Faltings Finiteness Theorems

Transcription

1 Faltings Finiteness Theorems Michael Lipnowski Introduction: How Everything Fits Together This note outlines Faltings proof of the finiteness theorems for abelian varieties and curves. Let K be a number field and S a finite set of places of K. We will demonstrate the following, in order: There are only finitely many isogeny classes of abelian varieties over K of dimension g with a polarization of degree d and with good reduction outside of S. This ultimately relies on the Tate Conjecture and the semisimplicity of the action of G K on V l (A) for an abelian variety A/K. (Shafarevich conjecture) There are only finitely many isomorphism classes of abelian varieties over K of a given dimension g, with polarization of degree d, which have good reduction outside S. Faltings amazing insight was that this could be done by understanding how the Faltings height h(a) varies for A/K an abelian variety within a fixed isogeny class. Let B A be a K-isogeny. Then exp(2[k : Q](h(B) h(a)) (read: change in height under isogeny ) is a rational number. For any individual prime l, the l-adic valuation of exp(2[k : Q](h(B) h(a)) can be controlled by the Tate Conjecture. Using results of Raynaud, Faltings shows that for large l, depending only on the field K and the places S where A has good reduction, the l-adic valuation doesn t change at all. These together imply that h(b) remains bounded within the isogeny class of A. It is a technically difficult but true fact that h is a Weil height, i.e. that there are only finitely many isomorphism classes of B/K of abelian varieties of fixed dimension g with h(b) any fixed upper bound. This was explained in Akshay s and Peter s talks; it rests on the existence of enough Siegel modular forms, defined over Z, to separate points in the moduli space A g,n of principally polarized abelian varieties of dimension g with full level n structure. As Brandon explained in his talk, this fact is also essential in Faltings proof of the Tate conjectures. There are thus only finitely many isomorphism classes of abelian varieties over K, equipped with a polarization of fixed degree, isogenous to A. Combined with finiteness of the number of isogeny classes with good reduction outside S, this proves the Shafarevich Conjecture. 1

2 Let s get started! Preliminaries Let K be a number field, O K its ring of integers. Let A 1, A 2 Spec(O K ) be semiabelian schemes of relative dimension g with proper generic fiber with s : Spec(O K ) A 1 the zero section. Let φ : A 1 A 2 be an isogeny between them with kernel G/O K. Lemma 5. h(a 2 ) = h(a 1 ) + 1/2 log(deg(φ)) [K : Q] log(#s (Ω 1 G/O K )). We will make good use of this formula. Proofs of Finiteness Theorems Theorem 5. Let S be a finite set of places of number field K. Then there are only finitely many isogeny classes of abelian varieties of a given dimension g with good reduction outside S. Proof. Let A 1, A 2 /K be an abelian varieties with good reduction outside S. Let M End Zl (T l (A 1 )) End Zl (T l (A 2 )) be the Z l algebra spanned by the image of π = G K. M is a Z l module of rank 8g 2 which acts on End Zl (T l (A 1 )) and End Zl (T l (A 2 ). By the Tate Conjecture, A 1 and A 2 are isogenous iff T r(m T l (A 1 )) = T r(m T l (A 2 )) for all m M, i.e. iff their Tate modules are Z l [π] isomorphic. Thus, it suffices to prove this for a set of Z l -module generators of M, which is the same as a set of Z l -module generators for M/lM by Nakayama. A fortiori, the image of ρ : π (M/lM) generates M/lM as a Z l -module. Also note that #(M/lM) l 8g2. ρ is unramified outside of S and l. Thus, letting K be the composite, inside K, of all extensions of K which are unramified outside of S and l AND are of degree 8g 2, then K is finite (Minkowski) and ρ factors through G K /K. Hence, it suffices to show that T r(g T l (A 1 )) = T r(g T l (A 2 )) for a set of coset representatives for G K /K = G K /G K (or any element in the conjugacy class of such a representative). By Chebotarev, we can choose these lifts to be finitely many representatives of Frobenius conjugacy classes F rob v1,..., F rob vr for v 1,..., v r / S {l}. But T r(f rob v T l (A 1 )) = T r(f rob v T l (A 2 )) whenever the action of F rob v on Het(A 1 1 ) and Het(A 1 2 ) have the same characteristic polynomial. By the Weil conjectures, there are only finitely many possibilities for said characteristic polynomials. These characteristic polynomials correspond precisely to the isogeny classes of A/K of dimension g with good reduction outside S. Theorem 6 ((Weak) Shafarevich Conjecture). Let S be a finite set of places of K, d > 0. There are only finitely many isomorphism classes of abelian varieties over K of a given dimension, with polarization of degree d, which have good reduction outside of S. 2

3 Proof. By Theorem 5, we may assume that all abelian varieties in question are isogenous to a fixed A. By extending the ground field, we can also assume that all B s extend to semiabelian schemes over Spec(O K ) and that d = 1. Let φ : B A be an isogeny. This induces an isogeny φ : B 0 A 0 between the connected components of their Neron models, with kernel G/O K. By the change in height formula, exp(2[k : Q](h(B) h(a))) = deg(φ)[k:q] (#s Ω 1 G/O K ) 2, We will bound the l-adic valuation of this change in height for each l. Namely, For any individual l, we will show that the change in height, among such semiabelian, principally polarized B which are isogenous to A, can be uniformly bounded. For large l, large relative to the places S of bad reduction and the ramification of K/Q, we will show that h(b) = h(a). A useful observation, for what follows, is that height does not change l-adically for l deg(φ). Indeed, G is killed by multiplication by deg(φ) which equals the order of ker(φ). Since [n] on G induces multiplication by n on s Ω 1 G/O K, we see that s Ω 1 G/O K is supported over primes dividing deg(φ). Staring at the change in height formula, we see that exp(2[k : Q](h(B) h(a))) is prime to l, i.e. there is no l-adic change in height. For small l: Suppose T l (B 1 ) φ T l (B 2 ) as Z l [π]-modules, π = G K, for two abelian varieties B 1, B 2 /K. Then by the Tate Conjecture, there is some isogeny fi a i = φ for some f i Hom(B 1, B 2 ), a i Z l. But the set Isom(T l (B 1 ), T l (B 2 )) is open Hom π (T l (B 1 ), T l (B 2 )) with its l-adic topology. Thus, for good l-adic approximations ã i Z to the a i, T l (g) is an isomorphism for g = ã i f i. Since T l (g) is an isomorphism, det(t l (g)) : det(t l (B 1 )) det(t l (B 2 )) is an isomorphism as well. But det(t l (g)) = deg(g) (Mumford, p.180), and so g is an isogeny of degree prime to l. Thus, the l-adic change in height exp(2[k : Q](h(B 1 ) h(b 2 ))) is zero. Suppose that we can show that there are only finitely many isomorphism classes of π-invariant sublattices of V l (A). By the Tate conjecture, these correspond to the Tate modules of finitely many viable B, say B 1,..., B n. But then for arbitrary B in the isogeny class of A, T l (B) = T l (B i ) for some i, whence exp(2[k : Q](h(B) h(a))) = exp(2[k : Q](h(B) h(b i ))) exp(2[k : Q](h(B i ) h(a))) whose l-adic valuation equals that of exp(2[k : Q](h(B i ) h(a))), independent of B. Now we ll prove that there are only finitely many isomorphism classes of π-invariant sublattices. 3

4 By the Tate Conjecture, End Ql [π](v l (A)) is semisimple. Thus, it decomposes as f First note that if L 1 L2 is a Z l [π]-isomorphism of π-stable sublattices of V = V l (A), then it induces an automorphism of V. By Schur s Lemma, Aut Ql [π](v ) = End Ql (D 1 ).. End Ql (D n ) for some division algebras D i /Q l. Make a finite unramified ground field extension F/Q l so that each D i -splits, i.e. D i = EndF (V i,1 )... End F (V i,m ) for subspaces V i,1,..., V i,m V F. Then the V i,m are absolutely irreducible representations of π and Aut π (V i,m ) = F. We will show that there are only finitely many F -homothety classes of π-stable lattices in each U = W i,j (and hence finitely many isomorphism classes of π stable lattices in V F ). As above, the F -span of π is all of End F (U). Thus, the O F -span of π contains α N End OF (M), where M is some lattice in U and α is the uniformizer of O F. Now take any π-stable lattice L U. By scaling appropriately, we may find a homothetic lattice, which we also call L, such that L M and There is a vector v L such that v M but v / αm. But then M L α N End OF (M)v = α N M. There can be only finitely many such L. Thus, it suffices to show that the base change map L L Zl O F from π-isomorphism classes of lattices in W to π-isomorphism classes of lattices in W F is finite to one. For this, it suffices to show that H 1 (G F/Ql, Aut OF [π](l F )) is finite. Since there are no homomorphisms between non-isomorphic representations among the V i,j V F, it suffices to restrict our attention to the lattice L = L F W, where W is the isotypic component of some absolutely irreducible F -representation of π. Supose W = W... W for some absolutely irreducible W. Then Aut OF [π](l ) = (O F )n S n as G F/Ql -groups. Corresponding to the exact sequence 1 (O F )n (O F )n S n S n 1, there is an exact sequence of pointed sets H 1 (G F/Ql, (O F )n ) H 1 (G F/Ql, (O F )n S n ) H 1 (G F/Ql, S n ) (Serre, Ch, 1, 5). Since the flanking terms are finite, the middle term must be finite as well. Hence, H 1 (G F/Ql, Aut OF [π](l F )) is finite after all. For large l: This explanation is from Rebecca s notes, but is assembled here for convenience. We may assume that all B in the isogeny class of A have semiabelian reduction over O K. Since Faltings height is invariant under base change, this assumption is harmless. Choose l large enough so that A has good reduction for all places v l of K and so that l is unramified in O K. 4

5 Let φ : B 1 B 2 be an isogeny of degree l h between abelian varieties over K isogenous to A. By filtering the kernel, we may assume that multiplication by l kills G = ker φ. We want to show that h(b 1 ) = h(b 2 ). Note that φ extends to a map φ : B 0 1 B 0 2 between the connected components of the Neron models of B 1 and B 2 over O K,(l). Since good reduction is a property of isogeny classes, B 1, B 2 also have good reduction over l and so φ is an isogeny of abelian schemes. Thus, the kernel G is a finite flat group scheme over O K,(l) with generic fiber G K = G. Our immediate goal is to use Raynaud s results to relate #s (Ω 1 G/O K,(l) ), a mysterious constituent in the change of Faltings height formula, to the Galois action on G(K). Let π = Gal(Q/Q), π = Gal(K/K). Consider the following representations over Z/lZ : V l = T l (B 1 )/lt l (B 1 ) = B 1 [l](k) Ṽl = Ind π π(v l ) W l = G(K) V l W l = Ind π π(w l ) Let [K : Q] = m. Note that W l is h-dimensional and W l is mh-dimensional. Consider the character χ = det( W l ) : π (Z/lZ). Because B 0 1 is semiabelian, the action of inertia I v, v l, on W l V l is unipotent. Hence, det(w l ) is trivial away from l. Hence, for any g I p, the action of det(g) on det( W l ) is simply by ɛ(g), where ɛ : π {±1} is the sign representation of π acting on the cosets π/ π. Thus, χ 0 = χɛ is unramified outside of l. By class field theory, any continuous character π (Z/lZ) unramified outside of l corresponds to a continuous character χ 0 : Q \A Q / p l Z p (Z/lZ). Also, any continuous character Z l = (1 + lz l ) (Z/lZ) (Z/lZ) must kill the 1-units 1 + lz l (since they form a pro-l group) and so factors through some dth power map (Z/lZ). Then χ 0 must be the dth power of the cyclotomic character (for some 0 d l 1): Claim 1. If #s (Ω 1 G/O K,(l) ) = l d, then d = d. χɛ = χ 0 = χ d l. Proof. Raynaud s results say that if H is a finite flat group scheme killed by l over a strictly henselian local ring R of mixed characteristic (0, l) with low ramification, then the Galois action on det(h) is τ v(d H/R) l, where τ l is the canonical tame character τ l : I t F l. Note that χ l Il = τ l. Thus, we are really interested in computing χɛ Il in terms of G. But 5

6 and Weil restriction O K,(l) /Z (l) Ind eπ π base change to Z un l (= R) Res π/il. Thus, we are actually interested in the Galois action on the generic fiber of G := Res OK,(l) /Z (l) (G) R = G i1 R... R G in, where there is one copy of G for each embedding i j : O K,(l) R. Applying Raynaud, we see that inertia acts by the character It follows that τ v(d G /R )/#G l = τ v(d G /R )/lmh l. d = v(d G /R)/#G = i v(d Gi /R)/#G i = i v(d Gi /R)/l h. On the other hand, by computations from Brandon s notes, #s Ω 1 G/O K = v l #s vω 1 G v/o K,v (essentially the Chinese remainder theorem) and Combining, we see that if l d = #s Ω 1 G/O K, then #s vω 1 G v/o K,v = #(O K,v /d Gv/O K,v ) 1/#Gv. d = v l f v v(d Gv/O Kv )/#G v. Making an unramified base change up to R preserves the valuation of the discriminant. Also, each of the f v embeddings of k v F l gives an embedding O K,(l) O K,v R. Running over all v l accounts for all possible embeddings O K,(l) R. Thus, d = v l = O K,v R i:o K,(l) R v(d GR /R)/#G R v(i G)/#i G = d. This is great news: we can now use global information (the Weil conjectures) to control the change in height. Define P i (T ) = det(t F rob p i Ind eπ πt l (A)). Then χ(f rob p ) = ±χ d l (F rob p) = ±p d is a zero of P mh (T ) modulo l. But by the Weil conjectures, the zeros of P mh (T ) are algebraic with absolute value p mh/2 under any complex embedding. We have the a priori crude bound d gm. 6

7 Indeed, since G is killed by l, s Ω 1 G/O K,(l) is a quotient of s Ω 1 B 0 1 [l]/o K,(l) which has order lmg. Thus, as long as we choose l large enough not to divide any of the values P i (±p j ) for this will force d = mh/2. 0 i 2gm 0 j gm j h/2 By the change in height formula, h(b 2 ) h(b 1 ) = 1 2 log(deg(φ)) 1 [K : Q] log(#s Ω 1 G/O K ) = 1 2 h log(l) 1 m d log(l) = 0 Thus, the height doesn t change under l power isogenies for large l! Combining with the earlier argument for small l, we see that the height remains bounded within isogeny classes. Thus, there can be only finitely many isomorphism classes of abelian varieties over K, equipped with a polarization of bounded degree, within the isogeny class of A. Combined with our earlier result (Theorem 5) on finiteness of the number of good-outside-s isogeny classes, this proves what we ve called the Shafarevich conjecture. Theorem ((Strong) Shafarevich Conjecture). Let S be a finite set of places of K. There are only finitely many isomorphism classes of abelian varieites over K of dimension g with good reduction outside S. Proof. See Corollary 6.6 in Brian s lecture 13 notes. This includes the statement that weak Shafarevich implies strong Shafarevich. References Bosch, Lutkebohmert, Raynaud. Neron Models. Faltings. Finiteness Theorems for Abelian Varieties. Mumford. Abelian Varieties. Serre. Galois Cohomology. 7