Geometric proofs of theorems of Ax-Kochen and Eršov

# Geometric proofs of theorems of Ax-Kochen and Eršov

Jan Denef Dedicated to the memory of Professor Jun-ichi Igusa KULeuven, Department of Mathematics, Celestijnenlaan 200 B, B-3001 Leuven (Heverlee), Belgium.
October 30, 2015
###### Abstract.

We give an algebraic geometric proof of the Theorem of Ax and Kochen on p-adic diophantine equations in many variables. Unlike Ax-Kochen’s proof, ours does not use any notions from mathematical logic and is based on weak toroidalization of morphisms. We also show how this geometric approach yields new proofs of the Ax-Kochen-Eršov transfer principle for local fields, and of quantifier elimination theorems of Basarab and Pas.

We thank Dan Abramovich, Steven Dale Cutkosky, and Kalle Karu for stimulating conversations and information.

## 1. Introduction

The purpose of the present paper is to give purely algebraic geometric proofs of the following theorem of Ax and Kochen [3] and of some other related results, such as quantifier elimination, which go back to Ax, Kochen, and Eršov.

###### Theorem 1.1.

Ax-Kochen’s Theorem on p-adic forms. For any positive integer there exists a positive integer such that for each prime , any homogeneous polynomial over the ring of p-adic integers, with degree and more than variables, has a nontrivial zero over .

The proof of Ax-Kochen, and also ours, starts from the well known elementary fact that the above theorem is true for replaced by the ring of formal power series over the field with elements (in this case one can take ). Thus, in order to prove Theorem 1.1, it suffices to prove the next theorem (applying it to the universal family of hypersurfaces of degree in projective space of dimension ).

###### Theorem 1.2.

Transfer of surjectivity. Let be a morphism of integral separable schemes of finite type over . For all large enough primes we have:

 f:X(Zp)→Y(Zp)

is surjective if and only if

 f:X(Fp[[t]])→Y(Fp[[t]])

is surjective.

In the present paper we prove Theorem 1.2 by using the Theorem of Abramovich and Karu [1] on Weak Toroidalization of Morphisms (extended to non-closed fields in [2]). Instead of weak toroidalization one could use Cutkosky’s Theorem on Local Monomialization [13], see Remark 5.4. In [19] we give a second algebraic geometric proof of Theorem 1.1, not using transfer to , by proving a conjecture of Colliot-Thélène [12]. Our proof [19] of this conjecture is also based on weak toroidalization of morphisms.

Ax and Kochen obtained Theorem 1.1 as a direct consequence of the following more powerful Transfer Principle due to Ax and Kochen [3] and (independently) Eršov [22, 23].

###### Theorem 1.3.

Ax-Kochen-Eršov Transfer Principle. Let be an assertion in the language of rings (see section 8.1 ). For all large enough primes we have the following. The assertion is true in if and only if it is true in .

Ax-Kochen and Eršov proved this Transfer Principle using methods from mathematical logic (model theory). An elementary but very ingenious proof has been given by Cohen [11](see also Weispfenning [33], and Pas [31]), but his method is still very much in the spirit of mathematical logic. In the present paper we also give a new proof of this Transfer Principle, again based on weak toroidalization.

Moreover, in section 8 we give a geometric proof of Basarab’s Quantifier Elimination Theorem [5] for henselian valuation rings of residue field characteristic large enough or zero. Basarab’s result is a refinement of a quantifier elimination theorem of Ax-Kochen [4], and is related to work of Delon [17] and Weispfenning [34]. His proof uses the methods of Ax-Kochen and Eršov, and is based on model theory. Basarab’s theorem directly implies the quantifier elimination theorem of Pas [31], which has several applications in arithmetic algebraic geometry. It enables to study certain integrals over local fields [18],[31],[7], in particular generalizations of Igusa’s local zeta functions [24, 25], and has several applications to motivic integration [21],[9, 10, 8]. This relates to work of Lichtin [27, 28], who was the first to apply monomialization of morphisms (i.e. local toroidalization) to study multivariate Igusa fiber integrals.

The present paper is about henselian valuation rings of residue field characteristic large enough or zero. Using multiplicative residues of higher order (i.e. with respect to certain proper ideals), instead of the multiplicative residues introduced in section 2, the method of the present paper can be easily adapted to give geometric proofs of quantifier elimination results of Basarab [5] and Pas [32] that are valid for henselian valuation rings of characteristic zero, without any restriction on the residue field. This approach can be much simplified in case of , for a fixed prime , using compactness, to get an easy proof, based on weak toroidalization, of Macintyre’s Quantifier Elimination Theorem [29]; this is done in [20].

Our paper is organized as follows. In section 2 we discuss multiplicative residues of elements in local integral domains. These also play a key role in Basarab’s paper [5] on quantifier elimination. In section 3 we formulate and prove a logarithmic version of Hensel’s Lemma. We did not fit it in the framework of logarithmic geometry, but this has been done more recently by Cao [6]. We recall the Weak Toroidalization Theorem in section 4. The heart of the present paper is section 5 where we state and prove what we call the Tameness Theorem 5.1. The Weak Toroidalization Theorem reduces its proof to the case of a log-smooth morphism where the Tameness Theorem is a direct consequence of the logarithmic Hensel’s Lemma. We prove Theorem 1.2 on transfer of surjectivity in section 7 as an easy consequence of the Tameness Theorem and Lemma 6.1 on transfer of residues. This lemma is stated and proved in section 6 as an easy application of embedded resolution of singularities. Finally, in section 8 we formulate and prove Basarab’s Quantifier Elimination Theorem 8.4 using the Tameness Theorem, and we prove the Ax-Kochen-Eršov Transfer Principle 1.3 as a direct consequence of Basarab’s Theorem.

Our motivation to develop an algebraic geometric approach to quantifier elimination for henselian valuation rings, comes from the above mentioned applications to the study of variants of the local zeta functions that Igusa introduced in [24, 25]. We are happy to dedicate this work to the memory of late Professor Jun-ichi Igusa.

### 1.4. Terminology and notation

In the present paper, will always denote a noetherian integral domain. A variety over is an integral separated scheme of finite type over . A rational function on a variety over is called regular at a point if it belongs to the local ring of at , and it is called regular if it is regular at each point of .

Uniformizing parameters over on a variety over , are regular rational functions on that induce an étale morphism to an affine space over .

A reduced strict normal crossings divisor over on a smooth variety over is a closed subset of such for any there exist uniformizing parameters over on an open neighborhood of , such that for any irreducible component of , containing , there is an which generates the ideal of in .

## 2. Multiplicative residues

Let be a noetherian integral domain, and a variety over . Let be any local -algebra which is an integral domain. We denote by its maximal ideal, by its field of fractions, and by the generic point of . For any -rational point on we denote by the -rational point on induced by . For any the pullback of to is denoted by . Moreover, for we write to say that .

###### Definition 2.1.

Let . The elements have same multiplicative residue if

 z′∈z(1+mA).

Let and let be rational functions on . The points have the same residues with respect to if and, for , the following two conditions hold.

1. The rational function is regular at if and only if it is regular at .

2. have same multiplicative residue if is regular at both and .

Instead of working with rational functions we can also work with locally principal closed subschemes of , i.e. subschemes whose sheaf of ideals is locally generated by a single element.

###### Definition 2.2.

Let and locally principal closed subschemes of . The points have the same residues with respect to if and, for , the following condition holds. Let be a generator for the ideal of at , then have same multiplicative residue.

###### Lemma 2.3.

Let be an affine variety over , and let be rational functions on . Then there exist regular rational functions on such that for any local R-algebra , which is an integral domain, and any we have the following. The points and have the same residues with respect to if they have the same residues with respect to .

Proof. This is clear, by taking for any finite list of regular rational functions on which satisfies the following condition. For each and each with regular at , there are elements and in this list with , and . Obviously, such a finite list exists if is affine.

###### Lemma 2.4.

Assume that is a noetherian normal integral domain. Let be a smooth variety over , and regular rational functions on . Let be a reduced strict normal crossings divisor over containing the zero locus of each , for . Then, for any local R-algebra , which is an integral domain, and any we have the following. The points and have the same residues with respect to if they have the same residues with respect to the irreducible components of .

Proof. Assume that have the same residues with respect to the irreducible components of . Set . Then also . Let be the irreducible components of that contain . Since has normal crossings over , there exist uniformizing parameters over on an open neighborhood of , with , such that generates the ideal of in for . We can write each , for , as a monomial in times a unit in , because is normal (being smooth over a normal ring). Obviously, this implies the lemma, because and have same multiplicative residue if .

### 2.5. The structure MR(A) of multiplicative residues

Assume now that , thus is any local integral domain. Denote by the set of equivalence classes of the equivalence relation on . The equivalence class of an element is called the multiplicative residue of and is denoted by . Note that is a commutative monoid with identity and with multiplication induced by the one on . Moreover is equipped with the natural multiplicative map onto the residue field . This map induces an isomorphism from the group of elements of which have an inverse, onto the multiplicative group of the residue field. Indeed the elements of are precisely the multiplicative residues of the units of . Note that is the unique element of the monoid that multiplied with any element of this monoid equals itself. We denote by the binary composition law on defined as follows. For , the composition is the unique in with .

Let be a any local integral domain. We call a bijection an isomorphism if it is compatible with multiplication and if there exists a (necessarily unique) field isomorphism such that . These conditions are equivalent to the requirement that is compatible with multiplication and . If and are valuation rings, then obviously such an isomorphism also induces an isomorphism of ordered groups between the value groups of and .

## 3. Logarithmic Hensel’s Lemma

Let be a noetherian normal integral domain. Let be a morphism of smooth varieties over , and , reduced strict normal crossings divisors over with . Denote the irreducible components of and by and . Note that and are normal, because they are smooth over the normal ring .

### 3.1. Log-smoothness

Let . Choose uniformizing parameters over on an open neighborhood of in so that locally at the locus of is . Choose uniformizing parameters over on an open neighborhood of in so that locally at the locus of is . Recall that uniformizing parameters over on a variety over , are regular rational functions on that induce an étale morphism to an affine space over . Since is normal, each such uniformizing parameter generates a prime ideal or is a unit in the local ring of any point of .

Definition The morphism is called log-smooth at with respect to , if the logarithmic jacobian

 (∂logf∗(yj)∂logxi(P))i=1,…,n,j=1,…,m

(considered as a matrix over the residue field of ) has rank equal to the relative dimension of at .
Here as usual denotes the rational function on .
Note that belongs to , because can be written as a unit in times a monomial in , since and is normal. Note also that the above definition of log-smoothness does not depend on the choice of the uniformizing parameters .

###### Logarithmic Hensel’s Lemma 3.2.

Let be any henselian local -algebra which is an integral domain, and its maximal ideal. Let . Assume that f is log-smooth at with respect to . Then, any having the same residues with respect to as , is the image under of an with the same residues as with respect to .

Proof. By restricting to suitable open neighborhoods of and we can make the following two assumptions.

a:

There exist uniformizing parameters over on such that the locus of is , and such that the locus of each is irreducible or empty.

b:

There exist uniformizing parameters over on such that the locus of is , and such that the locus of each is irreducible or empty.

#### 3.2.1. Changing coordinates

Let be the étale morphism induced by , and let be the étale morphism induced by . Consider the morphisms

 α:AnA→AnA:(X1,…,Xn)↦(x1(a0)X1,…,xn(a0)Xn), β:AmA→AmA:(Y1,…,Ym)↦(y1(f(a0))Y1,…,ym(f(a0))Ym).

Set , , and denote by the morphism induced by . Denote by and the morphisms obtained by base change of and as defined by the following two cartesian squares:

We think of and as coordinate changes induced by and , although these are not open immersions.

Denote the pull-back to (through ) of the standard affine coordinates on by . These are uniformizing parameters over on . Similarly, denote the pull-back to (through ) of the standard affine coordinates on by . These are uniformizing parameters over on . By construction we have

 (1) α∗X(xi)=xi(a0)x′i,β∗Y(yj)=yj(f(a0))y′j.

Note that uniquely lifts to a point with . Moreover uniquely lifts to a point with , because has the same residues as with respect to . By (1) we have

 (2) x′i(a′0)=1,y′j(b′)≡1modmA.

#### 3.2.2. Lifting the morphism f

We claim that there exists a morphism of schemes over such that the following diagram commutes:

 X′\lx@stackrelf′⟶Y′↓αX↓βYXA\lx@stackrelfA⟶YA

To prove this we only have to show for that is divisible by in .

By assumptions a and b, and because is normal and , there exist non-negative integers , for and , and units such that

 (3) f∗(yj)=ujn∏i=1xei,ji,

for . Hence is divisible by in . Thus by (1), is divisible by in .

On the other hand, evaluating(3) on the rational point yields

 yj(f(a0))=uj(a0)n∏i=1xi(a0)ei,j∈A.

Since is a unit in , we get that divides . Thus divides in . This proves the claim.

#### 3.2.3. Applying the classical Hensel’s Lemma

Note that , because and

 y′j(f′(a′0))=1≡y′j(b′)modmA,

by the second equality in (1) and the congruence in (2).

Note also that the morphism is smooth at . Indeed this follows from the log-smoothness of at , because, by the equations in (1), the jacobian matrix of at , with respect to the uniformizing parameters and , equals the logarithmic jacobian of at with respect to .

Hence, by the classical Hensel’s Lemma for smooth morphisms, can be lifted to a point with and .

#### 3.2.4. Conclusion of the proof of Logarithmic Hensel’s Lemma.

Put . Then , and

 a=αX(a′)≡αX(a′0)=a0modmA.

Moreover by (1) we have

 xi(a)=α∗X(xi)(a′)=xi(a0)x′i(a′).

But by (2), since . Hence and have the same multiplicative residue. This finishes the proof of the Logarithmic Hensel’s Lemma.

## 4. Toroidalization of morphisms

###### Definition 4.1.

Let be a field of characteristic zero. Let be a dominant morphism of nonsingular varieties over , and , reduced strict normal crossings divisors over .
We call toroidal with respect to and if , and if, after base change to an algebraic closure of , for each closed point of there exist uniformizing parameters for and uniformizing parameters for such that the following three conditions hold.

1. Locally at , is the locus of .

2. Locally at , is the locus of .

3. The morphism gives the as monomials in the .

Here we say that elements of a local ring , containing its residue field, are uniformizing parameters for if these elements minus their images in the residue field, form a system of regular parameters for .

###### Remark 4.2.

Note that, using the notation in the above definition, if is toroidal with respect to and , then is log-smooth with respect to and at each point of . The converse is also true, by the work of Kazuya Kato on logarithmic geometry (but we will not use this fact in the present paper).

The following theorem is a small extension, proved in [2], of the Weak Toroidalization Theorem of Abramovich and Karu [1].

###### Weak Toroidalization Theorem 4.3.

Let be a field of characteristic zero. Let be a dominant morphism of varieties over , and let be a proper closed subset. Then there exist nonsingular quasi-projective varieties , over , and a commutative diagram

 D⊂X′\lx@stackrelmX→X↓f′↓fE⊂Y′\lx@stackrelmY→Y

with , projective birational morphisms over , and reduced strict normal crossings divisors over , such that

1. is a quasi-projective morphism over and is toroidal with respect to and ,

2. is a divisor on , and is contained in ,

3. the restriction of the morphism to is an open embedding.

In the present paper, assertion (3) in the above theorem will not be used. The theorem is very much related to (but not implied by) Cutkosky’s Theorem on Local Monomialization of Morphisms (Theorem 1.3 in [13]). It is conjectured that we can take and to be compositions of blow-ups of non-singular subvarieties. This conjecture is a weakening of the Conjecture on (Strong) Toroidalization [1],[14]. Finally we mention that recently Illusie and Temkin obtained a result (Theorem 3.9 of [26]) which is more general than the above Theorem 4.3, and that Cutkosky [15] extended his Local Monomialization Theorem to complex and real analytic maps.

## 5. The Tameness Theorem

Let be a noetherian integral domain. In this section we assume that has characteristic zero. Let be a morphism of varieties over .

###### Tameness Theorem 5.1.

Given rational functions on , there exist rational functions on , and , such that for any -algebra A which is a henselian valuation ring we have the following. Any having the same residues with respect to as an image , with , is itself an image of an with the same residues as with respect to .

###### Remark 5.2.

In the statement of the Tameness Theorem we can choose the rational functions to be regular on , if is affine. Indeed, this is a direct consequence of Lemma 2.3.

### 5.3. Proof of the Tameness Theorem

The Tameness Theorem can be proved easily by using Basarab’s Elimination of Quantifiers [5]. Basarab’s work is based on model theory using the same methods as Ax-Kochen and Eršov. In the next subsections (5.3.1) up to (5.3.5) we present a purely algebraic geometric proof of the Tameness Theorem.

#### 5.3.1. Preliminary reductions.

Let be the field of fractions of . Our proof of the Tameness Theorem is by induction on the dimension of .

Covering with a finite number of affine open subschemes, we see that in order to prove the Tameness Theorem we may suppose that is affine. Moreover we may also suppose that is dominant. Indeed assume that is affine and let be the Zariski closure of . Assume that the Tameness Theorem for the dominant morphism , induced by , holds for given rational functions on and suitable chosen rational functions on . By Lemma 2.3 we can actually choose the so that they are regular rational functions on . Then obviously the Tameness Theorem for holds for the given and any finite list of regular rational functions on which contains an extension to for each of the regular rational functions on , and which contains a sequence of functions whose zero locus equals . It is clear that such a finite list exists because is affine.

Covering with a finite number of affine open subschemes, and using Lemma 2.3, we can further assume that is affine and that are regular rational functions on .

Finally we can suppose that is a normal ring, because any finitely generated subring of becomes normal after inverting a suitable non-zero element (see e.g. section 32 of Matsumura [30]).

#### 5.3.2. Applying the Weak Toroidalization Theorem.

Thus we assume that is normal, that and are affine, that is dominant, and that are regular rational functions on . Moreover we suppose that no one of the given rational functions on is identically zero, because we can discard those that are identically zero. Let be the union of the zero loci of the for .

Applying the Weak Toroidalization Theorem 4.3 and Remark 4.2, to the base change over of the above , , , and , we obtain a suitable , smooth quasi-projective varieties and over , and a commutative diagram of morphisms over

 D⊂X′\lx@stackrelmX→X0:=X⊗RR[Δ−1]↓f′↓fE⊂Y′\lx@stackrelmY→Y0:=Y⊗RR[Δ−1]

with , projective birational morphisms, , reduced strict normal crossings divisors over with , such that the following two conditions hold.

1. is log-smooth with respect to , , at each point of .

2. is a divisor on and is contained in .

Choose a nonempty open subscheme such that the rational map is regular on . Moreover choose a nonempty open subscheme such that the rational map is regular on , and such that is disjoint from and .

#### 5.3.3. Applying the Logarithmic Hensel’s Lemma.

Choose rational functions on such that the following two conditions hold.

1. For each point of and for each irreducible component of , at least one of the elements , or belongs to and generates the ideal of in .

2. For any field which is an -algebra, and any two distinct points , there is a such that and .

One easily verifies that condition (2) can be satisfied using that is quasi-projective over .

We claim that the Logarithmic Hensel’s Lemma for the morphism implies that the Tameness Theorem for the morphism is true for the given regular rational functions and the chosen rational functions , under the additional assumptions that and . Here, as before, denotes the generic point of .

To verify this claim we argue as follows. Note that lifts to a with , and that lifts to a with , because is a valuation ring, and are proper morphisms, and the rational maps , and are regular at respectively , and . Whence , since is disjoint from . Note also that is regular at , since . Hence any rational function on which is regular at , respectively , is also regular at , respectively , considered as rational function on , since . Denoting, as before, the maximal ideal of by , this implies that , by condition (2) with , and the hypothesis that and have the same residues with respect to . Hence, using condition (1), we obtain that and have the same residues with respect to the irreducible components of .
We can now apply the Logarithmic Hensel’s Lemma to the log-smooth morphism , to get a point with , having the same residues as with respect to the irreducible components of . Hence and have the same residues with respect to , because of condition (2) in subsection 5.3.2 and Lemma 2.4. Set . It is now obvious that satisfies the conclusion of the Tameness Theorem, because are regular.
Thus we have now verified our claim that the Tameness Theorem for the morphism is true for the given regular rational functions and the chosen rational functions , under the additional assumptions that and .

Next we enlarge the list of rational functions on by adjoining a sequence of regular rational functions on whose zero locus on equals . This is possible since is affine. Then the condition is automatically satisfied if , because then , since , and because has the same residues as with respect to .
Thus we have now proved the Tameness Theorem for the given regular rational functions and the chosen rational functions , under the additional assumption that . It remains to treat the case .

#### 5.3.4. Using the induction hypothesis.

Let be an irreducible component of . Note that . Let be the restriction to S of the regular rational function , for . By the induction hypothesis the Tameness Theorem is true for the morphism , induced by , and the list of rational functions . This yields a list of rational functions on satisfying the Tameness Theorem for the restriction of to .

#### 5.3.5. Conclusion of the proof of the Tameness Theorem.

Replace the list by its union with the lists of rational functions on obtained as above for each irreducible component of . Then it is clear that the Tameness Theorem for holds for the given and the new list . This finishes the proof of the Tameness Theorem.

###### Remark 5.4.

To prove the Tameness Theorem in the special case that the list of rational functions on is empty, we can use Cutkosky’s Local Monomialization Theorem [13] instead of weak toroidalization. This special case is sufficient to prove Theorems 1.2 and 1.1. Very recently Cutkosky [16] proved a stronger version of his Local Monomialization Theorem (with an additional requirement similar to (2) in 4.3) that suffices to prove the Tameness Theorem in general.

## 6. Transfer of residues

In the next lemma we use the notation of the beginning of section 2 and the terminology of subsection 2.5. Our proof of this lemma is an easy application of embedded resolution of singularities and does not depend on the Weak Toroidalization Theorem or the Tameness Theorem.

###### Lemma 6.1.

Transfer of residues. Let be a variety over , and regular rational functions on . Then there exists a positive integer N such that for any two henselian valuation rings and , having residue field characteristic or zero, and admitting an isomorphism , we have the following. For any there exists such that , identifying with via , and such that , for .

Proof. Let be the union of the zero loci of the regular rational functions on . By using embedded resolution of and induction on , and by inverting finitely many primes, we can assume that is smooth over and that is a strict normal crossings divisor over . Note that this reduction requires and to be valuation rings in order to apply the valuative criterion of properness to the resolution morphism. Moreover, covering with finitely many suitable open subschemes, we can further assume that there exist uniformizing parameters over on such that is the locus of and such that the locus of each is irreducible or empty. Then each is a monomial in the ’s times a unit in ), because is normal. Hence it suffices to prove the lemma for replaced by the uniformizing parameters . Thus we can assume that are uniformizing parameters over on . But then the lemma is a direct consequence of Hensel’s Lemma for the étale morphism induced by . Indeed, for , choose such that . Then, by Hensel’s Lemma, there exists such that and , identifying with via . This rational point satisfies the requirements of the lemma.

## 7. Transfer of surjectivity

In this section we prove Theorem 1.2 on transfer of surjectivity. We use the terminology of subsection 2.5. For any prime number there is an obvious unique isomorphism such that, for any integer and any unit in , we have .

###### Remark 7.1.

By using the isomorphism to identify multiplicative residues of elements of with multiplicative residues of elements of , one can give an obvious meaning to Definition 2.1, with a variety over , when and . Thus it is well defined when we say that and have the same residues with respect to given rational functions on . We will use this terminology throughout the following subsection.

### 7.2. Proof of Theorem 1.2 on transfer of surjectivity.

Covering with a finite number of affine open subschemes, we can assume that is affine. By the Tameness Theorem 5.1, with , and by Remark 5.2, there exist regular rational functions on , and , such that for any prime , which does not divide , the following is true for both and . Any having the same residues with respect to as an image , with , is itself an image of an .

Let be any prime which does not divide , and which is large enough as required by the statement of Lemma 6.1, on transfer of residues, for both the list of regular rational functions