We consider zeta functions with values in the Grothendieck ring of Chow motives. Investigating the $\lambda$–structure of this ring, we deduce a functional equation for the zeta function of abelian varieties. Furthermore, we show that the property of having a rational zeta function satisfying a functional equation is preserved under products.

Let $X$ be a proper smooth variety over a field $k$ of characteristic $0$ and $Y$ an effective divisor on $X$ with multiplicity. We introduce a generalized Albanese variety Alb$(X,Y)$ of $X$ of modulus $Y$, as higher dimensional analogue of the generalized Jacobian with modulus of Rosenlicht-Serre. Our construction is algebraic. For $k=ℂ$ we give a Hodge theoretic description.

Let $k$ be a field. We compute the set ${\left[{𝐏}^1,{𝐏}^1\right]}^{\mathrm{N}}$ ofnaivehomotopy classes of pointed $k$-scheme endomorphisms of the projective line ${𝐏}^1$. Our result compares well with Morel’s computation in [11] of thegroup${\left[{𝐏}^1,{𝐏}^1\right]}^{{𝐀}^1}$ of ${𝐀}^1$-homotopy classes of pointed endomorphisms of ${𝐏}^1$: the set ${\left[{𝐏}^1,{𝐏}^1\right]}^{\mathrm{N}}$ admits an a priori monoid structure such that the canonical map ${\left[{𝐏}^1,{𝐏}^1\right]}^{\mathrm{N}}\to {\left[{𝐏}^1,{𝐏}^1\right]}^{{𝐀}^1}$ is a group completion.

We relate $R$-equivalence on tori with Voevodsky’s theory of homotopy invariant Nisnevich sheaves with transfers and effective motivic complexes.