Reduction of Kummer surfaces modulo 2 in the non-supersingular case

We obtain necessary and sufficient conditions for the good reduction of Kummer surfaces attached to abelian surfaces with non-supersingular reduction when the residue field is perfect of characteristic 2. In this case, good reduction with an algebraic space model is equivalent to good reduction with a scheme model, which we explicitly construct.


Introduction
To an abelian surface A over a field K, one can attach the associated Kummer surface Kum(A), defined as the minimal desingularisation of the quotient of A by the antipodal involution ι(P ) = −P .If the characteristic of K is not 2, then Kum(A) is a K3 surface.Quartic surfaces with sixteen nodes in P 3 are classically studied singular models of Kummer surfaces associated to Jacobians of genus 2 curves.When K is a number field, such Kummer surfaces, as well as those attached to products of elliptic curves, form classes of K3 surfaces most accessible for studying their arithmetic; see for example [HS16,SZ17] and references therein.
If the characteristic of K is 2, then the geometry of Kum(A) depends on the 2-rank of A, that is, the integer r ∈ {0, 1, 2} such that A[2] has 2 r points over an algebraic closure of K.The abelian surface A is called ordinary if r = 2 and supersingular if r = 0; in the remaining case, where r = 1, A will be called almost ordinary.Then Kum(A) is a K3 surface if and only if A is not supersingular.This was proved by Shioda when A is a product of elliptic curves [Shi74] and by Katsura in general [Kat78].In contrast, if A is supersingular, then Kum(A) is geometrically rational.Katsura also points out another striking difference between these cases: if A is not supersingular, then the singularities of A/ι are rational double points, which are four geometric points of type D 4 when A is ordinary, and two geometric points of type D 8 when A is almost ordinary, giving rise to sixteen rational curves in Kum(A).On the other hand, if A is supersingular, then the unique singular point of A/ι is an elliptic singularity.In the non-supersingular case, the 2-rank of A also controls the height of the K3 surface Kum(A), which is 1 in the ordinary case and 2 in the almost ordinary case; see Remark 6.5.
In this paper we are interested in necessary and sufficient conditions for good reduction of Kummer surfaces.We will therefore take K to be a complete discretely valued field of characteristic zero, with ring of integers O K and perfect residue field k.Given an abelian surface A over K, we say that X = Kum(A) has good reduction if there exists a scheme or algebraic space X smooth and proper over O K with generic fibre X K X.Note that in this situation, the special fibre X k is necessarily a K3 surface since it must have trivial canonical bundle ω X k O X k , (1) and (coherent) Euler characteristic χ(O X k ) = 2.Although for general K3 surfaces, good reduction requires models which are algebraic spaces (see for example [Mat15a,Example 5.2]), in the cases considered in this paper, it turns out that schemes suffice.
If char(k) 2, then Kum(A) has good reduction if and only if there exists a quadratic twist A χ of A such that A χ has good reduction.Indeed, if Kum(A) has good reduction, one can show that the inertia group of K acts on H 1 ét (A K , Q ) via a quadratic character and then apply the classical Néron-Ogg-Shafarevich criterion (see the proof of Theorem A.1, which is certainly well known to the experts).Conversely, any quadratic twist of A satisfies Kum(A χ ) Kum(A), so replacing A with A χ , we can assume that A has good reduction.In this case, the Néron model A/O K of A/K is an abelian scheme with generic fibre A K A. We can then form the quotient A/ι and blow up the singular subscheme to obtain a smooth model of Kum(A) over O K ; see [Mat15b,Lemma 4.2] and [Ove21,Proposition 3.11].
If char(k) = 2, one direction of this argument still works: if Kum(A) has good reduction, then a quadratic twist A χ of A has good reduction, and Kum(A) Kum(A χ ).But the argument in the other direction breaks down, essentially because the singular subscheme of A/ι is no longer étale over O K .We are therefore left with the following question: suppose that char(k) = 2, and let A/K be an abelian surface with good reduction A/O K .When does Kum(A) have good reduction?
Our first result is a construction, in the non-supersingular case, of an explicit smooth model of Kum(A) over the ring of integers of a finite extension of K that trivialises the Galois action on the 2-torsion subgroup A [2].We use a local equation of the special fibre at each singular point, but not much else.Our method relies on the crucial fact that blowing up a (singular!)section commutes with specialisation to each fibre, provided the section meets both fibres at rational double points; see Proposition 5.1.As a consequence we obtain the following.
Theorem 1. Assume that char(k) = 2, and let A/K be an abelian surface with good, non-supersingular reduction.Then Kum(A) has potentially good reduction with a scheme model.
An important feature of our model is that it establishes a bijection between the sets of sixteen geometric exceptional curves of the special and generic fibres. (2)A weaker result concerning good reduction with an algebraic space model, and without an explicit bound on the degree of the field extension required, follows from Artin's simultaneous resolution of singularities [Art74].This can be applied because A/ι is flat over O K , and its fibres are normal varieties with at worst rational double points.
To explain more refined results than Theorem 1, let us fix an algebraic closure K of K, with residue field k, and let Γ K denote the Galois group of K/K.We therefore have the exact sequence • is the connected component of the identity of the 2-torsion subscheme A[2] ⊂ A (see Section 4.1 below).
Theorem 2. Assume that char(k) = 2, and let A/K be an abelian surface with good, ordinary reduction.Then Kum(A) has good reduction over K if and only if the exact sequence (1.1) of Γ K -modules splits.Moreover, in this case Kum(A) has good reduction with a scheme model.
Another way of phrasing the given condition is as the splitting of the connected-étale sequence of finite flat group schemes over O K .It is a consequence of Cartier duality that the splitting of (1.1) implies in particular that the Γ K -module A[2](K) is unramified (see Section 4.1); however, the converse fails -there exist examples where A[2](K) is unramified but the exact sequence (1.1) is non-split (see Example 6.4).
(2) This bijection, however, does not describe the images of these curves under the specialisation map on Néron-Severi groups.
In particular, this gives an example of a K3 surface over Q 2 which only attains good reduction over a non-trivial unramified extension. (3) We also have similar results in the almost ordinary case.
Theorem 3. Assume that char(k) = 2, and let A/K be an abelian surface with good, almost ordinary reduction.Then Kum(A) has good reduction over K if and only if the Γ K -module A[2](K) is trivial.Moreover, in this case Kum(A) has good reduction with a scheme model.
Again, it is easy to see that this condition is not vacuous: there exist examples where the The 'if' directions of Theorems 2 and 3 are proved by explicitly constructing a smooth model for Kum(A), by resolving the singularities of the quotient scheme A/ι as in the proof of Theorem 1.The hypotheses on A[2](K) are exactly what is required to make this construction work over O K itself.
The 'only if' directions of Theorems 2 and 3 are rather more involved, and the proofs use (the easy direction of) the main result of [CLL19].(Note that since we have potentially good reduction by Theorem 1, the crucial Hypothesis ( ) of [CLL19] is always satisfied in our case.)Indeed, we have an isomorphism A k /ι (A/ι) k (see Proposition 5.6), and this implies the key observation that Kum(A k ) is the 'canonical reduction' of Kum(A) in the terminology introduced in [CLL19] (see Section 6 for the definition).By [CLL19, Theorem 1.6], good reduction of Kum(A) with an algebraic space model is therefore equivalent to the existence of an isomorphism between the Galois representations in the -adic étale cohomology groups of Kum(A k ) and Kum(A), where is any odd prime.We compare the Galois action on these cohomology groups by explicitly calculating the Galois action on the exceptional curves of Kum(A k ) and Kum(A).
We finish this paper by applying our method to give a necessary and sufficient condition for good reduction of 'twisted' Kummer surfaces, that is, Kummer surfaces associated to 2-coverings of abelian surfaces; see Theorem 7.4.Again, we only have results in the case of non-supersingular reduction.Indeed, the approach of this paper does not seem to be suitable for studying good reduction of Kummer surfaces attached to abelian surfaces with good, supersingular reduction.In the non-supersingular case, we make crucial use of the fact that a smooth model for Kum(A) over O K can be obtained by resolving the singularities of the obvious singular model A/ι.This completely breaks down in the supersingular case, where no such resolution of A/ι can possibly provide a smooth model for Kum(A).

Notation and conventions
Throughout, O K is a complete discrete valuation ring with field of fractions K, maximal ideal m K , and residue field k = O K /m K .We assume that char(K) = 0 and that k is perfect with char(k) = 2.
Let K be an algebraic closure of K, and write Γ K = Gal(K/K).The residue field of the maximal unramified extension K nr ⊂ K is an algebraic closure k of k.We write Γ k = Gal( k/k) = Gal(K nr /K), and we have the usual inertia exact sequence Let G be the (abstract) group Z/2.For any abelian group object A (in any category), we write ι : A → A for the involution ι(x) = −x, which we also think of as an action of G.

Geometry in the ordinary case
For this section, we will let A/k be an ordinary abelian variety.Thus A[2]( k) (Z/2) 2 , and the quotient variety A/G has four (geometric) singular points, all of type D 1 4 ; see [Sch09,table on p. 144].These points are in a natural Γ k -equivariant bijection with A[2]( k).The exceptional divisor of the minimal resolution Kum(A) → A/G therefore contains sixteen rational curves, occurring in four (geometrically) connected components.Our goal here is to calculate the Galois action on these curves and in particular prove the following result.

Proposition 2.1. The sixteen exceptional curves of
We achieve this by first considering the case where k is algebraically closed and giving a geometric description of how these singular points arise.This description will then be easily seen to be Galois equivariant.Note that the factor A[2]( k) arises from the four singular points of A/G.It is therefore enough to complete around the image of the identity O ∈ A and prove that the exceptional curves of the minimal resolution of the resulting D 1 4 singularity are naturally indexed by A ∨ [2]( k).So suppose for now that k is algebraically closed, and let O A,O be the completed local ring of A at the identity element O ∈ A.

Proposition 2.2. There exists an isomorphism
, and the formal group law ∆ satisfies Proof.As in the proof of [Kat78, Lemma 4], the classification of formal group laws over k implies that there is an isomorphism of formal groups (2.1) where E is any ordinary elliptic curve over k.For example, we can take E to be the curve Note that the involution ι on E is defined by ι(x, y) = (x, y + x).The function z = x/y is a local parameter at the origin of the group law O ∈ E, so that the completed local ring O E,O is isomorphic to k z .We have ι(z) = z(1 + z) −1 .In view of (2.1), this proves the claim concerning the action of ι.For the claim concerning the form of ∆, we simply use the formula on the top of p. 115 of [Sil86].
Let S = Spec( O A,O ), let T = S/G, and let q : S → T be the quotient morphism.It follows from [Sch09, Proposition 1.1], which is simply a statement of the main result of [Art75], that where q is given by Let O ∈ S denote the closed point and q(O) ∈ T its image in T ; thus O = V (u, v) and q(O) = V (x, y, z) as closed subschemes of S and T , respectively.Let be the blowups at O and q(O), respectively.Explicitly, we have and .
It is clear that S (1) is smooth.By [Kat78, Proposition 3(i)] or [Sch09, Proposition 5.1], the singular point q(O) is a rational double point of type D 1 4 (for the classification of rational double points in all characteristics, see [Art77]).By [Sch09, Corollary 1.7], T (1) is normal with at most rational double points as singularities, and the minimal desingularisation of T factors through T (1) .We let D (1) ⊂ S (1) and E (1) ⊂ T (1) denote the reduced exceptional subschemes of S (1) → S and T (1) → T , respectively.
Proof.Let S → S be the blowup of the fibre q −1 (q(O)) ⊂ S. By the universal property of blowing up, we have a commutative diagram O becomes invertible on S (1) .By the universal property of blowing up, there is a unique factorisation S (1) → S → S, which therefore provides the required regular map S (1) → T (1) extending q.Explicitly, the map q is given by This map can therefore be identified with the relative Frobenius and is thus a universal homeomorphism.
Via the natural identification D (1) = P(T O A), where T O A is the tangent space to A at O, these three points therefore correspond to three distinguished tangent directions to A at O. We can give an alternative description of these tangent directions as follows.
Since k has characteristic 2, the inclusion A[2] ⊂ A induces an isomorphism T O (A[2]) → T O A, and the formula for ∆ given in Proposition 2.2 gives A[2] • µ ⊕2 2 .We therefore obtain three subgroups of A[2] • isomorphic to µ 2 , and the tangent spaces to these three subgroups give three preferred tangent directions to A at O. Lemma 2.4.These three directions are precisely those coming from the three singular points of T (1) .
We now drop the assumption that k is algebraically closed.Then we can form S = Spec( O A,O ), T := S/G, q : S → T , and T (1) := Bl q(O) T as above, and thus T (1) has precisely three singular points over k.The description of these three singular points as arising from the three subgroups of Ak[2] • respects the Galois action, and so we obtain the following.
Corollary 2.5.Let A be an ordinary abelian surface over k.Then there is an isomorphism of k-schemes

Proof. The three subgroups of Ak[2]
• isomorphic to µ 2 canonically correspond to the three non-zero elements of the abelian group Hom(A ∨ [2]( k), F 2 ), which we can identify canonically with If we now consider the minimal resolution Kum(A) → A/G, then we can apply the description given in Corollary 2.5 around each of the four singular points of A/G.Since these singular points correspond precisely to the points of A[2]( k), this completes the proof of Proposition 2.1.

Geometry in the almost ordinary case
We now want to carry out a similar analysis when A is an almost ordinary abelian surface over k.We have The exceptional divisor of the minimal resolution Kum(A) → A/G contains sixteen rational curves, occurring in two (geometrically) connected components.The main result of this section is then the following.
Proposition 3.1.The sixteen exceptional curves of Kum(A) → A/G are all k-rational.

Translating by the unique non-identity point of
we see that it suffices to prove the rationality of the eight exceptional curves of the minimal resolution of the formal D 2 8 singularity Spec( O A,O )/G.If we look at the associated Dynkin diagram, we see that the only possible Galois action is to interchange the two curves corresponding to the nodes at the two 'short ends' of the diagram.Thus we see straight away that at least six of the exceptional curves are k-rational, and the remaining two are defined (at worst) over a quadratic extension of k.

Formal groups and 2-divisible groups
To prove Proposition 3.1, we decompose the formal group of A.

Lemma 3.2. There is a k-isomorphism of formal groups
that of its dual.If we let D denote the Cartier duality functor for p-divisible groups (so that and invoking [Tat67, Proposition 1] then gives the result. which is defined by the ideal generated by u + ι(u) = a and is therefore of rank val u (a) over k.But since G h has height h, we know that G h [2] is of rank 2 h over k, and the result follows.
Corollary 3.4.Let x = u • ι(u) and a = u + ι(u).The quotient G h /ι is given by Spf(k x ), the element a lies in k x and satisfies val x (a) = 2 h−1 , and u satisfies the equation Proof.Arguing as in the proof of [Art75, Lemma 1], we see that k u is finite free over k x of rank dim k k u /(x) = 2.In particular, this implies that k((u))/k((x)) is finite of degree 2, from which we deduce that the inclusion k((x)) ⊂ k((u)) G is an equality, and hence and val u (x) = 2; hence val x (a) = 2 h−1 , as required.Finally the fact that u 2 + au + x = 0 is a straightforward check.

A naïve resolution of A/G
Having decomposed the formal group of A, we can then explicitly resolve the singular surface Spec( O A,O )/G in the most naïve way, by repeatedly blowing up the 'worst' singularity.Let us write S = Spec( O A,O ) = Spec(k u, v ), where we may use the results of Section 3.1 above to choose u and v in such a way that G acts via From Corollary 3.4, we obtain u 2 + au + x = 0, where a = x (x) for some (x) ∈ k x × .Similarly, we have v 2 + bv + y = 0, where b = y 2 η(y) for some η(y) ∈ k y × .By [Sch09, Proposition 1.1], we see that the quotient T := S/G is given by .
Write q : S → T for the quotient morphism.Then q(O) is the unique singular point of T and is of type D 2 8 .

Lemma 3.5. There exist a cubic extension k /k and a change of co-ordinates
.
Proof.If we make the given substitution, then we obtain the equation and for this to be of the required form, we need Solving these equations gives Thus we need to be able to take a cube root of η ∈ k y × , which we can always do over the (at worst) cubic extension k( 3 η(0)).
We already know that all the exceptional curves in the minimal resolution of T are rational over a quadratic extension of k.To prove that they are k-rational, we are therefore allowed to make a cubic extension of k and hence assume that Let T (1) be the blowup of T at q(O) = V (x, y, z).Thus .
Via direct calculation, we can show that T (1) has precisely two singular points, both defined over k, and given in co-ordinates (x, y, z, [X : 2 = (0, 0, 0, [0 : 1 : 0]).These are of type A 1 and D 1 6 , respectively.Indeed, if we set y = y/x and z = z/x, we can explicitly describe the formal completion and if we set x = x/y and z = z/y, then the formal completion T (1) is given by .
To perform explicit calculations on T (2) , we write x = x/y and z = z/y and take the formal completion Further explicit calculations show that T (2) has three singular points 2 , and Q (2) 3 .The first of these, 1 , is obtained simply as the strict transform of 2 and is given in co-ordinates (x , y, z , [X : Y : Z ]) by (0, 0, 0, [1 : 0 : 0]).It is therefore k-rational, and it is not difficult to show, again via explicit calculation, that it is of type A 1 .
This completes the proof of Proposition 3.1.

The Galois action on Kummer surfaces
We can now use the above results to describe, in the non-supersingular case, the Galois action on the cohomology of Kummer surfaces over K and k.We will assume that k is perfect of characteristic 2 and let A/O K be a relative abelian surface with generic fibre A := A K .

Preliminaries on Cartier duality
The 2-torsion subscheme A[2] is a finite flat group scheme over O K ; we let A[2] • be the connected component of the identity.By [Tat97, Section (3.7)(I)], we have the connected-étale exact sequence of finite If we base change from O K to k, the connected-étale sequence splits, and we have and this is the unique étale group scheme over k whose k-points are in Γ k -equivariant bijection with A k [2]( k); see for example [Tat97, Section (3.7)(IV)].Thus over O K , we deduce that which are all isomorphisms of (unramified) Γ K -modules.
Lemma 4.1.The connected-étale sequence Proof.Clearly if the connected-étale sequence splits, then the sequence of Γ K -module splits.Conversely, if the sequence of Γ K -modules splits, we obtain a finite flat group scheme ét has order 4. The Cartier dual of a commutative finite étale group O K -scheme annihilated by 2 is connected; hence Similarly, we have , and we also have a canonical isomorphism of Thus the connected-étale sequence gives rise to an exact sequence of Γ K -modules In particular, we have the following statement.

Galois action over the residue field
If A k is not supersingular, then the minimal resolution Kum(A k ) → A k /G is a K3 surface by [Kat78, Theorem B].There are sixteen (geometric) exceptional curves, which appear in four disjoint D 4 configurations when A k is ordinary, and two disjoint D 8 configurations when A k is almost ordinary.The results of Sections 2 and 3 then describe the Galois action on these curves: if A k is ordinary, the exceptional curves are indexed Galois-equivariantly by A ∨ [2]( k) × A[2]( k), and if A k is almost ordinary, they are all k-rational.Definition 4.3.Suppose that a group Γ acts on a finite set X.For a prime , we write Q X for the associated Thanks to the results of Sections 2 and 3, we can then describe the Then for any odd prime , there is an isomorphism Corollary 4.5.Suppose that A k is almost ordinary.Then for any odd prime , there is an isomorphism

Galois action over the fraction field
We now consider the Γ K -representation in the cohomology group H 2 ét (Kum(A) K , Q ) of the generic fibre.In this case, we have an isomorphism of Γ K -modules on the 2-torsion points of A. A detailed explanation of this isomorphism can be found in [Ove21, Lemma 4.1].
We will want to compare the Galois representations in H 2 ét (Kum(Ak), Q ) and H 2 ét (Kum(A) K , Q ), at least when A k is non-supersingular.In the almost ordinary case, this is entirely straightforward, so we will concentrate on the case where A k is ordinary.
In this case, if we let Let P ⊂ GL(W ) be the parabolic subgroup which leaves invariant the Theorem 4.6.Suppose that A k is ordinary.Then the following properties are equivalent: (1) There is an isomorphism of (2) The extension of F 2 [Γ K ]-modules (4.1) is split. ( If these properties hold, then the Proof.The implications (1) ⇒ (3) and (2) ⇒ (1) are clear.Either (1) or (2) implies that the representation of Γ K in W is unramified because the representation of Γ K in V is unramified.It remains to show that (3) implies (2).Choosing a section V 2 → W of the projection W → V 2 induces a section σ of the projection π : P → GL(V 1 ) × GL(V 2 ).Thus we have a split exact sequence of groups where R u (P ) is the unipotent radical of P .We need to show that if The pullback of (4.2) with respect to the inclusion where We note that the action of (A, B) Since this group is annihilated by 2, by the standard restriction-corestriction argument, it is enough to prove that Then H is contained in a subgroup of S 3 × S 3 isomorphic to the product of Z/2 ⊂ S 3 and Z/2 ⊂ S 3 .The F 2 -vector space V 1 has a basis whose elements are permuted by Z/2, and similarly for V 2 .This gives a basis of V 1 ⊗ F 2 V 2 whose elements are permuted by

Explicit smooth models
Let A/O K be an abelian scheme of relative dimension 2 whose reduction A k is non-supersingular.Let A := A K be the generic fibre of A. In this section, we show how to construct an explicit smooth model for Kum(A) under suitable assumptions on the 2-torsion A[2](K) as a Γ K -module.In particular, this will result in a proof of Theorem 1.

Blowups and specialisation
Let X → Spec(O K ) be a flat morphism of finite type, of relative dimension 2, and with normal, integral fibres.It is not true in general that blowing up a closed subscheme of X commutes with base change to k, even if the centre is flat over O K .However, we do have the following result.
Proposition 5.1.Suppose that Z ⊂ X is an O K -section such that both fibres of X have an isolated rational double point at Z. Then Bl Z (X ) k Bl Z k (X k ).
Proof.Let I ⊂ O X be the ideal of Z and I k ⊂ O X k the ideal of Z k in X k .Thus I k is just the image of I inside O X k , and similarly each power I n k is the image of the corresponding power  , so we need to show that the natural map k is an isomorphism for all n.Since this map is clearly surjective, we see that the natural map To see this, we first observe that each O X /I n is finite over O K .Indeed, O X /I is finite over O K , and I /I 2 is a finitely generated O X /I -module.Hence the surjective map (I /I 2 ) ⊗n → I n /I n+1 shows that each I n /I n+1 is a finitely generated O K -module, and thus so is each O X /I n by induction on n.
By the structure theorem for modules over a principal ideal domain, we see that each O X /I n is isomorphic to an O K -module of the form K .Thus flatness over O K is equivalent to having s = 0, which in turn is equivalent to the equality of dimensions (5.1) are both maximal ideals defining isolated rational double points on normal surfaces, it follows from Lemma 5.2 below that both sides of (5.1) are equal to n 2 .

Lemma 5.2. Let (A, m A ) be a complete Noetherian local ring, normal of Krull dimension 2, with a rational double point at m A . Then dim A/m A (m n
A /m n+1 A ) = 2n + 1. Proof.We may assume that the residue field A/m A is algebraically closed.In this case, the computation is done in Theorem 4 and Corollary 6 of [Art66].This result has the following important consequence.Lemma 5.3.Let x ∈ X (k), and assume that X k has a rational double point of type A 1 at x. Then there exists at most one (geometric) rational double point of X K specialising to x.
Remark 5.4.If P : Spec(K) → X K is a point, then there exist a finite extension L/K and a finitely generated O K -algebra R ⊂ L such that R[1/2] = L and the scheme-theoretic image of P is given by a closed immersion Spec(R) → X .Since the normalisation of R is equal to O L , the special fibre Spec(R ⊗ O K k) has a single point.We say that P specialises to x if the closed immersion Spec(R) → X sends the unique point of Spec(R Proof.Suppose towards a contradiction that there are two distinct such points P 1 , P 2 .Let L/K be a finite extension over which both are defined, with ring of integers O L and residue field k L .Let X O L be the base change of X to O L .Thus X O L is flat over O L with normal fibres, its special fibre X k L has a rational double point of type A 1 at x, and both P 1 and P 2 specialise to this point.
Let P 1 denote the scheme-theoretic closure of P 1 inside X O L .This is therefore an O L -section of X O L , and we let X O L denote the blowup of X along this section.Thus the special fibre X k L is smooth over k L at every point in the fibre over x.Now, we let P 2 denote the closure of P 2 inside X O L and P 2 ⊂ X O L its strict transform.Then P 2 is an O L -section of X O L such that the special fibre X k L is smooth at P 2,k L , but the generic fibre X L is singular at P 2,L .This is the contradiction we seek.
Remark 5.5.More generally, if we have a collection of rational double points, we can talk about their 'total degree' as being the sum of the lower indices in the ADE classification.We can then use Lemma 5.3 to prove by induction that the total degree of a collection of rational double points cannot increase under specialisation.We will not need this more general result.

Blowing up A/G along the étale part of 2-torsion
We now let Y := A/G be the quotient scheme by the involution ι and let q : A → Y be the quotient morphism.Note that Y is flat over O K .Our aim is to show that under appropriate conditions one can explicitly construct relative (smooth) Kummer surfaces by resolving the relative quotient surface Y .
We begin by describing the fibres of Y → Spec(O K ).Since quotients of quasi-projective schemes by finite group actions commute with flat base change, (4) we get a natural identification A/G → Y K .In fact, the same is true on the special fibre.

Proposition 5.6. The natural map
Proof.In view of compatibility of quotients by G with flat base change, we may assume that k is algebraically closed.Since the G-action is free on A \ A[2], we obtain an isomorphism Once more appealing to compatibility with flat base change, it therefore suffices to prove the analogous statement after base changing to the formal completion ).Up to translation, and possibly enlarging K if necessary to ensure that the map  G is injective; we need to show that it is in fact an isomorphism.In other words, we need to show that is surjective.But this is straightforward since every element of k u, v G can be written as a series in Let us now assume that A k is non-supersingular and that the exact sequence of Γ K -modules Lemma 5.7.The natural map σ (A[2] ét ) → Z is an isomorphism.
Proof.The claim can be checked after making a finite extension of K, so we may assume that σ (A[2] ét ) ⊂ A consists of either four (in the ordinary case) or two (in the almost ordinary case) disjoint O K -sections.Their images remain disjoint in Y , and so we may reduce to considering the scheme-theoretic image in Y of a single O K -section in A, where the claim is clear.
(4) This is essentially because the ring of functions on the quotient can be expressed as a kernel; see for example the argument in Section (4.24) on p. 59 of [EvdGM].
Let Y (1) be the blowup of Y in Z.It then follows from Proposition 5.1 that the special fibre Y (1) k is the blowup of Y k in its reduced singular locus.

The ordinary case
We now assume further that A k is ordinary.Thus both the generic fibre Y In other words, if we let Z (1) ⊂ Y (1) denote the scheme-theoretic closure of the twelve (geometric) singular points on Y (1) K , then Z (1) is finite étale over O K of degree 12 and intersects both the generic and special fibres in their reduced singular loci.We now let X be the blowup of Y (1) in Z (1) .Theorem 5.8.Suppose that A k is ordinary and the exact sequence ét gives rise to a smooth and projective scheme X /O K , equipped with an action of A[2] ét , whose fibres are the minimal desingularisations of the fibres of A/G.In particular, Proof.The statement about the special fibre follows from the results of Section 2; in particular, X k is smooth.The generic fibre X K is clearly the Kummer surface attached to A K , so it is smooth.We can see that It is projective because it is an iterated blowup of the projective O K -scheme Y .Finally, X /O K is smooth since it is flat with smooth fibres; see [Stacks,Lemma 01V8].

The almost ordinary case
The construction of an explicit smooth resolution of Y = A/G is slightly more involved in the almost ordinary case.Here, we assume that A[2](K) = A[2](K), and we show that we can perform the 'naïve' resolution described in Section 3.2 on the relative surface Y .In fact, we will replace A with the formal completion S = Spec( O A,O k ) at the zero section of the special fibre.We let T = S/G and write q : S → T for the quotient map.We will describe a resolution of S via an explicit sequence of blowups.To obtain a resolution of Y , we simply perform the same sequence of blowups and then translate the whole procedure by the point of A[2](K) given by the image of the non-identity point of First of all, note that any point in the kernel of the reduction map A(K) → A(k) can naturally be thought of as a K-point of S.Moreover, any such point extends uniquely to a section Spec(O K ) → S. In particular, all eight points of A[2] • (K) give rise to sections of S, and the images of these sections under q intersect the generic fibre T K precisely at its eight singular points.The fibre product T (1) := Y (1) × Y T is then the blowup of T along q(O).
We therefore know that the (reduced) singular locus of T (1) K consists of seven K-rational points of type A 1 , and the (reduced) singular locus of T (1) k consists of two k-rational points, one of type A 1 and one of type D 1 6 .By Lemma 5.3, there exists at least one singular point on T (1) (in the notation of Section 3).Let Q (1) 2 ⊂ T (1) denote the closure of this point and T (2) the blowup of T (1) along Q (1) 2 .Now the (reduced) singular locus of T (2) K consists of six K-rational points of type A 1 , and the (reduced) singular locus of T (2) k consists of three k-rational points, two of type A 1 and one of type D 0 4 .By Lemma 5.3, there exists at least one singular point on T (2) K specialising to the D 0 4 -singularity Q (2) 3 .We let Q (2) 3 ⊂ T (2) denote the closure of this point and T (3) the blowup of T (2) along Q (2) 3 .We therefore see that both the general and special fibres of T (3) have precisely five singular points, all of type A 1 , and the singular subscheme of T (3) is finite étale of degree 5 over O K .Now, as in the ordinary case, we may blow up the singular subscheme of T (3) to obtain a scheme which is formally smooth over O K .We therefore obtain the following analogue of Theorem 5.8.Theorem 5.9.Suppose that A k is almost ordinary and that A[2](K) is trivial as a Γ K -module.Then there exists a smooth and projective scheme X /O K , equipped with an action of A[2] ét , whose fibres are the minimal desingularisations of the fibres of A/G.In particular, X K Kum(A K ) and X k Kum(A k ).
Remark 5.10.It is not completely transparent exactly where the hypothesis that However, a more careful examination of the proof shows that: (1) the two sections Q (1) 2 and Q (2) 3 that we blew up above, (2) the chosen splitting of all combine to give a set of K-rational

Good reduction criterion
In this section, we complete the proofs of Theorems 1, 2, and 3. Since we will want to use the main result of [CLL19], we first need to explain the 'canonical reduction' of a K3 surface which plays a key role there.

Canonical reduction
Suppose that X/K is a K3 surface which attains good reduction after a finite and unramified extension of K.In general, this does not imply that X has good reduction over K itself; see for example [LM18,Section 7].However, it is still possible to produce a K3 surface X 0 /k as the 'reduction' of X, in a way that is unique up to k-isomorphism.
The key result that allows us to do this is [LM18, Proposition 4.7(2)], which says that if X 1 and X 2 are smooth models of our K3 surface X over O K , that is, smooth and proper algebraic spaces over O K with generic fibres identified with X, then X 1 and X 2 are connected by a sequence of flopping contractions and their inverses.In particular, this implies that the canonical rational map X 1 X 2 given by the identity on generic fibres is an isomorphism away from a finite collection of curves on the special fibres of X 1 and X 2 .We may therefore restrict this map to obtain a birational map X 1,k X 2,k between these special fibres.As remarked in the introduction, these special fibres must be K3 surfaces (since they have trivial canonical bundle and coherent Euler characteristic 2), and hence this restricted birational map is in fact an isomorphism.
This has the following important consequence.Suppose that L/K is a finite and unramified Galois extension, with induced residue field extension k L /k.If X /O L is a smooth model for X L , then we may consider the natural Gal(L/K)-action on X L as a rational action on X .By the above discussion, this is defined away from a finite collection of curves on X k L .We can therefore restrict this rational action to the special fibre X k L , and again, as in the above discussion, this restricted rational action is regular.Hence we may form the quotient X 0 := X k L / Gal(L/K), which is a K3 surface over k by the theory of Galois descent.Moreover, if we had any two such smooth models X 1 and X 2 over O L , then the identity map between the generic fibres induces a Gal(L/K)-equivariant isomorphism between their special fibres.Thus the K3 surface X 0 , up to isomorphism, does not depend on the choice of model X /O L , or indeed on the choice of L. Definition 6.1.The K3 surface X 0 over k is called the canonical reduction of X.
In our case the canonical reduction appears in the following way.Lemma 6.2.Let Y /O K be a flat, projective scheme with normal, 2-dimensional fibres, such that the minimal resolution X of Y K is a K3 surface.Assume that there exist a finite and unramified Galois extension L/K and a proper birational morphism X → Y O L such that X is smooth over O L with generic fibre isomorphic to X L .Then the minimal resolution of Y k is the canonical reduction of X.
Proof.The Galois group Gal(L/K) acts naturally on X L and Y L , and the morphism X L → Y L is Gal(L/K)equivariant.The action of Gal(L/K) on X L extends uniquely to a rational action of Gal(L/K) on X .The morphism X → Y O L is thus Gal(L/K)-equivariant.As we have seen, the rational action of Gal(L/K) on X restricts to a regular action on X k L .We therefore obtain a Gal(k L /k)-equivariant birational morphism which by definition is the canonical reduction of X, is the minimal resolution of Y k .

The proofs
We can now prove our main theorems.Theorem 6.3.Let A be an abelian surface over K with good, non-supersingular reduction.Then we have the following statements: (1) The Kummer surface Kum(A) attains good reduction with a scheme model after a finite field extension is unramified, then the extension L/K in (1) can be chosen to be unramified.
(3) (a) If A has ordinary reduction, then Kum(A) has good reduction over K if and only if the exact sequence Proof.Parts (1) and (2), as well as 'if' statements of part (3) immediately follow from Theorems 5.8 and 5.9.
In particular, we see that the K3 surface Kum(A) satisfies Hypothesis ( ) in the terminology of [LM18] and [CLL19].
For the 'only if' direction of part (3), we first note that the hypothesis that Kum(A) has good reduction implies that A[2](K) is an unramified Γ K -module.We then consider the singular model A/ι for Kum(A), which by the results of Section 5 has a simultaneous resolution after an unramified extension of K. Hence Lemma 6.2 shows that Kum(A k ) is the canonical reduction of Kum(A).
It therefore follows from [CLL19, Theorem 1.6] that good reduction of Kum(A) over K is equivalent to the existence of an isomorphism of Γ K -representations ) for any odd prime .Indeed, given this isomorphism, we have an induced isomorphism on respective semisimplifications and therefore an isomorphism ét (Ak, Q ) since both sides are semisimple and H 2 ét (A K , Q ) H 2 ét (Ak, Q ).Hence we can apply Theorem 4.6 in the case of ordinary reduction or Corollary 4.5 in the case of almost ordinary reduction.
Example 6.4.The condition appearing in Theorem 6.3(3a) is not automatic, even if we assume that A[2](K) is unramified.To see this, we let E/Z 2 be the elliptic curve defined by If we reduce modulo 2, we get the curve E F 2 : y 2 + xy = x 3 + 1, which is smooth over F 2 , and hence E is indeed an elliptic curve over Z 2 .Note that E F 2 has the 2-torsion point (x, y) = (0, 1) and is therefore ordinary.If we make the change of co-ordinates y = 1 2 (y − x), then E := E Q 2 can be defined by the equation and the right-hand side factors as Since 20 is not a square in Q 2 , it follows that E(Q 2 ) has precisely one point of exact order 2, namely (x, y) = − 1 4 , 1 8 .However, the full 2-torsion of E is defined over the unramified extension In particular, if we take A = E × Z 2 E, then A := A Q 2 is an abelian surface with good ordinary reduction, and the exact sequence modules and is non-split.Thus Kum(A) has good reduction over Q 2 ( √ 5) but not over Q 2 .
We leave it to the reader to find an example of an abelian surface A/K with good, almost ordinary reduction, such that A[2](K) is unramified but non-trivial as a Γ K -module.Remark 6.5.In the case where Kum(A) has good reduction, we can ask what the height is of the reduced K3 surface over k, that is, the height of its formal Brauer group over k as defined in [AM77].This is the same as the height of the Kummer surface Kum(A k ) and can be detected in the slopes (roughly speaking, the valuations of the Frobenius eigenvalues) of the geometric crystalline cohomology groups of Kum(A k ).After replacing k with k, and using crystalline cohomology relative to W := W ( k), we have an isomorphism of F-isocrystals Q .This allows us to simply read off the slopes of H 2 cris (Kum(Ak)/W ) Q from those of Indeed, if A k has 2-rank r ∈ {1, 2}, then H 1 cris (Ak/W ) Q has slopes 0, 1 2 , 1 with multiplicities r, 4 − 2r, r, respectively.If we therefore set h := 3 − r, then H 2 cris (Ak/W ) Q has slopes 1 − 1 h , 1, 1 + 1 h with multiplicities h, 6 − 2h, h, respectively.Since W (−1) Q has slope 1, we deduce from [Ill79, Section II.7.2] that Kum(A k ) has height 1 when A k is ordinary, and height 2 when A k is almost ordinary.

Twisted Kummer surfaces
We can also play the same game with Kummer surfaces obtained via twisting.Indeed, let A/K be an abelian surface, and let Z be a K-torsor for A where A[2] acts diagonally on the product, is a K-torsor for A. There is an action of G on A Z , coming from the G-action on A, and we can form the quotient A Z /G.Let Kum(A Z ) be the minimal resolution of A Z /G.This is a K3 surface over K; indeed, it is K-isomorphic to Kum(A).Alternatively, we could use the fact that the A[2]-action on A gives rise to an action on Kum(A) and then form the twist (Kum(A) Since translations by elements of A(K) act trivially on the cohomology of A, we have canonical isomorphisms of Γ K -modules of Γ K -representations.
Lemma 7.1.Suppose Kum(A Z ) has good reduction, and let be an odd prime.Then both A[2](K) and H 2 ét (A K , Q ) are unramified as Γ K -modules, and It follows that some quadratic twist of A has good reduction (see the appendix).Since quadratic twists of A do not change the K-isomorphism class of Kum(A Z ), we will therefore assume that A has good reduction, with Néron model A/O K .The connected-étale sequence for A[2] then gives rise to an exact sequence We write π : Z → Z ét for the quotient morphism.Lemma 7.2.Suppose that there exists an isomorphism of K-schemes Z A[2] • K × K Z ét .Then the morphism π : Z → Z ét has a section.Remark 7.3.The lemma is not immediate since the second projection defined by the isomorphism Z where each L i is a finite field extension of K.If L i = K for some i, then both Z and Z ét are trivial torsors, and hence the claim follows from Theorem 4.6.We may therefore assume either that m = 1 and L 1 is a quartic extension of K, or that m = 2 and L 1 and L 2 are (not necessarily distinct) quadratic extensions of K. Since K , gives rise to a K-morphism Z ét → Z, and hence to K-morphisms Spec(L i ) → Z φ(i) → Spec(L φ(i) ) for some function (not necessarily a permutation) φ : {1, . . ., m} → {1, . . ., m}.
If φ is a permutation, then the composite map Z ét → Z → Z ét is an isomorphism, and we therefore obtain a section as claimed.But if φ is not a permutation, then we must have m = 2, L 1 L 2 are isomorphic quadratic extensions of K, and (after possibly reindexing) • K -torsor and therefore admits a section as claimed.The proof in the almost ordinary and supersingular cases is similar, but much easier.
Theorem 7.4.Let A be an abelian surface over K with good, non-supersingular reduction.Let Z be a K-torsor for A [2].
(1) The twisted Kummer surface Kum(A Z ) attains good reduction after a finite extension of K.
(2) If the étale K-scheme Z is unramified, then Kum(A Z ) attains good reduction after a finite unramified extension of K.
(3) (a) If A has ordinary reduction, then Kum(A Z ) has good reduction over K if and only if the étale K-scheme Z ét is unramified and the morphism π : Z → Z ét has a section.(b) If A has almost ordinary reduction, then Kum(A Z ) has good reduction over K if and only if the étale K-scheme Z ét is unramified, the morphism π : Z → Z ét has a section, and all points of A[2] • (K) are defined over the splitting field of Z ét .
(1) In the almost ordinary case, Z ét is a K-torsor for Z/2, and so its splitting field is either K itself or a quadratic extension of K.
(2) If Z is a trivial torsor, then the condition in part (3a) does not quite reduce to that in Theorem 6.3 since here we only require a scheme-theoretic section, whereas in Theorem 6.3 the section is required to be a group homomorphism.This implicitly proves (in the ordinary case) that the connected-étale sequence 0 −→ 0 has a section as Γ K -modules if and only if it has a section as Γ K -sets.The most direct proof that we know of this result goes through Theorem 4.6.
(3) It is not difficult to check that if Kum(A Z ) has good reduction, then it does so with a scheme model.Indeed, we can easily adapt the explicit constructions of Section 5 to produce scheme-theoretic models of our twisted Kummer surfaces.
Proof of Theorem 7.4.Parts (1) and (2) follow from Theorem 6.3.Indeed, Kum(A) and Kum(A Z ) are isomorphic over a finite extension L/K.If the K-scheme Z is unramified, then A[2](K) is unramified; hence L can be taken to be unramified over K. Now let us turn to part (3).We first claim that all hypotheses imply that Z extends to an étale A[2]-torsor Z/O K or, equivalently, that Z is unramified as an étale K-scheme.Indeed, if Kum(A Z ) has good reduction, then this follows from Lemma 7.1.On the other hand, if Z ét is unramified and π has a section, then the fact that π : Z → Z ét is a A[2] • K -torsor implies that Z A[2] • K × K Z ét .In the ordinary case, this implies directly that Z is unramified.In the almost ordinary case, we use the extra assumption that all points of A[2] • (K) are defined over the splitting field of Z ét to conclude this.
We may therefore assume that we have such a torsor Z.We let Z ét denote the pushout of Z along A[2] → A[2] ét ; thus Z ét K Z ét .The special fibres Z k and Z ét k are then étale torsors for A k [2] and A k [2] ét , respectively.Now, the connected-étale sequence for A k [2] splits canonically, and hence we may write • is trivial as an étale sheaf on Spec(k), any such torsor must be trivial, so we have There is a natural action of both A k [2] and A k [2] ét on A k , and we may therefore form the twisted Kummer surfaces Kum(A k,Z k ) and Kum(A k,Z ét k ), which are isomorphic since ) is the canonical reduction of Kum(A Z ).To see this, we can twist the abelian scheme A by the A[2]-torsor Z to form A Z .Now taking the quotient of A Z by the natural G-action gives rise to a flat O K -scheme A Z /G, with normal fibres, such that the minimal resolutions of the fibres are Kum(A Z ) and Kum(A k,Z k ), respectively.Thus we may apply Lemma 6.2.As Γ K -representations, we have as well as if A k is ordinary, and if A k is almost ordinary.Indeed, in the latter case we know that all sixteen exceptional curves of Kum(A k,Z ét k ) have to be defined over the (at most) quadratic extension of k trivialising Z ét k but not over any smaller field.We therefore deduce from [CLL19, Theorem 1.6] that Kum(A Z ) has good reduction over K if and only if there exists an isomorphism of (unramified) Γ K -sets in the ordinary case, and in the almost ordinary case.
In the ordinary case, if we have such an isomorphism, then the existence of a section was proved in Lemma 7.2.Conversely, if π has a section, then we know that Z A[2] • K × K Z ét since π : Z → Z ét is a trivial A[2] • K -torsor.In the almost ordinary case, if we have such an isomorphism Z(K) 8 i=1 Z ét (K), then any set-theoretic section will be Γ K -equivariant.The fact that all points of A[2] • (K) have to be defined over the splitting field of Z ét follows from applying Theorem 6.3 over this field.
Conversely, suppose that π has a section and all points of A[2] • (K) are defined over a splitting field L/K for Z ét .If L = K, then clearly both Z(K) and 8 i=1 Z ét (K) are trivial Γ K -sets.Otherwise, L/K is quadratic, and then both Z(K) and 8 i=1 Z ét (K) consist of sixteen points, none of which are fixed by Γ K , but all of which are fixed by the index 2 subgroup Γ L ⊂ Γ K .We can therefore directly construct a Γ K -equivariant bijection Z(K) 8 i=1 Z ét (K).It is possible for Kum(A Z ) to have good reduction over K even if Kum(A) does not.We give examples in both the ordinary and almost ordinary cases.
Example 7.6.To give an example where A has good, ordinary reduction, we take elliptic curves E 1 , E 2 over Z 2 with ordinary reduction, such that E 1 [2](Q 2 ) is a trivial Γ Q 2 -module but E 2 [2](Q 2 ) is a non-trivial but unramified Γ Q 2 -module.For example, we could take E 1 to be defined by y 2 + xy = x 3 − 4x − 1 and E 2 to be the curve from Example 6.4.
Let K/Q 2 be the unramified quadratic extension over which all Q 2 -points of E 2 [2] are defined (equivalently, over which the connected-étale sequence for E 2 [2] splits), and let σ : Z/2 E 1 [2] ét → E 1 [2] be a splitting of the connected-étale sequence of E 1 .We take A = E 1 × Z 2 E 2 and A = A Q 2 ; thus σ induces a map Z/2 → A[2].The quadratic extension K/Q 2 gives rise to a class [K] ∈ H 1 (Γ Q 2 , Z/2), and we let Z/Q 2 be an A[2]-torsor whose cohomology class is equal to the image of [K] in H 1 (Γ Q 2 , A[2]).
To complete the proof, we now choose χ ∈ H 1 (Γ K , Z/2) = Hom(Γ K , Z/2) to be a lifting of the restriction of ρ to I K .Then the representation of Γ K in H 1 ét (A χ K , Q ) restricts to the trivial representation of I K .
as the image of the splitting.Taking the closure of G inside A[2] then gives a finite flat group scheme G ⊂ A[2] of order |A[2] ét | mapping isomorphically onto A[2] ét .This gives a splitting of the connected-étale sequence.Let A ∨ /O K be the dual abelian surface, which exists by [BLR90, Theorem 8.5].By [Oda69, Corollary 1.3], the finite flat group k-schemes A[2] and A ∨ [2] are Cartier duals of each other; that is, surjective, we may therefore replace A with its completion at the zero section O k ∈ A k on the special fibre.Thus we have an action of G on O A,O k O K u, v , and we may choose the local parameters u and v in such a way that the description of the action and its quotient given in [Sch09, Proposition 1.1] and [Art75, Theorem, p. 60] holds modulo m K .The quotient

K
and the special fibre Y (1) k contain twelve (geometric) singular points, all of which are rational double points of type A 1 .Each of the twelve singular points on Y (1) K has to specialise to a singular point on Y (1) k , and by Lemma 5.3 each singular point on Y (1) k is specialised to by at most one singular point on Y (1) K .
modules splits.If this condition holds, then the Γ K -module A[2](K) is unramified and Kum(A) has good reduction over K with a scheme model.(b) If A has almost ordinary reduction, then Kum(A) has good reduction over K if and only if the Γ K -module A[2](K) is trivial.If this condition holds, then Kum(A) has good reduction over K with a scheme model.
and A/G has two geometric singular points, both k-rational and both of type D 2 8 ; see [Sch09, table on p. 144].
canonically by [Tat97, last paragraph on p. 142].Thus by Cartier duality, we obtain a