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Cyclotomic integers and finite geometry. (English) Zbl 0939.05016

The author presents a new approach to the study of combinatorial structures via group ring equations.

In a fundamental paper of R. J. Turyn [Character sums and difference sets, Pac. J. Math. 15, 319-346 (1965; Zbl 0135.05403)] it is shown that the character method for the study of group ring equations works nicely under the so-called self-conjugacy condition. Recall that an integer $n$ is called self-conjugate modulo $m$ if all prime ideals above $n$ in the $m$th cyclotomic field $ℚ\left({\xi }_{m}\right)$ (with ${\xi }_{m}:={e}^{2\pi i/m}$) are invariant under complex conjugation. Under this condition all cyclotomic integers in $ℚ\left({\xi }_{m}\right)$ of absolute value ${n}^{t/2}$ can be determined for any integer $t\ge 1$. The complete knowledge of the cyclotomic integers of prescribed absolute value is the key ingredient making the character method work so well under the self-conjugacy condition. However, self-conjugacy is a very severe restriction, i.e. the self-conjugacy method fails in almost all cases, because the ‘probability’ that $n$ is self-conjugate modulo $m$ decreases exponentially fast in the number of distinct prime divisors of $n$ and $m$. Knowing the cyclotomic integers of prescribed absolute value completely would lead to an almost complete determination of the class group of the underlying cyclotomic field modulo the class group of its maximal real subfield. This, however, is a problem of algebraic number theory that appears to be far beyond the scope of the methods known today. This demonstrates the needs for more general results about cyclotomic integers of prescribed absolute value.

The author presents a new approach to the absolute value problem. Exploiting the decomposition groups of prime ideals the following key result is proved, where $F\left(m,n\right)$ is an integral valued function the definition of which is too involved to be restated here. Theorem 3.5. Assume $X\overline{X}=n$ for $X\in ℤ\left[{\xi }_{m}\right]$ where $n$ and $m$ are positive integers. Then $X{\xi }_{m}^{j}\in ℤ\left[{\xi }_{F\left(m,n\right)}\right]$ for some $j$.

This reduction to subfields is the key to obtain a general bound on the absolute value of cyclotomic integers, the upshot of which is Theorem 4.2. Let $X\in ℤ\left[{\xi }_{m}\right]$ be of the form $X={\sum }_{i=0}^{m-1}{a}_{i}{\xi }_{m}^{i}$ where ${a}_{0},\cdots ,{a}_{m-1}$ are integers with $0\le {a}_{i}\le C$ for some constant $C$. Furthermore, assume that $X\overline{X}=n$ is an integer. Then $n\le {2}^{s-1}{C}^{2}F\left(m,n\right)$ where $s$ is the number of distinct odd prime divisors of $m$. If the assumption on the coefficients ${a}_{i}$ is replaced by $|{a}_{i}|\le C$, then $n\le {2}^{t}{C}^{2}F\left(m,n\right)$ where $t$ is the number of distinct prime divisors of $m$.

The preceeding theorems are applied to derive a new general exponent bound for difference sets in groups: Theorem 5.2. Assume the existence of a $\left(v,k,\lambda ,n\right)$-difference set $D$ in a group $G$. If $U$ is a normal subgroup of $G$ such that $G/U$ is cyclic of order $e$, then $e\le v{\left({2}^{s-1}F\left(e,n\right)/n\right)}^{1/2}$ where $s$ is the number of distinct odd prime divisors of $e$. This result is then applied to all parameter series corresponding to known difference sets with $\text{gcd}\left(v,n\right)>1$, i.e. to Hadamard, McFarland, Spence and Chen/Davis/Jedwab parameters. The results obtained here have strong implications for the (non-)existence of circulant Hadamard matrices, which is outlined at the end of Section 6. Section 7 presents a general exponent bound for groups containing relative difference sets. With this at hand, the author derives strong necessary conditions for the existence of quasiregular projective planes which in turn lead to asymptotic exponent bounds for abelian groups admitting planar functions. Finally, in Section 8, the methods developed in the paper are utilized for the study of group invariant weighing matrices.

##### MSC:
 05B10 Difference sets 05B20 Matrices (incidence, Hadamard, etc.)