Rationality of motivic zeta functions for curves with finite abelian group actions (Q2376579)

Let \(X\) be a variety over a finite field \(\mathbb{F}\) and let \(\text{Sym}^n(X)\) be the \(n\)th symmetric power of \(X\). A famous result by \textit{B. Dwork} [Am. J. Math. 82, 631--648 (1960; Zbl 0173.48501)] states that the zeta-function \[ Z_X(t)= \sum_i |\text{Sym}^n(X)| t^n \] is a rational function on \(t\). If \(k\) is any field one can consider the Grothendieck ring \(A(k)= K_0({\mathcal V}_k)\) for varieties over \(k\), i. e., the ring of \(\mathbb{Z}\)-combinations of isomorphism classes of \(k\)-varieties modulo the relations \[ [X]=[Y]+ [X-Y] \] for closed varieties \(Y\subset X\). Then the motivic zeta function is defined, as a power series in the ring \(A(k)\), by \[ \zeta_X(t)= \sum_i [\text{Sym}^n(X)] t^n.\tag{1} \] By a result of \textit{M. Kapranov} [``The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups'', \url{arXiv:math/0001005}], if \(X\) is a curve then the function in (1) is rational, while the same statement is false for higher-dimensional varieties. The aim of this paper is to prove the rationality of motivic zeta functions for curves with a finite abelian group action. Let \(G\) be a fixed algebraic group and let \({\mathcal V}^G_k\) be the category of varieties with \(G\)-actions. An object in \({\mathcal V}^G_k\) is a pair \((V,\sigma)\), with \(X\in{\mathcal V}_k\) and \(\sigma: G\times X\to X\) an algebraic group action of \(G\) on \(X\). The Grothendieck group \(K_0({\mathcal V}_k)\) is the free abelian group on isomorphism classes in \({\mathcal V}^G_k\)1 modulo the relations \[ [X,\sigma]= [Y,\tau]+ [X- Y,\sigma], \] where \((Y,\tau)\) is a closed \(G\)-invariant subspace of \((X, \sigma)\). Multiplication in \(K_0({\mathcal V}^G_k)\) is defined by \[ [X,\sigma] [Y,\tau]= [X\times Y,\,\sigma\times\tau], \] where \(\sigma\times\tau: G\times X\times Y\to X\times Y\) is defined by \((\sigma\times\tau)(g,x,y)= (\sigma(g,x), \tau(g,y))\). Let \(A(k,G)\) be the Grothendieck ring obtained in such a way. Then the author proves the following: Theorem 1. Let \(G\) be finite abelian group of order \(m\) and let \(C\) be a non-singular projective curve over an algebraically closed field \(k\) of characteristic \(p\), with \(p\) not dividing \(m\). Let \(\sigma: G\times C\times C\) be a group action on \(C\). Then the motivic zeta function \[ \zeta_{(C,\sigma)}= \sum_i [\text{Sym}^n(C,\sigma)] t^n \] is rational, as a power series in the ring \(A(k,G)\).