1 The Fundamental Group of Abelian Varieties Contents Renjie Lyu 1 Tate Modules 1 2 The Fundamental Group of An Abelian Variety 2 3 Application 4 1 Ta...

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Contents 1 Tate Modules

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2 The Fundamental Group of An Abelian Variety

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3 Application

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1 Tate Modules Let A be a g-dimensional abelian variety over a field k. Let ` be a prime number different from char(k). Denote by ks the separable closure of the field k. For any nonnegative integer n, we set a group A[`n ](ks ) := Hom(Z /`n Z, A(ks )). equipped with a natural Galois action by the absolute Galois group Gal(ks /k). Hence, A[`n ](ks ) is the group of the `n -torsion points in the group A(ks ) of ks -points of A. The multiplication by ` morphism on A induces a homomorphism of groups ` : A[`n+1 ](ks ) → A[`n ](ks ) which is Gal(ks /k)-equivariant. Note that with these homomorphisms the collection {A[`n ](ks )} forms a projective system of abelian groups with Gal(ks /k)-action. Definition 1.1. Let A be a g-dimensional abelian variety over a field k, let k ⊂ ks be a separable closure of the field k, and let ` be a prime number different from char(k). We define the Tate module T` A of A to be n T` A = ← lim −{A[` ](ks )}n∈Z≥0 . n

If char(k) = p > 0 then we define the Tate-p-module Tp,´et A to be n ¯ Tp,´et A := ← lim −{A[p ](k)}n∈Z≥0 n

where the homomorphisms are multiplication by p and k¯ is the algebraic closure of the field k.

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2 The Fundamental Group of An Abelian Variety Theorem 2.1 (Lang-Serre). Let X be an abelian variety over a field k with an identity eX ∈ X(k). Assume that Y is a complete variety over the field k with a k-point eY ∈ Y (k). If f : Y → X is an ´etale covering with f (eY ) = eX then Y has the structure of an abelian variety such that the morphism f is a separable isogeny. In order to prove this theorem we need the following two lemmas. Lemma 2.2. Let X be a complete variety over a field k. Suppose given a k-point e ∈ X(k) and a k-morphism m : X × X → X such that m(x, e) = x = m(e, x) for all x ∈ X. Then X is an abelian variety with group law m and the identity point e. Proof. See [EvdGM, Chapter X, Proposition 10.34]. Lemma 2.3. Let Z be a k-variety, let Y be an integral k-scheme of finite type, and let f : Y → Z be a smooth proper morphism of k-schemes. If there exists a section s : Z → Y of the morphism f then all fibres of f are irreducible. Proof. See [EvdGM, Chapter X, Lemma 10.35] Proof of Theorem 2.1. By the Lemma 2.2, it suffices to construct a group law mY : Y × Y → Y . Let ΓX ⊂ X × X × X be the graph of the multiplication mX : X × X → X, and let Γ0Y be the pullback of ΓX via the morphism f × f × f : Y × Y × Y → X × X × X. We write ΓY ⊂ Γ0Y for the connected component containing the point (eY , eY , eY ). In the following, we show that the projection q12 : ΓY → Y × Y from ΓY to the first two factors is an isomorphism. In particular, we can define the desired group law by taking −1 , where q3 is the projection to the third factor. mY := q3 ◦ q12 There is a natural commutative diagram ΓY

ΓX

q12

p12

/Y ×Y

f ×f

/ X × X.

Recall that the graph Γ0Y is ´etale over ΓX and the connected component ΓY is an open subset of ΓY , which implies that the left hand arrow is ´etale. Further, it follows that the morphism q12 is ´etale since the bottom arrow is an isomorphism and f × f is ´etale. We claim that the finite ´etale morphism q12 is an isomorphism. In fact, let q2 : ΓY → Y be the composition of q12 and the projection p2 : Y × Y → Y to the second factor. We have a section of the morphism q2 induced by the identification eY × Y

/Y ×Y ∼

%

Y

2

p2

and the section s1 (eY , y) = (eY , y, y) of q12 over eY × Y (Note that s1 is a section that maps into Γ0Y and the image s1 (eY × Y ) ∩ ΓY 6= ∅. Thus it is contained in ΓY ). It follows from the Lemma 2.3 that any fiber of the morphism q2 is irreducible. We define −1 Z := q2−1 (eY ) = q12 (Y ×eY ) the irreducible fiber of q2 over the origin eY . Then it induces the pullback r : Z → Y × eY of the morphism q12 , which is an finite etale morphism with the same degree of q12 . On the other hand, we have a section s2 (y, eY ) = (y, eY , y) of the morphism r. It follows that r is an isomorphism and we conclude that q12 is an isomorphism too. Definition 2.4. (Grothendieck) Let X be a scheme. Fix an algebraically closed field Ω and a geometric point x ¯. We define a functor Fx¯ : FEt/X → Sets from the category of finite ´etale morphisms over X to the category of sets by giving Fx¯ (f : Y → X) = {y ∈ Y (Ω)|f (y) = x ¯}. In particular, assume that X is a connected locally noetherian scheme with a geometric point x ¯. Then the ´etale fundamental group π1´et (X, x ¯) is defined to be the automorphism group of the functor Fx¯ . Theorem 2.5. (Grothendieck) Assume that X is a connected locally noetherian scheme with a geometric point x ¯. Then π1 := π1´et (X, x ¯) is a pro-finite group, and the functor Fx¯ induces an equivalence of categories eq FEt/X −→ finite π1 − sets Corollary 2.6. Let A be an abelian variety over the field k with the origin e ∈ A(k), and let ks be a separable closure of the field k. Regard e as a geometric point, i.e., there is an algebraically closed field Ω including k. Then we have the canonical isomorphisms Y T` A if char(k) = 0, ` Y π1´et (Aks , e) ' lim ←−A[n](ks ) ' T` A if char(k) = p > 0, n Tp,´et A × `6=p

where the projective limit run over all maps A[nm](ks ) → A[n](ks ) given by multiplication by m, and where ` runs over the prime numbers. Further, there exists a canonical isomorphism π1´et (A, e) ' π1´et (Aks , e) o Gal(ks /k), where the Galois group acts on π1´et (Aks , e) through the action on the projective system {A[n](ks )}n∈Z Proof. For simplicity, we denote by π = π1´et (X, x ¯) the ´etale fundamental group for any scheme X with a geometric point x ¯. By the Theorem 2.5, we have π1´et (Aks , e) = lim ←−(π/H)

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for all open subgroups, i.e., closed subgroups with finite index. It follows from the equivalence of categories in the Theorem 2.5that each open subgroup H associates to a ´etale covering fH : YH → X. By the Theorem of Lang-Serre, the k-variety YH is an abelian variety and fH is a separable isogeny. Therefore, by [EvdGM, Proposition 5.6], it follows that the kernel Ker(fH ) is a ´etale k-group scheme and ∼

YH / Ker(fH ) − → X. In particular, a separable isogeny is a Galois covering [EvdGM, Galois Covering 10.33]. Denote by I the set of isomorphism classes of separable isogenies f : Y → X over X. Two isogenies f : Y → X and f 0 : Y 0 → X are isomorphic if there an isomorphism of abelian varieties α : Y → Y 0 such that f 0 ◦ α = f . We give a partial order on I by dominance. We say f dominants f 0 , denote by f ≥ f 0 if there exists a homomorphism of abelian varieties h : Y → Y 0 such that f 0 ◦ h = f . In particular, the induced homomorphism of group schemes Ker(f ) Ker(f 0 ) gives a projective system {Ker(f )(ks )}f ∈I . It follows that π ' lim ←−{Ker(f )(ks )}. f ∈I

If n is a positive integer then [n]X : X → X factors as f

g

X− → X/X[n]loc → − X where f is purely inseparable and g is separable. Here, the local group scheme X[n]loc is the identity component of kernel X[n], which fits into a short exact sequence of group schemes 1 → X[n]loc → X[n] → X[n]et → 1 see [EvdGM, Proposition 4.45]. If char(k) = 0 or char(k) = p > 0 and p - n then X[n]loc = {id} and [n]X is separable. For the rest of the proof, we write g = [n]sep . Let I 0 ⊂ I be the subsets of isogenies [n]sep for n ∈ Z≥1 . Then I 0 is cofinal in I, in fact, if f : Y → X is a separable isogeny of degree d, then there exist an isogeny g : X → Y such that f ◦ g = [d]X . It follows from [EvdGM, Corollary 5.8] that [d]sep dominates f . Therefore, we have π ' lim lim ←− {Ker(f )(ks )} = ← −X[n](ks ). f ∈I 0

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3 Application In this section, we show a fundamental relation between the `-adic cohomology of an abelian variety A and its Tate module T` A. Proposition 3.1. Let A be an abelian variety over a field k, and let k ⊂ ks be a separable algebraic closure. Assume that ` is a prime number with ` 6= char(k). Then we have H 1 (Aks , Z` ) ' Hom(T` A, Z` )

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as Z` -modules with continuous action of Gal(ks /k). Proof. In general, if X is a complete variety over the k with a geometric point x ¯, then there is an isomorphism H 1 (Xks , Z` ) ∼ ¯), Z` ) = Homcont (π1´et (Xks , x where Homcont (, ) means the continuous group homormorphisms. For the details, see Milne’s online notes [Mil80, Example 11.4]. Then the homomorphism Gal(ks /k) → Out(π1´et (Xks , x ¯)) induces a homomorphism Gal(ks /k) → Aut(π1´et (Xks , x ¯)ab ). It gives a continuous Galois action on Homcont (π1´et (Xks , x ¯)ab , Z` ). In our case, it follows that Y H 1 (Aks , Z` ) ∼ = Homcont (π1´et (Aks , e)ab , Z` ) = Hom( T` A, Z` ) = Hom(T` A, Z` ) `

since the ´etale fundamental group π1´et (Aks , e) = phism Z`0 → Z` is trivial if `0 6= `.

Q

` T` A

is abelian and a group homor-

References [EvdGM] Bas Edixhoven, Gerard van der Geer, and Ben Moonen. Abelian varieties. Book project, available on Ben Moonen’s home page. [Mil80]

´ James S. Milne. Etale cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980.

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