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Journal of the London Mathematical Society Advance Access published online on April 27, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdp003
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© 2009 London Mathematical Society

On ground fields of arithmetic hyperbolic reflection groups. III

Viacheslav V. Nikulin

Department of Pure Mathematics
The University of Liverpool
Liverpool
L69 3BX
United Kingdom

Steklov Mathematical Institute of Russian Academy of Sciences
ul. Gubkina 8
GSP-1
Moscow 117966
Russia
vvnikulin@list.ru

The paper continues from the work of Nikulin. Using our methods of 1980 and 1981, we define some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions at least 3, and we give good upper bounds for their degrees (over Q). This extends the earlier results of Nikulin for dimensions at least 4. This finally delivers a possibility, in principle, of effective finite classification of maximal arithmetic hyperbolic reflection groups (more generally, of reflective hyperbolic lattices) in all dimensions. Our results also give another proof of finiteness in dimension 3. In fact, using our methods, we show that finiteness in dimension 3 follows from finiteness in dimension 2.

2000 Mathematics Subject Classification 20F55, 51F15, 22E40.

Dedicated to I. R. Shafarevich for his 85th birthday

This paper was written with the financial support of EPSRC, United Kingdom (grant no. EP/D061997/1).

Received February 24, 2008; revised December 9, 2008;
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