Trace formula for rings of Witt vectors (Q2363522)

In the paper under review, the authors generalize Pulita exponential series and apply them to establish an analog of the trace formula for rings of Witt vectors. Recall that the zeta function of an algebraic variety \(V\) defined over a finite field \({\mathbb F}_q\), where \(q = p^s\) and \(p\) is an odd prime, is given by \[ \zeta(V, t) = e^{\sum_{r \geq 1} N_r \frac{t^r}{r}}, \] where \(N_r\) is the number of \({\mathbb F}_{q^{r-}}\) points on \(V\). In his classical work \textit{B. Dwork} [Am. J. Math. 82, 631--648 (1960; Zbl 0173.48501)] used the expression of the number \(N_r\) by multiplicative and additive characters to prove the rationality of \(\zeta\), analytically expressed the additive character on \({\mathbb F}_{q^r}\) in terms of the series \(e^{\pi (x - x^q)}\) (where \(\pi = \sqrt[p-1]{-p}\)), and used the trace formula to express \(N_r\). The authors prove an analog of Dwork's trace formula for Gauss sums on Witt vectors over \({\mathbb F}_q\) of finite length \(2\). The main tool used in the argument is a generalisation of \textit{A. Pulita}'s exponential series [Math. Ann. 337, No. 3, 489--555 (2007; Zbl 1125.12001)].