On Rational Points of Varieties over Local Fields having a Model with Tame Quotient Singularities

We study rational points on a smooth variety X over a complete local field K with algebraically closed residue field, and models of X with tame quotient singularities. If a model of X is the quotient of a Galois action on a weak N\'eron model of the base change of X to a tame Galois extension of K, then we construct a canonical weak N\'eron model of X with a map to this model, and examine its special fiber. As an application we get examples of singular models of X such that X has K-rational points specializing to a singular point of this model. Moreover we obtain formulas for the motivic Serre invariant and the rational volume, and the existence of K-rational points on certain K-varieties with potential good reduction.


Introduction
In this article we study smooth and proper varieties over a complete local field K with algebraically closed residue field k with regard to the existence of K-rational points. A standard way to detect rational points of varieties over complete local fields is to look at models. A model of a K-variety X is an integral, flat scheme X over the ring of integers O K such that the generic fiber of X is isomorphic to X. There is a natural map X (O K ) → X(K), and a specialization map X (O K ) → X k (k), where X k ⊂ X is the special fiber. If X is a proper O K -scheme, then the natural map X (O K ) → X(K) is a bijection. As O K is Henselian, the specialization map is surjective whenever X is smooth over S := Spec(O K ). If X is regular, then every O K -point of X factors through the smooth locus of X over S, see [BLR90, Chapter 3.1, Proposition 2]. Hence if a K-variety X has a regular and proper model X → S, then X has a K-rational point if and only if the special fiber of the smooth locus of X over S is not empty. But if X is not regular, then there may exist O K -points intersecting the singular locus of X over S, see Example 4.6. The existence of weak Néron models plays an important role in the study of rational points. A weak Néron model of a smooth K-variety X is a smooth and separated model Z of X, such that the natural map from Z(O K ) to X(K) is a bijection. Hence if X admits a weak Néron model, then X has a K-rational point if and only if the special fiber of this weak Néron model is not empty. It is known that every smooth and proper K-variety has a weak Néron model, see [BLR90,Chapter 3.5, Theorem 2]. But in general a weak Néron model is not unique. The smooth locus over S of a regular, proper model of a smooth, proper K-variety X is a weak Néron model of X. There is a way to obtain a weak Néron model from any proper model, the so called Néron smoothening, see [BLR90,Chapter 3], which is constructed by blowing up singular points having sections through them. But given a singular point, it is hard to decide a priori whether there is a section containing that point. Therefore the Néron smoothening does not yield a straightforward method for constructing a weak Néron model from an arbitrary singular model.
In this article we consider the following situation. Let X be a K-variety, let L/K be a Galois extension, and let X L be the base change of X to L. Then G := Gal(L/K) acts on X L such that X L /G ∼ = X. Consider a model Y of X L with a good G-action, i.e. an action such that every orbit is contained in an affine open subscheme of Y, extending this action on X L . Then the quotient X := Y/G is an O K -scheme and in fact a model of X. In general X will have tame quotient singularities, and there can be O K -points through the singular locus, see Example 2.7 and Example 4.6. Note that interesting models of X L with an action as required really exist and appear naturally. For example models of X L obtained from models of X by base change and normalization have such a G-action, and these are exactly the techniques used to construct a model with semistable reduction in the semistable reduction theorem, see [Liu02, Chapter 10, Proposition 4.6]. Moreover, we show in Theorem 2.9 that if X is a proper and smooth K-variety, then there is always a weak Néron model of X L to which the Galois action on X L extends. To construct such a weak Néron model, we show in particular that the Néron smoothing as constructed in [BLR90,Chapter 3] is compatible with actions of the Galois group as described above.
As the model X obtained by taking the quotient is singular in general and has sections through the singular locus, neither X nor its smooth locus over S will be a weak Néron model of X. But there is a way to construct a weak Néron model of X out of a weak Néron model of X L with a G-action extending the Galois action on the generic fiber. In this context we show the following theorem.
Theorem. (Theorem 3.1) Let L/K be a tame Galois extension, let X be a smooth K-variety, and let X L be the base change of X to L. Let Y be a smooth model of X L with a G := Gal(L/K)-action extending the Galois action on X L . Let X := Y/G be the quotient. Then there is a smooth model Z of X and a separated S := Spec(O K )-morphism Φ : Z → X , such that the induced map Z(O K ) → X (O K ) is a bijection, and such that for all smooth, integral S-schemes V and all dominant S-morphisms Ψ : V → X there is a unique S-morphism Ψ ′ : V → Z such that Φ • Ψ ′ = Ψ. In particular Z is unique. If Y is a weak Néron model of X L , then Z is a weak Néron model of X.
In fact, Z is the fixed locus of some G-action on the Weil restriction of Y to S, see Construction 3.1. The construction goes back to [Edi92], where it is used in the context of abelian varieties and Néron models. Note that the uniqueness of Z with its properties is interesting, because in general a weak Néron model is, in contrast to a Néron model, not unique. The theorem and its proof also yield an explicit description of a weak Néron model Z of X. Having this description at hand, we can examine its special fiber Z k which is important for finding K-rational points of X. We show the following key lemma.
Lemma. (Lemma 4.1) Let Y G be the fixed locus of the G-action on Y. Then there is a k-morphism b : Z k → Y G such that for any point y ∈ Y G with residue field κ(y) the inverse image of y is isomorphic to A m κ(y) as κ(y)-schemes for some m ∈ N. To show this lemma we use the explicit description of Z and of the G-action on the complete local ring of a fixed point, which is examined in Lemma 7.1 and Lemma 7.5. There are some interesting applications of the key lemma. For example we deduce from it that the quotient X has O K -points if and only if Y G = ∅, see Corollary 4.4. In fact these O K -point will pass through the image of Y G in X , which in general will be singular. Hence we obtain examples of singular models with section through the singular locus.
We can use the obtained results also to study certain motivic invariants, the motivic Serre invariant and the rational volume. The motivic Serre invariant S(X) of a Kvariety X is defined to be the class of the special fiber of a weak Néron model of X in some quotient of the Grothendieck ring of varieties, namely in K OK 0 (Var k )/(L − 1), see Definition 5.2. The Serre invariant is interesting in the context of rational points, because it vanishes if X has no K-rational point. From the key lemma we deduce the following theorem.
Theorem. (Theorem 5.2) Let X be a smooth, proper K-variety. Let L/K be a tame Galois extension, X L the base change of X to L. Let Y be a weak Néron model of X L with a good G := Gal(L/K)-action extending the Galois action on X L . Then . The rational volume s(X) of a K-variety X is defined to be the Euler characteristic with proper support and coefficients in Q l , l = char(k) a prime, of the special fiber of a weak Néron model of X. The rational volume vanishes if X has no K-rational point, too.
Theorem. (Theorem 5.4) Let X be a smooth, proper K-variety, and let L/K be a tame Galois extension of degree q r , q a prime. Then s(X) = s(X L ) mod q.
The proof of this theorem uses the fact that there is always a weak Néron model of X L with an action of Gal(L/K) extending the Galois action on X L , see Theorem 2.9, as well as the equation for the Serre invariant (Theorem 5.2). Moreover, we use the fact that for a scheme of finite type V over some field with a good action of a q-group G, we have χ c (V ) = χ c (V G ) mod q. This argument goes back to [Ser09, Section 7.2]. Finally, we can deduce the existence of rational points for some varieties with potential good reduction. By definition, a K-variety X has potential good reduction if there is a Galois extension L/K such that the base change of X to L admits a smooth and proper model.
Corollary. (Corollary 6.1) Let X be a smooth, proper K-variety with potential good reduction after a base change of order q r , q = char(k) a prime. If the Euler characteristic of X with coefficients in Q l , l = char(k) a prime, does not vanish modulo q, then X has a K-rational point.
To prove this corollary we use Theorem 5.4, and the fact that the Euler characteristic with coefficients in Q l is constant on the fibers of a smooth and proper morphism. In addition, we obtain a similar result for the Euler characteristic with coefficients in the structure sheaf, see Corollary 6.2. In this Corollary we need to assume that there is a tame Galois extension L/K of prime degree, such that there is a smooth and proper model of X L with a good G-action extending the Galois action on X L , because we cannot use the results concerning the motivic invariants. We show directly that the G-action on this smooth and proper model of X L has a closed fixed point, and use Corollary 4.4 to conclude that in this case the model obtained by taking the quotient will have an O K -point inducing a K-point of X.
Acknowledgements. The results contained in this article are part of the author's dissertation, written under the supervision of Hélène Esnault. My thesis was supported by the SFB/TR45 "Periods, moduli spaces and arithmetic of algebraic varieties" of the DFG (German Research Foundation). I would like to thank the members of the "Essener Seminar für Algebraische Geometrie und Arithmetik" for their support. In particular, I am very thankful to Andre Chatzistamatiou for the numerous discussions we had, and for all the suggestions he made. I thank Johannes Nicaise and Olivier Wittenberg for reading my thesis and for helping me with important remarks. I am very grateful to Hélène Esnault for the time and ideas she gave me and for her constant support.

Conventions.
A variety over a field F is a geometrically integral, separated Fscheme of finite type over F . We assume that an integral scheme is connected. All schemes are assumed to be noetherian. If U is a V -scheme, Spec(F ) → V any point. We set U F := U × V Spec(F ).
In the entire article, let K be a complete local field with ring of integers O K , S := Spec(O K ), and residue field k. Assume that k is algebraically closed.

Models with Galois Actions
Definition 2.1. Let X be a K-variety. A model of X is an integral S-scheme X of finite type over S such that X K ∼ = X.
Remark 2.1. Let X be a non-empty K-variety, and let X → S be any model of X. Then X dominates S, so by [Har77, Chapter III, Proposition 9.7] X is flat over S.
Remark 2.2. Let ϕ : X → S be a model of a K-variety X. Then we have maps as follows induced by the universal property of the fiber product. Definition 2.2. A weak Néron model of a smooth K-variety X is a smooth and separated model X → S of X, such that the natural map X (O K ) → X(K) is a bijection.
Remark 2.3. Let X be a smooth K-variety attached with a weak Néron model X → S. Then X(K) = ∅ if and only if the special fiber X k of X → S is empty. This is true, because by definition the natural map X (O K ) → X(K) is a bijection, the specializing map X (O K ) → X k (k) is surjective by Remark 2.2, and k is algebraically closed.
Remark 2.4. A weak Néron model does not exist for all smooth K-varieties X. It follows from [BLR90, Chapter 3.5, Theorem 2] that a weak Néron model exists if X is proper over K.
Note that a weak Néron model is not unique. Take any weak Néron model, blow up a point in the special fiber, and then take the smooth locus of the obtained scheme. This is again a weak Néron model. Now fix a Galois extension L/K with Galois group G := Gal(L/K). Let O L be the ring of integers of L, T := Spec(O L ). Note that k is the residue field of L. For a general introduction to local fields and their Galois extensions we refer to [Ser79]. A Galois extension L/K is called tame, if the order of its Galois group is prime to char(k). From [Ser79, Chapter IV, Corollary 2 and Corollary 4] we get that the Galois group of a tame Galois extension L/K is always cyclic.
We now want to consider group actions of the Galois group. Therefore, recall the following facts concerning group actions of an abstract finite group G.
Let U be a scheme, Aut(U ) the abstract group of automorphisms of U . A G-action on U is given by a group homomorphism µ U : G → Aut(U ). If U is an affine scheme, i.e. U = Spec(A), then a group action on U is also given by a group homomorphism µ # U : G → Aut(A). Let U , V be schemes with G-actions. We call a morphism of schemes f : Remark 2.5. By definition of the Galois group, G acts on L, and K = L G . The G-action of L can be restricted to O L , and O G L = O K . We call this action the Galois action on O L . Note that Spec(L) ֒→ T is G-equivariant for these actions. Let X be a K-variety. As X is flat over K, by [Gro63, Exposé V, Proposition 1.9], G acts on X L such that X L → Spec(L) is G-invariant and X L /G ∼ = X. We call this action the Galois action on X L .
Remark 2.6. Let X be a K-variety. Let ϕ : Y → T be a model of X L with a good G-action. Assume that X L ֒→ Y is G-equivariant for the action on Y and the Galois action on X L , i.e. the G-action on Y extends the Galois action on X L . Take any h ∈ G, and let g ∈ Aut(Y) and g T ∈ Aut(T ) be its images. As the maps X L ֒→ Y, X L → Spec(L), and Spec(L) ֒→ T are Gal(L/K)-equivariant, we obtain that g T • ϕ • g −1 | XL = ϕ | XL . As X L ⊂ Y is open and dense, Y is reduced, and T is separated, [GW10, Corollary 9.9] implies that g T • ϕ • g −1 = ϕ, i.e. ϕ is G-equivariant. Let π : Y → X := Y/G be the quotient. Using that the maps in the square on the left hand side are G-equivariant, we get the following big commutative diagram.
Note that X is an S-scheme of finite type by [Gro63, Exposé V, Proposition 1.5]. As X is a quotient by a finite group of the integral scheme Y, it is integral, too. As Spec(L) ֒→ T is flat, by [Gro63, Exposé V, Proposition 1.9] we obtain Hence X → S is a model of X.
In general the quotient X will be singular. To see this, look at the following example.
Example 2.7. Let k be an algebraically closed field with char(k) = 2, and set K := k((s)), L := k((t)) with t 2 = s. Hence L/K is a tame Galois extension with Galois group G = Z/2Z. The Galois action on k((t)) is given by α : k((t)) → k((t)); P (t) → P (−t). Using the fact that t is invertible in k((t)), one shows that the map given by sending t to t and y to xt is a G-equivariant isomorphism. Hence Y is a model of X L with a G-action extending the Galois action on is singular in (0, 0, 0), so by Remark 2.6 it X is a singular model of X.
Remark 2.8. In order to get a projective example, replace in Example 2.7 X = A 1 For a given K-variety X and a Galois extension L/K, there exist interesting models of X L with a good action of the Galois group as in Remark 2.6. In this context we can show the following theorem. Proof. In order to prove this theorem, we need to recall how a weak Néron model is constructed. The main tool of showing that weak Néron models actually exist is the so called Néron smoothening.
Definition 2.3. Let X be a smooth K-variety, and let X → S be a model of X. A Néron smoothening of X is a proper S-morphism f : X ′ → X such that f is an isomorphism on the generic fibers, and such that the canonical map Sm(X ′ /S)(S) → X ′ (S) is bijective. Here Sm(X ′ /S) is the smooth locus of X ′ over S.
In order to prove Theorem 2.9 we need the following Lemma.
Lemma 2.10. Let Y be a smooth L-variety, let Y → T be a model of Y with a good G-action, and assume that the structure map ϕ : Y → T is G-equivariant for this action and the Galois action on T . Then there exists a projective Néron smoothening f : Y ′ → Y, and a good G-action on Y ′ such that f is G-equivariant.
Proof. By [BLR90, Chapter 3.1, Theorem 3] there exists a Néron smoothening f : Y ′ → Y, which consists of a finite sequence of blowups with centers in the special fibers. We need to construct a G-action on Y ′ such that f is G-equivariant. Note that if we blow up an integral scheme U with a good G-action in a closed Ginvariant subscheme V ⊂ U , and denote by u : U ′ → U the blowup, then there is a G-action on U ′ such that u is G-equivariant. The reason for this is the following. Let h ∈ G be any element, So by the universal property of blowup, see [Har77, Chapter II, Corollary 7.15], there exists a unique g U ′ ∈ Aut(U ′ ) such that u • g U ′ = g U • u. This way we can define the required group action on U ′ , and u is G-equivariant by construction. Consider f , which is a sequence of blowups, i. e. we have Here the f i are blowups of some closed subschemes V i ⊂ Y i . One checks in the proof of [BLR90, Chapter 3.4, Theorem 2] that all the V i are obtained using the same construction. Hence if we show that V : and we can conclude inductively on the length of the sequence of blowups.
One can check in [BLR90, Chapter 3.4, Theorem 2] that V is constructed as follows.
As the Gaction on V i is given by isomorphisms, regular points are mapped to regular points, Using the same argument as in the induction, we can show that V t = V is G-invariant, and this is what we wanted to show. We still need to show that the G-action on Y is good. So take any orbit in Y ′ .
Its image under f will be contained in an open affine subset U ⊂ Y. As f is projective, f −1 (U ) is projective over U and contains our orbit, which is finite, because G is finite. By [Liu02, Chapter 3, Proposition 3.36.b] there is an affine subset U ′ ⊂ f −1 (U ) containing every finite set of points. Hence the action on Y ′ is good.
In [Nic12] the following similar theorem in the context of formal schemes is proven: Theorem. Any generically smooth, flat, separated formal O L -scheme X ∞ , topologically of finite type over O L , endowed with a good G-action compatible with the G-action on O L , admits a G-equivariant Néron smoothening. Now we are finally ready to prove Theorem 2.9. So let X be a smooth, proper Kvariety. In particular X is a separated S-scheme of finite type, hence by Nagata's embedding Theorem, see [GW10,Theorem 12.70], there exists a proper, integral S-scheme X and an immersion X ֒→ X over S which is schematically dense. As X is proper over K, X K is in fact isomorphic to X. Altogether, X → S is a proper model of X. Set X T := X × S T , and let Φ : X T → T be the projection to T . Note that Φ is proper, and X T × T Spec(L) = X L . By Remark 2.1, X is flat over S, therefore X T is flat over T . Hence there cannot be a connected component of X T only supported on the special fiber. But the generic fiber X L of X T is connected, hence X T is connected. Hence one can check locally that X T is integral, which is straightforward to check. Altogether, Φ : X T → T is a proper model of X. As X → S is flat, by [Gro63, Exposé V, Proposition 1.9] there exists a good Gaction on X T such that Φ is G-equivariant, and X T /G ∼ = X . This G-action extends the Galois action on X L by construction. By Lemma 2.10 there exists a projective Néron smoothening f : Hence ϕ : Y → T is a smooth and separated model of X L . As Φ and f are proper, by the valuative criterion of properness the natural map We still need to show that there is a good G-action on Y extending the Galois action on X L . As f is G-invariant for the G-action on Y ′ and X T , the G-action Y ′ extends the Galois action on X L . So it suffices to show that this G-action restricts to Y, i.e. that Y ⊂ Y ′ is G-invariant. To show this, we can simply use the proof in the base case of the induction in Lemma 2.10. Note that the action is good for the following reason. Take any orbit in Y. As the action on In [EN11, Proposition 4.5] the following similar statement is proven.
Proposition. Let G be any finite group, X a smooth and proper K-variety, endowed with a good G-action. Then there is a weak Néron model X → S of X endowed with a good G-action, such that X ֒→ X is G-equivariant.

A Canonical Weak Néron Model of a Quotient Scheme
Theorem 3.1. Let L/K be a tame Galois extension, G := Gal(L/K). Let O L be the ring of integers of L, T := Spec(O L ). Let X be a smooth K-variety, and let ϕ : Y → T be a smooth model of X L with a good G-action extending the Galois action on X L . Let X := Y/G be the quotient. Then there is a smooth model Z → S of X and a separated S-morphism Φ : Z → X , such that the induced map Z(S) → X (S) is a bijection, and such that for all smooth integral S-schemes V and all dominant S-morphisms Ψ : V → X there is a unique S-morphism Ψ ′ : V → Z making the following diagram commutative.
In particular Z is unique with this property up to a unique isomorphism. If Y → T is a weak Néron model of X L , then Z → S is a weak Néron model of X.
Proof. The proof consists of six steps. First we will give the construction of Z as a functor of schemes, then we construct Φ as a morphism of functors. In the third step we will show that Z is represented by a smooth S-scheme. Thereafter we show the properties of Φ, namely that it is separated and that the map Z(S) → X (S) induced by Φ is an isomorphism. Afterwards we show the universal property. In the final step we consider the case that Y is a weak Néron model of X L .
Construction of Z. We now construct Z. The construction can be found in [Edi92], where it is used in the context of abelian varieties.
Definition 3.1. The Weil restriction of a T -scheme U to S is defined as the functor Definition 3.2. Let V be an S-scheme with a G-action, such that the structure map is G-equivariant for this action and the trivial action on S. We define the functor of fixed points by By [Edi92, Proposition 3.1] this functor is represented by a subscheme of V . We call this scheme the fixed locus of the G-action on V .
Note that G is a finite cyclic group, because L/K is a tame Galois extension. Therefore every G-action is given by one automorphism.
Construction 3.1. [Edi92, Construction 2.4 and Theorem 4.2] Fix a generator of G, and let g ∈ Aut(Y) and g T ∈ Aut(T ) be its images. Theñ It is easy to see thatg is an S-morphism. Therefore the structure map Res T /S (Y) → S is G-equivariant for the G-action on Res T /S (Y) and the trivial G-action on S. Define Construction of Φ. View X and Z as functors from the category of flat S-schemes to the category of sets. We now construct a morphism of functors Φ : Z → X . As soon as we will have shown that Z is in fact representable by a flat S-scheme, this will yield an S-morphism of schemes by Yoneda's lemma for the category of flat S-schemes.
We need to construct maps Φ(W ) : Z(W ) = Hom T (W × S T, Y) G → X (W ) for all flat W ∈ (Sch /S), and show that they are functorial. Take any f ∈ Z(W ). Let π : Y → X be the quotient map. We have the following commutative diagram.
Hence π • f is G-equivariant for the G-action on W × S T and the trivial action on X , and therefore, by the universal property of the quotient p W : W × S T → W we obtain a unique f ′ ∈ X (W ) making the diagram above commutative. We set Φ(W )(f ) := f ′ . It is easy to check that this map is functorial.
View X K ∼ = X as a presheaf on the category of K-schemes. We now construct an inverse map of functors Φ | ZK −1 : Take any h ∈ X(W ), and consider the following diagram with p L and p W the projection maps.
As h is a K-morphism, the diagram commutes, and hence the universal property of fiber product induces a unique h * ∈ Hom L (W L , X L ) with π • h * = h • p W . Using that h * is unique, one can easily show that it is actually G-equivariant, hence we may set Φ | ZK −1 (W )(h) := h * . It is straightforward to check functoriality and the Representability of Z. Now we are ready to show that Z is actually represented by an S-scheme. Unfortunately we cannot show that Res T /S (Y) is representable using [BLR90, Chapter 7.6, Theorem 4], because as Y does not need to be quasiprojective, we cannot show that every finite set of points in Y is contained in an affine subset of Y. Therefore we show directly that Z is representable by using We have seen that X ∼ = Z K , and X is integral by assumption. As Z is smooth over S, it is reduced, and flat over S, so there is no irreducible component only supported on the special fiber. Altogether Z is integral. This yields that Z → S is a smooth model of X.
Properties of Φ. In order to show that Φ is separated, take any valuation ring R with quotient field Q, and any two morphisms f 1 , f 2 ∈ Hom(Spec(R), Z) such that Take any σ G ∈ X (S). Let σ ′ G : Y × X S → X be the pullback of σ G , π ′ : π −1 (S) → S the pullback of π, and set ϕ ′ := ϕ • σ ′ G . By the universal property of the fiber product, we have a one to one correspondence of sections σ of ϕ with π • σ = σ G • π T , and sections σ ′ of ϕ ′ with π ′ • σ ′ = π T . Note that π T • ϕ ′ = π ′ , π T is separated, and π ′ is proper, because π is a quotient map and hence proper. So ϕ ′ is proper by [GW10,Proposition 12.58]. Hence without loss of generality we may assume that ϕ is proper. By assumption, Y L ∼ = X × S Spec(L), hence σ G induces a unique section σ ′ of ϕ | YL with π • σ ′ = σ G • π T | Spec(L) . As ϕ is proper, we get a unique section σ of ϕ with σ | Spec(L) = σ ′ . As S is separated, π • σ = σ G • π T . We still need to show that σ ∈ Z(S), i. e. that σ = g • σ • g T −1 . Therefore one shows that g • σ • g T −1 is a section of ϕ and π •g •σ •g T −1 = σ G •π T , and one concludes using the uniqueness of σ with this properties. Hence σ is the unique element in Z(S) with Φ(S)(σ) = σ G , i.e. Φ(S) is bijective.
Universal property. Now let V be a smooth, integral S-scheme and let Ψ : V → X be a dominant S-morphism. Assume that there exists a Ψ ′ : V → Z such that Φ • Ψ ′ = Ψ. As Ψ is an S-morphism, it maps V K to X ∼ = X K . We have already seen that Φ| ZK : Z K → X is an isomorphism with inverse map Φ | ZK −1 . Therefore As V K is open and dense in V, V is reduced, and Z is separated over X , Ψ ′ is unique on V by [GW10, Corollary 9.9]. Now we construct Ψ ′ . First we need to show some facts concerning Y and the G-action on Y. Consider X T := X × S T and the following diagram.
Here p X and p T are the projection maps. Note that the diagram commutes, so there is a unique h with p T • h = ϕ and p X • h = π. As p X and π are finite, by [GW10, Proposition 12.11] h is finite, too. As X is flat over S, the G-action on T induces a G-action on X T such that p T is G-equivariant and X T /G = X , see [Gro63, Exposé V, Proposition 1.9]. As ϕ and p T are G-equivariant, h is G-equivariant, too. Let n : X n T → X T be the normalization. By assumption, Y is integral and smooth over T , so in particular normal. As Y L = X L = X L , h is generically an isomorphism, and therefore dominant. So the universal property of normalization induces a unique morphism s : Y → X n T such that n • s = h. Note that s is finite, because h and n are finite, and an isomorphism on X L ⊂ Y. Altogether s is a finite birational morphism between integral normal schemes. That means, by [GW10, Corollary 12.88] it is an isomorphism. So we may assume that h = n and Y = X n T . Back to V and Ψ. Consider the following cartesian diagram.
with V T := V × S T = V × X X T , π V and p the projection maps. As V is smooth over S, so in particular flat, the G-action on T induces a G-action on V T such that By construction p is G-equivariant. It might happen that V T is not connected. Let V T = U 1 ⊔ · · · ⊔ U m , with U i ⊂ V T the connected components. As V = V T /G is connected, G acts transitively on the connected components. As Ψ is dominant, the same holds for p. Note that X T is connected, because it is flat over T and generically isomorphic to the L-variety X L . Hence there exists at least one component U i such that p | Ui is dominant. As G acts transitively on V T and p is G-equivariant, p| Uj is dominant for every component U j . By assumption V is smooth over S, so V T is smooth over T . Hence every component U i of V T is normal. So by the universal property of normalization there are unique morphisms Ψ T | Ui : U i → Y such that n • Ψ T | Ui = p | Ui . This defines a unique morphism Ψ T on all of V T such that n • Ψ T = p. As p and n are G-equivariant, Ψ T is G-equivariant, too. Take any W ∈ (Sch /S), f ∈ V(W ). By the universal property of the fiber product, It is easy to check that this defines a map of functors, so we obtain an S-morphism Ψ ∈ Hom S (V, Z).
We still need to check that Ψ = Φ • Ψ ′ . Therefore it suffices to check that for all One observes that Φ(Ψ ′ (f )) = Ψ • f = Ψ(f ), which we wanted to show.
We still need to check that Z is unique up to a unique isomorphism with its properties. Assume there is a Z ′ and a morphism Φ ′ : Z ′ → X having the same properties as Z and Φ. So we get unique morphisms α : Z → Z ′ and α ′ : The case that Y is a weak Néron model. Assume that ϕ : Y → T is a weak Néron model of X L . Hence ϕ is separated, so by [Gro63, Exposé V, Proposition 1.5] X is separated over S. As Φ is separated, Z → S is separated, too. Hence to show that Z → S is a weak Néron model of X, we still need to show that is a bijection. This map is injective, because Y is a separated T -scheme. Take any σ ′ ∈ Z K (K). As Y is a weak Néron model of X L , Y(T ) ∼ = X L (L), so there is a σ ∈ Hom T (T, Y) with σ | Spec(L) = σ ′ . As g −1 T maps Spec(L) to itself, we get g • σ • g −1 T | Spec(L) σ | Spec(L) . As Y is a separated T -scheme, g • σ • g −1 T = σ, i. e. σ ∈ Z(S). Hence the map Z(S) → Z K (K) is surjective.

In [Edi92, Theorem 4.2] the following statement is proven:
Theorem. Let L/K be a tame Galois extension, O L the ring of integers of L, and T := Spec(O L ). Let X be an abelian variety over K. Then there is a good Gal(L/K)-action on the Néron model ϕ : Y → T of X L extending the Galois action on X L , and Z → S given by Construction 3.1 is the Néron model of X.
Note that the Néron model of an Abelian variety is uniquely determined by a universal property. In [Edi92] this universal property is used to show that Z is the Néron model of X. As we do not have a universal property for weak Néron models in general, we had to use different methods to prove the universal property in Theorem 3.1. Moreover, Néron models of Abelian Varieties are quasi-projective. In Theorem 3.1 we do not assume that Y is quasi-projective, which makes the proof of the representability of Z less straightforward.
Remark 3.2. If we do not assume that ϕ is smooth in Theorem 3.1, we can modify Construction 3.1 by considering Sm(Y/S), the smooth locus of Y over S, instead of Y. This is well defined, because the G-action restricts to Sm(Y/S). We will get a smooth model of X with an S-morphism Φ as in Theorem 3.1. Note that the map Φ(S) : Z(S) → X (S) will be injective in this case, but in general not surjective. Nevertheless, if we assume that the smooth locus of Y over S is a weak Néron model of X L , the modified Z will be a weak Néron model of X. This is in particular the case if Y is regular and ϕ is proper. Lemma 4.1. Assumption and notation as in Theorem 3.1. Let Y G be the fixed locus of the G-action on Y. Then there is a k-morphism b : Z k → Y G such that for every point y ∈ Y G with residue field κ(y) the inverse image of y is isomorphic to A m κ(y) as κ(y)-schemes for some m ∈ N.
Proof. As L/K is a tame Galois extension, G = Z/rZ with r prime to char(k). Let the G-action on Y be given by g ∈ Aut(Y), and that on T by g T ∈ Aut(T ).
To construct b, let W ∈ (Sch /k) be any k-scheme, w : W → Spec(k) the structure map. Recall the construction of Z in Construction 3.1. Set Here the first equation holds, because f is G-equivariant, and the second, because the action on the special fiber It is obvious that b is functorial, so we get the required k-morphism.
Let y ∈ Y G be any point with residue field κ(y), j y : Spec(κ(y)) ֒→ Y G ⊂ Y be the immersion of the point y. Note that b −1 (y) is defined by the following cartesian Take any affine κ(y)-scheme W = Spec(A) ∈ (Sch /κ(y)) with structure map ω : W → Spec(κ(y)). By the universal property of the fiber product we obtain To compute k ⊗ R G R we use Lemma 7.5. This lemma also implies that for a primitive r-th root of unity µ ∈ k ⊂ κ(y). Note that . One observes that f sends all points in Spec(A[t]/(t r )) to y ∈ Y, so it factors uniquely through Spec(O Y,y ), i. e. there is a unique morphismf such that the following diagram commutes.
Let r y := i # y : O Y,y → κ(y) be the residue map. Note that j • i y = j y . As As y lies in Y G , there is an induced G-action on Spec(O Y,y ) given by some map g ∈ Aut(Spec(O Y,y )) withg r = id and α y ∈ Aut(O Y,y ) with α r y = id, respectively, such that j is G-equivariant.
Here a ∈ b −1 (y)(W ) as described before, ρ 1 and ρ 2 are the morphisms we get from the definition of tensor product, and i 0 : Hence by the universal property of tensor product there is a uniqueã such that diagram (2) commutes.
κ(y)), such that ρ 1 and ρ 2 are G-equivariant, see Lemma 7.5. As α −1 • a • α y = a, we get and, using that G acts trivially on i 0 (A), we obtain Asã is unique with these properties, α −1 •ã •α y =ã. Denote byr : /(t r ) the canonical map given by the properties of the tensor product. We haver(t) = t. The R-structure of A[t]/(t r ) is given byr • ρ R , with ρ R : R → k ⊗ R G R the canonical map. The R-structure of O Y,y is given by β y := (ϕ • j) # . As a is an R-morphism, we obtain the following commutative diagram.
By Remark 7.4, R G ⊂ R is a local subring having the same residue field as R. As It is easy to check that the following diagram commutes.
k 5 U g g P P P P P P P P P P P P P P O O Hence the universal property of tensor product induces a unique k-morphismβ y withβ y • ρ R = ρ 1 • β y . Looking at diagram (3) again, we get As ρ R is surjective,r =ã •β y , i. e.ã preserves the k[t]/(t r )-structure given byβ y on O Y,y ⊗ O G Y ,y κ(y), and on A[t]/(t r ) given byr.
This map is bijective, because there is an obvious inverse map. It is easy to check that it is functorial.
Remark 4.2. If we do not assume that the Galois extension L/K is tame, then we cannot show Lemma 4.1. This is because Lemma 7.1 is wrong in this case, see Example 7.3, and hence we cannot show Lemma 7.5, which is the main ingredient of the proof of Lemma 4.1. It would be very interesting to know what happens in the non-tame case.
Remark 4.3. Note that if we do not assume that k =k, but that L over K is totally ramified, and that k contains all r-th primitive roots of unity, one can still show Lemma 4.1. There should also be no problem to replace O K by a Henselian ring.  Proof. As Y G = ∅, there is a closed fixed point y ∈ Y. Note that G acts on Spec(Ô Y,y ) given by some α y ∈ Aut(Ô Y,y ) with α r y = id, such that the natural map j : Spec(Ô Y,y ) → Y is G-equivariant. As L/K is a tame Galois extension, G is a cyclic group of order prime to char(k), so by Lemma 7.1, G acts on R := O L given by some α R ∈ Aut(R) sending a generator t of the maximal ideal in R to µt, with µ ∈ R a primitive r-th root of unity. Note thatÔ Y,y is an R-module via β y := (ϕ • j) # , and β y is G-equivariant. As ϕ is smooth, and the residue field of R is equal to the residue field ofÔ Y,y , [Gro67, Proposition 17.5.3] implies thatÔ Y,y ∼ = R[[x 1 , . . . ,x n ]] as R-module for somex 1 , . . . ,x n ∈Ô Y,y . Note that t,x 1 , . . . ,x n form a regular system of parameters ofÔ Y,y . As α y (t) = α R (t) = µt, by Lemma 7.1 we may choose a system of parameters x 0 , . . . , x n with α y (x i ) = µ ℓi x i for some ℓ i ∈ N, such that x 0 = t. SoÔ Y,y ∼ = R[[x 1 , . . . ,x n ]] ∼ = R[[x 1 , . . . , x n ]] as Rmodules. Let I ⊂Ô Y,y be the ideal generated by x 1 , . . . , x n . Note that α y (I) ⊂ I. So the quotient map is a G-equivariant retraction of β y . Thereforeσ # is a section of ϕ•j, and σ := j •σ # is a section of ϕ. As bothσ and j are G-equivariant, the same holds for σ. Let the G-action on T be given by g T ∈ Aut(T ), that on Y by g ∈ Aut(Y). Let π : Y → X and π T : T → S be the quotients. Let ϕ G : X → S be the structure map of X as S-scheme. Every element in X (O K ) is given by a section of ϕ G . As σ is G-invariant and π is a quotient map, π • σ • g T = π • g • σ = π • σ. So by the universal property of the quotient π T : T → S, there exists a unique σ G : S → X such that π • σ = σ G • π T . Furthermore, As π T is an epimorphism, ϕ G • σ G = id S , i.e. σ G is a section of ϕ G .
Note that the image of a closed fixed point y ∈ Sm(Y/T ) G in X is a singular point in general, so in fact we construct sections through singular points. Here is an example for such a section through a singular point. where U and V are separated k-schemes of finite type such that there exists a finite, surjective, purely inseparable k-morphism U → V . We still denote the image of A 1 k in K mod 0 (Var k ) by L. Remark 5.1. Let X be a smooth, separated K-variety without K-rational point. Then S(X) = 0. This holds, because in this case X viewed as an S-scheme is a weak Néron model of X, i. e. the special fiber of this weak Néron model is empty. Hence if S(X) = 0, then X has a K-rational point. Proof. By Theorem 3.1 we know that Z → S as constructed in Construction 3.1 is a weak Néron model of X. Hence by definition S(X) equals to the class of the special fiber Z k in K OK 0 (Var k )/(L − 1), and it suffices to show the following statement: Ui for some m i ∈ N, by proceeding in the following way. By [Edi92, Proposition 3.5], Y G is smooth over T G = Spec(k), hence in particular reduced. Replacing Y G by an open subset, we may assume that it is integral. Let η be the generic point of Y G with residue field κ(η). By Lemma 4.1 there is an isomorphism β : b −1 (η) → A m1 κ(η) over κ(η) for some m 1 ∈ N. As β is defined by finitely many rational functions over Y G , we find an open subset U 1 of Y G over which β is already defined. In particular b −1 (U 1 ) ∼ = A m1 U1 . Now one can proceed with Y G \ U 1 in the same way. The claim follows by noetherian induction using that Y G is of finite type over k. So using the scissor relations in the Grothendieck ring of k-varieties we get in K 0 (Var k )/(L − 1) that This proves Theorem 5.2.

Rational Volume.
Fact. [NS11b, Example 4.3 and Corollary 4.14] There exists a unique ring morphism (realization morphism) χ c : K OK 0 (Var k )/(L − 1) → Z that sends a class of a separated k-scheme U of finite type to the Euler characteristic with proper support with l = char(k) a prime. The map does not depend on the choice of l.
Definition 5.3. Let X be a smooth K-variety with weak Néron model. Then the rational volume of X is defined by Remark 5.3. Let X be a smooth K-variety without K-rational point. Then s(X) = 0. This holds, because by Remark 5.1, S(X) = 0, hence in particular s(X) = χ c (S(X)) = 0. So if s(X) = 0, then X has a K-rational point.
Theorem 5.4. Let X be a smooth, proper K-variety, and let L/K be a tame Galois extension, such that G := Gal(L/K) is a q-group, q = char(k) a prime. Then s(X L ) = s(X) mod q.
In particular, if s(X L ) does not vanish modulo q, then X has a K-rational point.
Proof. Let O L be the ring of integers of L, T := Spec(O L ). By Theorem 2.9 there is a weak Néron model ϕ : Y → T of X L with a good G-action on Y, extending the Galois action on X L . Hence Theorem 5.2 implies that . As X L ⊂ Y is G-invariant, the same holds for Y k , so the action of G on Y restricts to Y k . By [EN11, Proposition 5.4], for every variety U over a field F with a good G-action, χ c (U ) = χ c (U G ) mod q. This Proposition is based on an argument in [Ser09, Section 7.2]. In our case we get Assume now that s(X L ) = 0 mod q. This implies that s(X) = 0. But the rational volume of a smooth K-variety without K-rational point vanishes, see Remark 5.3, hence X has a K-rational point.

Rational Points on Certain Varieties with Potential Good Reduction
Definition 6.1. A smooth, proper K-variety X has potential good reduction (after a base change of order r) if there exists a Galois extension L/K (of degree r), such that X L has a smooth and proper model.
Corollary 6.1. Let X be a smooth, proper K-variety, which has potential good reduction after a base change of order q r , q = char(k) a prime. Then with K s a separable closure of K, l = char(k) a prime. In particular, if χ(X) does not vanish modulo q, then X has a K-rational point.
Proof. Let L/K be the field extension of degree q r , such that there is a smooth and proper model of X L . Let O L be the ring of integers of L, T := Spec(O L ), and ϕ : Y → T a smooth and proper model of X L , which is in particular a weak Néron model of X L . So by definition s(X L ) = χ c (Y k ). As ϕ is proper, Y k is proper over k, and hence the ordinary cohomology coincides with the cohomology with proper support, i. e. χ c (Y k ) = χ(Y k ). As ϕ is proper and smooth, by [Del77, Exposé V, Theorem 3.1] we get bijections between H i (Y k , Z/nZ), and H i (Y, Z/nZ), and H i (X L × Spec(L) Spec(L s ), Z/nZ) for all i, with L s a separable closure of L. Therefore we have for all i that Note that L s = K s , because L/K is a tame Galois extension. Therefore we get X L × Spec(L) Spec(L s ) = X × Spec(K) Spec(K s ) = X K s , hence This implies that s(X L ) = χ(X). Hence Theorem 5.4 implies the corollary.
Corollary 6.2. Let X be a smooth, proper K-variety, let L/K be a tame Galois extension of prime order q, and assume that there is a smooth and proper model of X L with a good G := Gal(L/K)-action extending the Galois action on X L , i.e. in particular X has potential good reduction after a base change of order q.
If χ(X, O X ) does not vanish modulo q, then X has a K-rational point.
Proof. Let O L be the ring of integers of L, T := Spec(O L ). Let ϕ : Y → T be the smooth and proper model of X L on which there is a good G-action extending the Galois action on X L . As X L ⊂ Y is G-invariant, the same holds for Y k ⊂ Y, hence the G-action on Y restricts to a good G-action on Y k . Let f : Y k → Y k /G be the quotient.
Assume that the action of G on Y has no fixed point, so in particular the action of G on Y k has no fixed point. As q is a prime, the action is free. So f is a finite, etale morphism of degree q by [Gro63, Exposé V, Corollaire 2.3]. Moreover Y k is smooth and proper over k, because ϕ is smooth and proper. As f isétale and finite, Y k /G is smooth and proper over k, too. As f isétale, f * (T Y k /G ) = T Y k , and therefore f * (td(T Y k /G )) = td(T Y k ). Let s : Y k → Spec(k) and s ′ : Y k /G → Spec(k) be the structure maps. We have s = s ′ • f . Using [Ful98, Corollary 15.2.2], and the projection formula in the third line, we obtain Note that H i (X L , O XL ) = H i (X L , O X ⊗ K L) = H i (X, O X ) ⊗ K L holds for all i ≥ 0. So in particular χ(X L , O XL ) = χ(X, O X ). As ϕ is smooth and proper, and T is connected, by [Gro61, Theorem 7.9.4.I] the Euler characteristic is constant on the fibers of ϕ, hence χ(Y k , O Y k ) = χ(X L , O XL ), which does not vanish modulo q. This is a contradiction, hence Y G = ∅. So Corollary 4.4 implies that X has a K-rational point.

Appendix
In this section we show two lemmas concerning tame cyclic actions on Henselian, regular, local rings. These results are used in Section 4. The following lemma should be known to the experts; a similar statement can be found in [Ser68].
Lemma 7.1. Let A be a regular, Henselian ring of dimension n with maximal ideal m, such that its residue field κ is a field of char(κ) ∤ r containing all r-th roots of unity, and let α ∈ Aut(A) with α r = id, such that the residual map on κ is trivial. There exists a regular system of parameters x 1 , . . . , x n ∈ m ⊂ A with α(x i ) = µ ℓi x i , µ ∈ A a primitive r-th root of unity, and ℓ i ∈ {0, . . . , r − 1}. If there are z 1 , . . . , z s ∈ m ⊂ A, such that thez 1 , . . . ,z s ∈ m/m 2 are linearly independent, and such that α(z i ) = µ ℓi z i for some ℓ i ∈ {0, . . . , r − 1}, then we may chose x i = z i for i ≤ s.
As A is a regular local ring of dimension n with residue field κ, m/m 2 is an ndimensional κ-vector space. Letᾱ ∈ Aut(m/m 2 ) be the automorphism induced by α. As the morphism on A/m = κ induced by α is trivial,ᾱ is a κ-linear map. For some algebraic closureκ of κ, there exists a basis of m/m 2 ⊗ κκ , such that the matrix corresponding toᾱ has Jordan normal form. Asᾱ r = id, all eigenvalues are r-th roots of unity, i. e. powers of µ, and as r = 0 ∈ κ ⊂κ, the matrix is already diagonal. But all r-th roots of unity are assumed to be in κ, soᾱ is diagonalizable, too. Therefore m/m 2 decomposes into eigenspaces E j . By assumption, for all i there exists a j such thatz i ∈ E j . Note that for all j one can choose a basis B j of E j such that for all i,z i ∈ ∪B j . This uses the fact that thez i are linearly independent. Set {x s+1 , . . .x n } := ∪B j \{z 1 , . . . ,z s }. As the E j are eigenspaces, we haveᾱ(x i ) = µ lix i for some l i ∈ {0, . . . , l − 1}. Choosex i ∈ A such thatx i mod m 2 =x i . As r is invertible in A, we can define x i for i > s as follows. Remark 7.2. In fact we do not need to assume in Lemma 7.1 that A is Henselian, but only that all the r-th roots of unity in κ lift to A. The same is true in Remark 7.4 and Lemma 7.5.
Let A be a ring, α ∈ Aut(A) with α r = id. Then α defines an action of G := Z/rZ on A, and the subring is called the ring of invariants.
Remark 7.4. Let A be as in Lemma 7.1. Then m G := m ∩A G ⊂ A G is an ideal and we have A G / m G ֒→ A/ m = κ. With a proof similar to the proof of Lemma 7.1, we can show that there exists a lifts ∈ A of s ∈ κ such that α(s) =s, i. e.s ∈ A G , and hence A G / m G = κ. Hence there is a ringhomomorphism A G → κ, and hence we may consider κ ⊗ A G A.