Partial zeta functions of algebraic varieties over finite fields (Q5927549)

The author introduces the notion of a partial zeta function which generalizes the classical zeta function of an algebraic variety defined over a finite field. NEWLINENEWLINENEWLINEMore precisely, let \(X\) be an affine algebraic variety over \(\mathbb F_q\) embedded in some affine space \(\mathbb A^n\). Let \(d_1, \cdots, d_n\) be n positive integers. For each \(k\), let NEWLINE\[NEWLINE N_{d_1,\cdots,d_n}(k,X)=\#\{x\in X(\overline{\mathbb F_q})\mid x_1\in\mathbb F_{q^{d_1k}},\cdots,x_n\in\mathbb F_{q^{d_nk}}\}, NEWLINE\]NEWLINE then the associated partial zeta function \(Z_{d_1,\cdots,d_n}(X,T)\) is defined as NEWLINE\[NEWLINE Z_{d_1,\cdots,d_n}(X,T)=\exp \big(\sum_{k=1}^{\infty}\frac{N_{d_1,\cdots,d_n}}{k}T^k \big). NEWLINE\]NEWLINE In the special case that \(d_1=\cdots=d_n=d\), the partial zeta function \(Z_{d,\cdots,d}(X,T)\) becomes the classical zeta function \(Z(X\otimes\mathbb F_q,T)\) which is rational and satisfies a suitable Riemann hypothesis. NEWLINENEWLINENEWLINEIn this paper, the author gives two approaches to the general structural properties of the partial zeta function in the direction of the Weil-type conjectures. NEWLINENEWLINENEWLINEThe first one, using an inductive fibred variety point of view, shows that the partial zeta function is rational when the sequence of positive integers \(\{d_1,\cdots,d_n\}\) can be rearranged such that \(d_1|d_2|\cdots|d_n\). NEWLINENEWLINENEWLINEThe second approach, due to Faltings, shows that the partial zeta function is always nearly rational. NEWLINENEWLINENEWLINEIn the last section, the author gives some examples and explains briefly their connections with other more classical problems.