Gauss Sums over Quasi-Frobenius Rings

Philippe Langevin and Patrick Solé

Date: March 1999

Université de Nice-Sophia-Antipolis, CNRS-URA 1376, Laboratoire I3S, Route des Colles, BP145, 06903 Sophia-Antipolis

Let $A$ be a commutative finite ring. The group of additive characters and the group of multiplicative characters of $A$ are respectively denoted by $\widehat{A^{+}}$ and $\widehat{A^{\times}}$. One defines the complete Gauss sum related to $\chi\in \widehat{A^{\times}}$ and $\psi\in \widehat{A^{+}}$ by :

$$G_A(\chi,\psi) = \sum_{x\in A} \chi(x) \psi(x),$$

and , for any subgroup $S$ of ${{A}^\times}$, the incomplete Gauss sum $G_S(\chi,\psi)$ is $\sum_{x\in S} \chi(x) \psi(x)$. These sums have been extensively studied in the context of the finite fields. In this paper, we examine the case of local, commutative quasi-Frobenius rings. Such a ring has at least one generating character $\mu$ such that for all $\psi\in {{A}^{+}}$ there exists $a\in A$ satisfying : $\psi(x) = \mu(ax)$ for any $x\in A$. The quasi-Frobenius rings are plentiful since finite fields, the quotient rings ${\bf Z}/m{\bf Z}$, and Galois rings are quasi-Frobenius rings. We suppose that $A$ is a local ring with maximal ideal $M$, commutative, and quasi-Frobenius that is not a field. The set $1+\hbox{ann}(M)$ is a subgroup of ${{A}^\times}$, called the strong units group of $A$and denoted by $U$. The following assertions hold

(1)

$A$ has one and only one minimal ideal.

(2)

$\hbox{ann}(M)$ has the same order, say $q$, than the residual field of $A$.

and lead to a first theorem.

Theorem 1   Let $\mu$ be a generating character and let $\chi\in \widehat{A^{\times}}$.

$$\vert G_A(\chi,\mu) \vert{}^2= \begin{cases} {\vert{A}\vert}, &\chi\not\perp U\\ 0,& \chi \perp U\\ \end{cases}$$

Corollary 1   Let $S$ be a subgroup of $A^\times$. Let $\chi\in \widehat{A^{\times}}$. If $\chi\perp U$ then the incomplete Gauss sum $G_S(\chi,\mu)$ has absolute value less than or equal to $\frac{q- \epsilon}q\sqrt{{\vert{A}\vert}}$, where $\epsilon= {\vert{U\cap S}\vert}$.

The ring $A$ is local and has one and only one multiplicative subgroup $T$ of order $q-1$. This is the Teichmüller subgroup of $A$. In the case where $A$ is a Galois ring of characteristic $p^\ell$ the absolute value of the incomplete sums $G_T(\chi,\mu)$ is not constant. We know that it is bounded above by $p^{\ell-1}\sqrt{q}$, when $f=2$ and $\ell=2$, the estimate of the corollary is slightly better.

Now, if we suppose that $A$ is a Galois ring of characteristic four , we get a non-archimedian estimate of these sums that generalizes the congruences of Stickelberger.

Theorem 2   Let $\xi$ be the principal $(q-1)$-root of $1$. Let $\wp$ be a prime ideal above $2$ in the ring ${\bf Z}[i,\xi]$ where $i^2=-1$. There exists a multiplicative character $\omega$ of order $q-1$ such that for any integer $a\in[0,q-1]$, the $\wp$-adic valuation of $G_{T}(\omega^a, \mu_A)$ is larger than the sum of the binary digits of $a$.