[Paper Review] Relative motives and the theory of pseudo-finite fields

Johannes Nicaise|arXiv (Cornell University)|Mar 9, 2004

Algebraic Geometry and Number Theory15 references4 citations

TL;DR

This paper generalizes motivic integration to the relative setting over a base variety $S$ over a field $k$ of characteristic zero, constructing a motivic realization morphism from the Grothendieck ring of pseudo-finite fields over $S$ to the tensor product of $\mathbb{Q}$ with the Grothendieck ring of constructible effective Chow motives over $S$. The key contribution is a geometric proof of relative quantifier elimination for pseudo-finite fields using Galois formulas, enabling a motivic incarnation of parameterized arithmetic integrals.

ABSTRACT

We generalize the motivic incarnation morphism from the theory of arithmetic integration to the relative case, where we work over a base variety S over a field k of characteristic zero. We develop a theory of constructible effective Chow motives over S, and we show how to associate a motive to any S-variety. We give a geometric proof of relative quantifier elimination for pseudo-finite fields, and we construct a morphism from the Grothendieck ring of the theory of pseudo-finite fields over S, to the tensor product of Q with the Grothendieck ring of constructible effective Chow motives. This morphism yields a motivic realization for the parametrized arithmetic motivic integrals of Cluckers-Loeser. Finally, we define relative arc and jet spaces, and the three relative motivic generating series.

Motivation & Objective

• To extend motivic integration to the relative case, where parameters are encoded in a base variety $S$ over a field $k$ of characteristic zero.
• To develop a theory of constructible effective Chow motives over $S$, enabling a motivic realization of arithmetic integrals with parameters.
• To provide a geometric proof of quantifier elimination for ring formulas over $S$ in the theory of pseudo-finite fields, using Galois formulas.
• To construct a morphism from the Grothendieck ring of pseudo-finite fields over $S$ to the Grothendieck ring of motives over $S$ tensored with $\mathbb{Q}$, realizing parameterized arithmetic integrals motivically.
• To define relative arc and jet spaces and three motivic Poincaré series (geometric, arithmetic, Igusa) in the relative setting.

Proposed method

• Constructs the category $CMot_S$ of constructible effective Chow motives over $S$ as a direct limit over finite stratifications of $S$ into smooth, irreducible locally closed subsets.
• Generalizes the motivic Chern character $\chi_c$ to a morphism $K_0(Var_S) \to K_0(CMot_S)$, whose image is denoted $K_0^{mot}(Var_S)$, using constructible resolution of singularities and split exact blow-up sequences.
• Introduces induction and restriction functors on the homotopy category of constructible motives with $G$-action, proving that the motive of a quotient $X/G$ equals the $G$-invariant part of the motive of $X$, via a character decomposition.
• Uses Galois formulas to give a purely geometric proof of relative quantifier elimination: any ring formula over $S$ is equivalent to a quantifier-free Galois formula over pseudo-finite fields.
• Defines relative arc and jet spaces $\mathcal{L}_n(X/S)$, and constructs three motivic Poincaré series: geometric, arithmetic, and Igusa, with coefficients in $K_0(Var_S)$ or $K_0^{mot}(Var_S)\otimes\mathbb{Q}$.
• Establishes base change properties for all three Poincaré series, showing compatibility with pullbacks along morphisms $W \to S$.

Experimental results

Research questions

• RQ1 How can motivic integration be generalized to the relative setting over a base variety $S$ instead of over a field $k$?
• RQ2 What is the geometric structure of definable sets in the theory of pseudo-finite fields over $S$, and can quantifier elimination be proven using Galois-theoretic geometry?
• RQ3 Can a motivic realization morphism $\chi_{(c)}: K_0(PFF_S) \to K_0^{mot}(Var_S)\otimes\mathbb{Q}$ be constructed, and what does it represent in terms of parameterized arithmetic integrals?
• RQ4 How do relative arc and jet spaces behave over $S$, and what are the properties of the associated motivic Poincaré series in the relative setting?
• RQ5 What is the uniformity of jet lifting properties over fibers of $X \to S$, and how does this affect the convergence and base change of Poincaré series?

Key findings

• A geometric proof of relative quantifier elimination is established: any ring formula over $S$ is equivalent to a quantifier-free Galois formula over pseudo-finite fields, with equivalence defined via definable bijections over $M$-valued points of $S$.
• The morphism $\chi_c: K_0(Var_S) \to K_0(CMot_S)$ is constructed, and its image $K_0^{mot}(Var_S)$ provides a motivic realization of the Grothendieck ring of $S$-varieties.
• The motivic realization morphism $\chi_{(c)}: K_0(PFF_S) \to K_0^{mot}(Var_S)\otimes\mathbb{Q}$ is defined, yielding a concrete motivic incarnation of parameterized arithmetic integrals.
• The relative geometric Poincaré series $P_{geom}(X/S;T)$ is well-defined in $K_0(Var_S)[[T]]$ due to uniformity in truncations, as shown by Theorem 9.7.
• The relative arithmetic Poincaré series $P_{arith}(X/S;T)$ is defined via $\chi_{(c)}$ applied to ring formulas $\psi_n$ defining $n$-jets, and satisfies base change: coefficients pull back compatibly under $W \to S$.
• The uniformity result (Theorem 9.7) ensures that for any $S$-scheme $X$ of finite type, the $n$-jet truncations stabilize uniformly, with $n' = cn + e$ for some $c,e > 0$, independent of the fiber.