The Néron Ogg Shafarevich criterion Erik Visse

The Néron Ogg Shafarevich criterion Erik Visse February 17, 2017 These are notes from the seminar on abelian varieties and good reductions held in Amsterdam late 2016 and early 2017. The website for the seminar can be found at https://staff.fnwi.uva.nl/w.p.bindt/seminar.html. This talk is for a large part based on Section 3 of Bart Litjens master s thesis [Lit14] 1 Galois theory of local fields Throughout this talk, let K be a non-archimedean local field with valuation v, ring of integers O K, maximal ideal m K, and residue field k = O K /m K. Let L be a Galois extension of K with valuation w, ring of integers O L, maximal ideal m L, and residue field l. In some but not all of the statement we will need k to be perfect. It would not hurt us to assume this to be the case throughout the talk, so we will. Remark 1.1. Remember that O K is a discrete valuation ring, hence its spectrum consists of two points, one of which is closed and is called the special point as opposed to the generic point. Given some morphism to the spectrum of a discrete valuation ring, we call the fibre above the special point the special fibre. Definition 1.2. The decomposition group D of the above extension is D(L/K) = {σ Gal(L/K) σ(m L ) = m L }. It is easily seen that the elements σ of the decomposition group leave the w-valuation of an element of L unchanged, i.e. w(σ(x)) = w(x) holds for each x L, and hence w(σ(x) x) min{w(σ(x)), w(x)} = w(x). We could require that for units of O L the inequality be strict. The automorphisms σ satisfying this form a subgroup known as the inertia subgroup. Definition 1.3. The inertia subgroup of D(L/K) is I(L/K) = {σ D x O L : σ(x) x m L }. This may seem a bit artificial at first, but the definition is motivated by the following: Proposition 1.4. There is an exact sequence 1 I(L/K) D(L/K) Gal(l/k) 1. The only hurdle in proving this is the surjectivity of the map D(L/K) Gal(l/k). Definition 1.5. Let X be a Gal(K s /K)-set. We say that X is unramified at v if I(K s /K) acts trivially on X, i.e. I ker(g Aut(X)). The goal of today is to treat the Néron Ogg Shafarevich criterion, which allows to test for good reduction at a place v by testing if certain sets are unramified at v. 1

2 Recap In this section we will review some of the concepts that have been treated in earlier lectures, and add some small results for motivation. 2.1 Tate modules and l-adic cohomology Definition 2.1. Let A be an abelian variety over a field F with separable closure F s. For a prime number l different from the characteristic of F, the l-adic Tate module is T l A = lim A[l n ](F s ). n Proposition 2.2. In the setting above, there is a canonical isomorphism of Galoismodules H 1 ét (A F s, Z l) = Hom cts (T l A, Z l ). Proposition 2.3. In the setting above, there is an isomorphism of graded Galoismodules H 1 ét (A F s, Z l) = H ét (A F s, Z l). Propositions 2.2 and 2.3 show that the Tate module of an abelian variety A essentially captures all of the l-adic cohomology of A. 2.2 Néron-models and the structure of connected group schemes Definition 2.4. Let X be a variety over a field F with ring of integers O F. A Néron model of X over O F is a smooth O F -scheme N(X), equipped with an isomorphism between X and the generic fibre η of N(X), such that for any smooth O F -scheme Y and morphism ϕ : Y F X, there exists a unique morphism Y N(X) extending ϕ. It s probably good to see the above definition in a picture: X ϕ η Y F Spec(F ) Spec(O F ) N(X)! Y Theorem 2.5. For any abelian variety A over a field F with ring of integers O F, its Néron-model exists, it is uniquely unique, and it is a quasi-projective commutative group scheme over O F. Proposition 2.6. An abelian scheme over a discrete valuation ring R is the Néronmodel of its generic fibre. 2

Remark 2.7. Notice that the Néron-model is not assumed to be proper. Properness of Néron-models is strongly connected with good reduction, as we will see in the next section. Theorem 2.8 (Chevalley structure theorem). Let G be a connected smooth group scheme over a perfect field F. Then there is a unique exact sequence 1 H G B 1 with H a connected smooth affine subgroup, and B an abelian variety. Proposition 2.9. Let H be a commutative affine group of finite type over a perfect field F. There is a unique exact sequence 1 D H U 1 with D of multiplicative type, and U unipotent. The sequence splits. 3 Good reduction To get an intuition about the concept of good reduction, let s consider elliptic curves over Q. Given such an elliptic curve E, we can write down a minimal Weierstrass equation y 2 = f(x) with coefficients in Z. We want to know if E modulo a prime p is again an elliptic curve, i.e. we want to check if we get a smooth curve, since the fixed point over Q gets reduced to a point over F p. Also the genus is automatically still 1 since we get a degree 3 curve in the projective plane. The resulting curve is found by reducing the polynomial f modulo p; smoothness can be checked with the Jacobi criterion. This boils down to knowing if the discriminant of f is divisible by p. If it is, then the reduced curve is not smooth, and vice versa. In general, given a variety over Q there is no canonical model; there is no analogue of the Weierstrass equation for elliptic curves. Therefore we need to do something more complicated. The first step is the realization that reduction modulo p is achieved by base-changing to F p. However, we run into trouble here since we cannot base-change from Q to F p directly. This is where Néron-models come in handy, since going first via Q p by base-changing, then Z p via the Néron-model, we can finally base-change to F p. Of course, this method only works when the Néron-model exists. They do not exist in complete generality and this is why good reduction is an active area of research for general varieties. Their existence for abelian varieties is what makes the current talk possible. Definition 3.1. For X a smooth and proper scheme over a local field K we say that X has good reduction at v if there exists a smooth and proper O K -scheme X whose generic fibre X OK Spec(K) is isomorphic to X. Since the model X is smooth and proper, its base-change to k is as well, so the definition makes sense. Proposition 3.2. Let A be an abelian variety over K. Then A has good reduction at v if and only if its Néron-model over O K is proper. 3

Proof. Assume that the Néron-model is proper. By its definition it satisfies all the requirements for good reduction as given in Definition 3.1. Hence A has good reduction at v. Conversely, assume that A has good reduction at v and let X be the associated proper scheme. Since Spec O K is connected and Noetherian, a result by Grothendieck [Mum65, Ch.6, Thm. 6.14] shows that X is in fact an abelian scheme. Proposition 2.6 says that X is the Néron-model of A. 4 The criterion of Néron Ogg Shafarevich The criterion that is treated in this section was published by Ogg for elliptic curves in 1967; Shafarevich is credited with knowing the result around the same time. For abelian varieties of dimension at least 2, the theorem was proven by Serre and Tate who were also the ones to have named it. Their reliance on work by Néron has earned him part of the credit of the theorem. For n coprime to char(k), we use the notation A n as shorthand for the set A n = Hom(Z/nZ, A(k s )) = A[n](k s ). Remember that k is the residue field of K. In the statement of the following theorem we are sloppy in our language. Whenever we say that an integer n is coprime to the characteristic of some field F, we mean to say char(f ) n. Theorem 4.1 (Néron Ogg Shafarevich criterion). For A an abelian variety over K, the following statements are equivalent: 1. A has good reduction at v, 2. A n is unramified at v for all n coprime to char(k), 3. A n is unramified for infinitely many n coprime to char(k), 4. T l (A) is unramified at v for some prime l char(k), 5. T l (A) is unramified at v for all primes l char(k). Before we start the proof of the theorem, let us remark that item 4 is equivalent to the statement that for all m 0 the set A l m is unramified at v. Moreover, we could have concluded item 5 after proving the equivalence of 1 and 4 since item 1 does not depend on any l. This last phenomenon is relatively common: many statements involving Z l for some l can a posteriori be strengthened to hold for all l, at least away from some field characteristic. This theorem implies many useful things, one of which is that having good reduction at v is invariant under isogeny. Proof of Theorem 4.1. Some easy implications can immediately be noticed. By the observation that we have just made, and by using the fact that every finite abelian 4

group is a product of cyclic prime-power order groups, we have 4 2 5 3. Following a suggestion from Wessel, the implication 1 2 should follow from the proper base-change theorem. This was attempted in the talk. Wessel wrote a short note on it that is available separately. Here is a different proof: if A has good reduction at v then N(A) is proper over O K and hence an abelian scheme. The special fibre N(A) k is then an abelian variety and its n-torsion (over k s ) is free of rank 2 dim(n(a) k ) = 2 dim(a). By Lemma 4.4 1 the part of A[n](k s ) that is fixed by I(K s /K) is of full rank and hence A[n](k s ) is unramified. This is exactly the statement of 2. We are left to show that the implication 3 1 holds; by Proposition 3.2 it is enough to show that N(A) is proper. We will need the following lemma: Lemma 4.2. Let R be a discrete valuation ring with residue field k and fraction field K. Let X be a smooth R-scheme whose generic fibre X K is geometrically connected and whose special fibre X k is proper and non-empty. Then X R is proper. We will apply this lemma to R = O K and X = N(A). Being (isomorphic to) the abelian variety A, the fibre N(A) K is geometrically connected. Hence all we need to do is to show that N(A) k is proper since it is obviously non-empty. Since by assumption of item 3 there are infinitely many integers n coprime to char(k) for which A n is unramified at v, for any integer c 1 there is such an n with n > c. In what follows we assume to have taken such an n > c. Since N(A) k is a group scheme (not necessarily connected), we may take its connected component of the identity N(A) 0 k. We take c = [N(A) k : N(A) 0 k ], which is finite because quasiprojective morphisms are of finite type and hence N(A) k has finitely many connected components. The index [N(A) k : N(A) 0 k ] is precisely their number. In analogy to A n we use the notation N(A) k [n](k s ) =: N(A) k,n. We will need two more lemmas, with the notation coming from Theorem 2.8 and Proposition 2.9 applied to the case G = N(A) 0 k. Lemma 4.3. The index [N(A) k,n : N(A) 0 k,n ] divides c. The group N(A)0 k,n is a free Z/nZ-module of rank dim(d) + 2 dim(b). Lemma 4.4. For n coprime to char(k), we have A I(Ks /K) n = N(A) k,n. To continue the proof, we consider the order of N(A) k,n. We have n 2 dim A = #N(A) k,n c n dim(d)+2 dim(b) dim(d)+2 dim(b)+1 < n 1 This lemma is stated later in the proof because the original attempted proof using proper basechange did not require it. This short proof of 1 2 was added after the talk and the author preferred not to change the order of the remainder of the material. 5

where the first inequality comes from Lemma 4.3 and the second from our choice of n > c. Hence we may conclude 2 dim(a) dim(d) + 2 dim(b). On the other hand, we have dim(d) + 2 dim(b) = rk ( ( N(A) 0 k,n) rk (N(A)k,n ) = rk and therefore 2 dim(a) = dim(d) + 2 dim(b). A I(Ks /K) n ) = rk(a n ) = 2 dim(a) However, applying Theorem 2.8 and Propositon 2.9 to N(A) 0 k, which has dimension dim(a), we conclude dim(a) = dim(u) + dim(d) + dim(b) and therefore ultimately dim(u) = dim(d) = 0. Since H in Theorem 2.8 and Proposition 2.9 is connected and of dimension 0, the group schemes N(A) 0 k and B are isomorphic and hence N(A) 0 k is an abelian variety, in particular it is proper over k. Since N(A) k N(A) 0 k is finite and therefore proper, so is N(A) k k. This was the last thing that needed to be shown, and this concludes the proof. Sketch of proof of Lemma 4.3. The first statement comes from playing around with indices of subgroups. The second statement comes from applying Theorem 2.8 to G = N(A) 0 k and taking the functor Hom(Z/nZ, ) of the resulting exact sequence. We get the sequence 0 H[n](k s ) N(A)[n] 0 k (ks ) B[n](k s ). One proves that the last map is surjective using n-divisibility of H(k s ). Using Proposition 2.9 and the fact that H[n](k s ) and B[n](k s ) are free Z/nZ-modules of ranks dim(d) and 2 dim(b) respectively finishes the proof. About proof of Lemma 4.4. The proof of this lemma is rather long and involved. It involves the maximal unramified extension K ur of K inside K s and the Néron mapping property: N(A) k (O K ur) = A(K ur ). The field K ur comes in since it is the fixed field of I(K s /K). References [Mum65] David Mumford, Geometric Invariant Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1965. 3 [SPa] [Mil13] [Lit14] Stacks Project authors, The Stacks Project. http://stacks.math.columbia.edu/. James S. Milne, Lectures on Etale Cohomology (v2.21), 2013. www.jmilne.org/math/. Bart Litjens, Good reduction of abelian varieties, 2014. Available on the seminar website https://staff.fnwi.uva.nl/w.p.bindt/seminar.html. (document) 6