On ground fields of arithmetic hyperbolic reflection groups. III

Abstract

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.