##
**Definable sets, motives and \(p\)-adic integrals.**
*(English)*
Zbl 1040.14010

In the paper under review, the authors considerably generalize several of their previously established methods and results in arithmetic geometry, and that in various directions. Namely, in a previous paper [J. Denef and F. Loeser, Invent. Math. 135, 201–232 (1999; Zbl 0928.14004)], they had developed a general theory of integration – called motivic integration – on the space of arcs of an algebraic variety \(X\) over a field \(k\) of characteristic zero. This theory, which they now refer to as “geometric” motivic integration, is here replaced by a suitable arithmetic analogue which appears to be much better adapted to the study of certain rationality questions concerning Poincaré series of arithmetic schemes. More precisely, the authors develop a different kind of motivic integration theory, the so-called “arithmetic” motivic integration, which is largely based on methods from general field arithmetic and first-order model theory in mathematical logic. On the other hand, the Poincaré series \(P_p(T)\) associated to any pair \((X,p)\),where \(X\) is a reduced and separated scheme of finite type over \(\mathbb{Z}\) and \(p\) is a prime number, is known to be a rational function. This fact was proved by the first author [J. Denef, Invent. Math. 77, 1–23 (1984; Zbl 0537.12011)]. Then, just a little later, J. Denef, J. Pas, and A. Macintyre proved, independently, that the degrees of the nominator and denominator of the rational function \(P_p(T)\) are bounded independently of the prime number \(p\). In view of these deep results, the basic task of the paper under review was to derive a much stronger uniformity result by constructing a canonical rational function \(P_{\text{ar}}(T)\) which specializes to \(P_p(T)\) for almost all prime numbers \(p\). In fact, it is precisely to this end that the authors develop their new “arithmetic” motivic integration theory, in the first part of the paper, which then turns out to be the right tool for constructing the universal rational function \(P_{\text{ar}}(T)\) by means of the Grothendieck ring of certain Chow motives. The paper consists of ten sections whose contents are as follows:

After a first section devoted to preliminaries on Grothendieck rings of Chow motives, the authors develop in section 2 what is needed, in the sequel, from the theory of Galois stratifications in field arithmetic. This and the construction of virtual motives from Galois stratifications associated to first-order logic formulas, as carried out in section 3, relies on the fundamental work of M. D. Fried and M. Jarden [“Field arithmetic”, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 11, Berlin (1986; Zbl 0625.12001)]. Here the authors also introduce a new class of objects, the so-called “definable subassignments”, which are more geometric in nature than first-order Galois formulas. Section 4 discusses definable subassignments for rings, and section 5 does the same for power series rings. These constructions are then used to define a measure on the set of definable subassignments of the space of germs of arcs of an arithmetic scheme \(X\) and this leads to the basic theory of arithmetic motivic integration developed in section 6. Applications to general rationality results for power series associated to algebraic varieties, in the spirit of the earlier works of J. Denef and F. Loeser mentioned above, are presented in section 7. The general results obtained here strikingly demonstrate the enormous power of the authors’ new arithmetic motivic integration theory, and this insight is even strengthened by the results exhibited in the remaining three sections. Namely, in section 8, the authors show that arithmetic motivic integration indeed specializes to \(p\)-adic integration.

Finally, the arithmetical Poincaré series \(P_{\text{ar}}(T)\) for an algebraic variety \(X\) over a field of characteristic zero is established in section 9 where it is also proved that \(P_{\text{ar}}(T)\) factually specializes to the usual \(p\)-adic Poincaré series when \(k\) is a number field.

As the authors point out, the arithmetical Poincaré series seems to contain much more information about the underlying variety \(X\) than the authors’ formerly studied geometric counterpart \(P_{\text{geom}}(T)\) defined by geometric motivic integration. This observation is demonstrated by means of a concrete example discussed in the concluding section 10. Actually, the authors compute the two different Poincaré series \(P_{\text{ar}}(T)\) and \(P_{\text{geom}}(T)\) for branches of plane algebraic curves, thereby showing that the poles of \(P_{\text{ar}}(T)\) completely determine the Puiseux pairs of the branch, whereas the series \(P_{\text{geom}}(T)\) only encodes the multiplicity at the origin.

All in all, the significance of the methods and results exhibited in this important paper can barely be overestimated, since these achievements unquestionably signify a major step forward in the theory of motivic integration within arithmetic geometry. Moreover, the various contributions of the two authors in this realm, over many years and in their entirety, must be seen as being pioneering and epoch-making in the field.

After a first section devoted to preliminaries on Grothendieck rings of Chow motives, the authors develop in section 2 what is needed, in the sequel, from the theory of Galois stratifications in field arithmetic. This and the construction of virtual motives from Galois stratifications associated to first-order logic formulas, as carried out in section 3, relies on the fundamental work of M. D. Fried and M. Jarden [“Field arithmetic”, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 11, Berlin (1986; Zbl 0625.12001)]. Here the authors also introduce a new class of objects, the so-called “definable subassignments”, which are more geometric in nature than first-order Galois formulas. Section 4 discusses definable subassignments for rings, and section 5 does the same for power series rings. These constructions are then used to define a measure on the set of definable subassignments of the space of germs of arcs of an arithmetic scheme \(X\) and this leads to the basic theory of arithmetic motivic integration developed in section 6. Applications to general rationality results for power series associated to algebraic varieties, in the spirit of the earlier works of J. Denef and F. Loeser mentioned above, are presented in section 7. The general results obtained here strikingly demonstrate the enormous power of the authors’ new arithmetic motivic integration theory, and this insight is even strengthened by the results exhibited in the remaining three sections. Namely, in section 8, the authors show that arithmetic motivic integration indeed specializes to \(p\)-adic integration.

Finally, the arithmetical Poincaré series \(P_{\text{ar}}(T)\) for an algebraic variety \(X\) over a field of characteristic zero is established in section 9 where it is also proved that \(P_{\text{ar}}(T)\) factually specializes to the usual \(p\)-adic Poincaré series when \(k\) is a number field.

As the authors point out, the arithmetical Poincaré series seems to contain much more information about the underlying variety \(X\) than the authors’ formerly studied geometric counterpart \(P_{\text{geom}}(T)\) defined by geometric motivic integration. This observation is demonstrated by means of a concrete example discussed in the concluding section 10. Actually, the authors compute the two different Poincaré series \(P_{\text{ar}}(T)\) and \(P_{\text{geom}}(T)\) for branches of plane algebraic curves, thereby showing that the poles of \(P_{\text{ar}}(T)\) completely determine the Puiseux pairs of the branch, whereas the series \(P_{\text{geom}}(T)\) only encodes the multiplicity at the origin.

All in all, the significance of the methods and results exhibited in this important paper can barely be overestimated, since these achievements unquestionably signify a major step forward in the theory of motivic integration within arithmetic geometry. Moreover, the various contributions of the two authors in this realm, over many years and in their entirety, must be seen as being pioneering and epoch-making in the field.

Reviewer: Werner Kleinert (Berlin)

### MSC:

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

14G20 | Local ground fields in algebraic geometry |

03C10 | Quantifier elimination, model completeness, and related topics |

03C98 | Applications of model theory |

03C68 | Other classical first-order model theory |

03C95 | Abstract model theory |

12E30 | Field arithmetic |

12L12 | Model theory of fields |

14G15 | Finite ground fields in algebraic geometry |

14G27 | Other nonalgebraically closed ground fields in algebraic geometry |

11G25 | Varieties over finite and local fields |

11S40 | Zeta functions and \(L\)-functions |

12L10 | Ultraproducts and field theory |

14F20 | Étale and other Grothendieck topologies and (co)homologies |

14G05 | Rational points |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14J20 | Arithmetic ground fields for surfaces or higher-dimensional varieties |

14F42 | Motivic cohomology; motivic homotopy theory |

### Keywords:

arithmetic geometry; motives; arithmetic motivic integration; field arithmetic; first-order model theory; arithmetical Poincaré series; zeta functions; rationality questions; Galois stratifications; definable subassignments; \(p\)-adic integration
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\textit{J. Denef} and \textit{F. Loeser}, J. Am. Math. Soc. 14, No. 2, 429--469 (2001; Zbl 1040.14010)

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