also reported on a result of Anna Cadoret: The dihedral group has a regular realization over \bbQab\bbQ_{\rm ab} with only inertia groups of order~22.

In addition to these survey talks seventeen participants were invited to report on their own achievements in 45 minutes talks. Altogether, the talks presented the impressive progress made in Field Arithmetic in recent years. The reader may find here extended abstracts of all talks. We hope they will be to the benefit of all of the participants as well as the fans of Field Arithmetic.

The organisers: Wulf-Dieter Geyer, Moshe Jarden, Florian Pop

Cite this article

Wulf-Dieter Geyer, Moshe Jarden, Florian Pop, The Arithmetic of Fields. Oberwolfach Rep. 3 (2006), no. 1 pp. 317–378

DOI 10.4171/OWR/2006/06