KIAS Number Theory Seminar



On the inverse Galois problem with restricted ramification 1 and 2

Joachim Koenig (Technion/Wurzburg)

November 14 (Tue), 2017, 4:00pm-6:00pm, Room 1424
November 16 (Thu), 2017, 4:00pm-6:00pm, Room 1424
(Please ignore November 15 in the KIAS webpage.)

Abstract: In this series of talks, I will consider various strong versions of the inverse Galois problem with added conditions on ramified primes. I will first discuss various versions of "minimal ramification problems" and review known results and conjectures. Here "minimal ramification" can be understood as the problem to minimize the number of primes ramifiying in an extension with a given Galois group, or also as the problem to minimize the ramification indices. I will also discuss relations to conjectures and heuristics about the distribution of Galois groups (such as the Malle-Bhargava heuristics) and the distribution of class groups (such as the Cohen-Lenstra heuristics). I will furthermore present new results on Galois realizations with restricted ramification indices, obtained in joined work with D.Neftin and J.Sonn. We use two essentially different approaches, one coming from Shavarevich's method for solvable group, and another one coming from geometric Galois theory and specialization of Galois coverings. Finally, I will present some results on so-called intersective polynomials. The notion of intersective polynomials, that is, integer polynomials with a root in every Q_p, but not in Q, was introduced by Sonn. Their construction is related to restricted-ramification problems. I will present conjectures and new results on the existence of such polynomials for prescribed Galois groups. This last talk in particular will be very easily accessible without a lot of theoretical background.



In each month, we usually meet at

  • Room 1423 or 1424,
  • 4 or 5PM, and
  • Thursdays or Fridays.

    However, there are also many exceptions. Please check the tentative schedule of upcoming seminars.

  • Please send the organizer an email if you are interested in giving a talk. If you want to receive regular email announcement, please send an email with your name/affiliation, in title of "subscribe", to the organizer.

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    The first number theory seminar started March 10th, 2005 and it became a regular seminar since then. For more information on the early history of the number theory seminar, see the article (by Soon-Yi Kang in KIAS News Letter, Fall 2005) and the following archive files.

       Archive: 2017 | 2016 | 2015 | 2014 | 2013 | 2012 | 2011 | 2010 | 2009 | 2008 | 2007 | 2006 | 2005 |

    Organizer:

    Chan-Ho Kim

    Research Fellow
    Korea Institute for Advanced Study (KIAS)
    85 Hoegiro,  Dongdaemun-gu, Seoul 130-722,
    Republic of Korea
    Tel: +82-2-958-3722, Fax: +82-2-958-3786
    Email: chanho at kias.re.kr

    Number Theory Group at KIAS:
    (email: +@kias.re.kr)

    Choi, Youn-Seo (y-choi2)
    Park, Chol (cpark)
    Kim, Eunmi (ekim)
    Kim, Chan-Ho (chanho)
    Kim, Kwang-Seob (kwang12)
    Kim, Wansu (wansukim)
    Ha, Junsoo (junsooha)
    Haan, Jae-ho (jaehohaan)

    (*) The picture is originally a PGF created by Jang Soo Kim and is under the creative commons licence.