Dembélé, Lassina (2009) Non-solvable Galois number fields ramified at 2, 3 and 5 only. In: Invited Speaker : Heilbronn Seminar, Bristol University, Bristol, 2 Dec 2009 (Unpublished)

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Abstract

In the mid 90s, Dick Gross made the following conjecture.

Conjecture: For every prime $p$, there exists a non-solvable Galois number field $K$ ramified at $p$ only.

For $p>=11$ this conjecture follows from results of Serre and Swinnerton-Dyer using mod $p$ Galois representations attached to classical modular forms. However, it a consequence of the Serre conjecture, now a theorem thanks to Khare and Wintenberger, et al, that classical modular forms cannot yield the case $p<=7$. In this talk, we show that the conjecture is true for $p=2,3$ and 5 using Galois representations attached to Hilbert modular forms. We will also explain the limitations of this technique for the prime $p=7$, and outline an alternative strategy using the unitary group U(3) attached to the extension Q(zeta_7)/Q(zeta_7)^+.

Item Type:	Conference Item (Paper)
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Official Date:	2 December 2009
Dates:	Date Event 2 December 2009 Completion
Status:	Not Peer Reviewed
Publication Status:	Unpublished
Version or Related Resource:	Dembélé, Lassina (2009). Non-solvable Galois number fields ramified at 2, 3 and 5 only. In: Invited Speaker : Cambridge Number Theory Seminar. Cambridge, 12 Jan 2010.
Conference Paper Type:	Paper
Title of Event:	Invited Speaker : Heilbronn Seminar, Bristol University
Type of Event:	Other
Location of Event:	Bristol
Date(s) of Event:	2 Dec 2009
Related URLs:	Other

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