Cyclotomic Integers and Finite Geometry

The most powerful method for the study of finite geometries with regular or quasiregular automorphism groups G is to translate their definition into an equation over the integral group ring Z[G] and to investigate this equation by applying complex representations of G. For the definitions and the basic facts, see Section 2. If G is abelian, this approach boils down to proving or disproving the existence of elements of Z[G] with 0-1 coefficients whose character values are cyclotomic integers of certain prescribed absolute values. Up to now there have been two general methods to tackle this problem, namely, Hall’s multiplier concept and Turyn’s selfconjugacy approach. However, both methods need severe technical assumptions and thus are not applicable to many classes of problems. Despite many efforts over a period of more than 30 years no general method has been found to overcome these difficulties. In this paper, we present a new approach to the study of combinatorial structures via group ring equations which works without any restrictive assumptions. In order to understand the method of the present paper it will be instructive to briefly discuss the self-conjugacy concept first. Turyn [51] demonstrated that the character method for the study of group ring equations works very nicely under the so-called self-conjugacy condition. An integer n is called self-conjugate modulo m if all prime ideals above n in the mth cyclotomic field Q(ξm) are invariant under complex conjugation. Under this condition it is possible to find all cyclotomic integers in Q(ξm) of absolute value n for any positive integer t. It is the complete knowledge of the cyclotomic integers of prescribed absolute value which makes the character method work so well under the self-conjugacy condition. Since Turyn’s fundamental work [51] there have been dozens of papers extending and refining his approach. However, all these results are restricted to the case of self-conjugacy, and that is a very severe restriction indeed. Namely, the “probability” that n is selfconjugate modulo m decreases exponentially fast in the number of distinct prime divisors of n and m; see Remark 2.2. This means that the self-conjugacy method fails in almost all cases. One may ask if it is possible to extend Turyn’s method in order to get rid of the self-conjugacy assumption. It turns out that in general this is impossible – at least with present day methods. The required complete knowledge of the cyclotomic integers of prescribed absolute value would yield an