The Fundamental Group

We compute the étale fundamental group of an abelian variety in characteristic zero.

Our exposition very loosely follows Anna Cadoret.

Theorem 1. Fix an abelian variety \(A\) over an algebraically closed field of characteristic zero. Then \[\pi_1^{\mathrm{\acute et}}(A;0)=\lim_n A[n].\]

We will not review the entire theory of the étale fundamental group, but we will recall its basic properties. The general theory is rather abstract and bland, so we will content ourselves with simply stating what we need. For a quick exposition, see (for example) here.

Definition 2. Fix a scheme \(X\), and choose a geometric point \(\overline x\hookrightarrow X\). Let \(\mathrm{F\acute Et}(X)\) denote the category of finite étale covers of \(X\). Then the étale fundamental group \(\pi_1^{\mathrm{\acute et}}(X;\overline x)\) is the automorphism group of the fiber functor \[(-)_{\overline x}\colon\mathrm{F\acute Et}(X)\to\mathrm{FinSet}\] given by sending a finite étale cover \(Y\to X\) to the fiber \(Y_{\overline x}\to\overline x\).

Remark 3. It turns out that the functor \((-)_{\overline x}\) upgrades into an equivalence of categories \[(-)_{\overline x}\colon\mathrm{F\acute Et}(X)\to\mathrm{FinSet}\left(\pi_1^{\mathrm{\acute et}}(X;\overline x)\right).\]

Example 4. Consider \(X=\operatorname{Spec}k\), where \(k\) is some field. Then the category of finite étale covers of \(X\) is the opposite category of the category of finite-dimensional separable \(k\)-algebras. By taking embeddings into a fixed algebraic closure (which corresponds to a chosen geometric point of \(X\)), one can check that \(\pi_1^{\mathrm{\acute et}}(X;\overline x)=\operatorname{Gal}(k^{\mathrm{sep}}/k)\).

This is a great definition, but it makes it very unclear how one might compute anything. To this end, it is enough to compute the automorphisms of \(F\) on a subcategory of \(\mathrm{F\acute Et}(X)\). In particular, it turns out to be enough to work with a cofinal sequence of Galois objects.

Definition 5. A finite étale cover \(Y\to X\) is Galois if and only if its automorphism group equals its degree.

Example 6. A Galois extension of fields produces a Galois cover.

Example 7. Let \(A\) be an abelian variety over a field \(k\). If an integer \(n\) is coprime to the characteristic of \(n\), then the multiplication-by-\(n\) map \([n]\colon A\to A\) is finite and smooth and thus a finite étale cover. It turns out to have degree \(n^{\dim A}\) with automorphism group isomorphic to its kernel (acting by translation), which is \[A[n]\cong(\mathbb Z/n\mathbb Z)^{2\dim A}.\] Thus, \([n]\colon A\to A\) is a Galois cover with automorphism group \(A[n]\).

Remark 8. Let \(\mathcal C\) be a cofinal subcategory of \(\mathrm{F\acute Et}(X)\) of Galois objects, where cofinal means that any object in \(\mathrm{F\acute Et}(X)\) admits a morphism from an object in \(\mathcal C\). It follows from the general theory that the natural map \[\pi_1^{\mathrm{\acute et}}(X;\overline x)\to\lim_{Y\in\mathcal C}\operatorname{Aut}_XY\] is an isomorphism.

The main input to Theorem 1 will be a classification of finite étale covers of abelian varieties.

Lemma 9. Let \(A\) be an abelian variety over an algebraically closed field \(k\). Then \(\pi_1^{\mathrm{\acute et}}(A;0)\) is abelian.

Proof. By a variant of the Eckmann–Hilton argument (see this blog post), it is enough to show that \(\pi_1^{\mathrm{\acute et}}(A;0)\) is a monoid object (with identity). (This implies that \(\pi_1^{\mathrm{\acute et}}(A;0)\) is a monoid object in the category of groups, which forces the group to be abelian.) Well, functoriality provides both a natural identity via \[\pi_1^{\mathrm{\acute et}}(\operatorname{Spec}k)\stackrel0\to\pi_1^{\mathrm{\acute et}}(A;0)\] and multiplication via \[\pi_1^{\mathrm{\acute et}}(A)\times\pi_1^{\mathrm{\acute et}}(A)\stackrel{(i_1,i_2)}\to\pi_1^{\mathrm{\acute et}}(A\times A)\stackrel m\to\pi_1^{\mathrm{\acute et}}(A).\] (All basepoints are the zero point.) All these maps are group homomorphisms, and functoriality can verify that the identity map is in fact an identity using the group law on \(A\). The claim follows. \(\blacksquare\)

Proposition 10. Let \(A\) be an abelian variety over an algebraically closed field. For any connected finite étale cover \(Y\to A\) and point \(\overline y\in Y_0\), the scheme \(Y\) can be given the structure of an abelian variety for which \(Y\to A\) is an isogeny.

Proof. The main point is to show that \(Y\) is an abelian variety. Note that \(Y\) is connected, \(Y\) is reduced because \(A\) is reduced, and \(Y\) is proper because \(Y\to A\) is finite. Thus, it is enough to show that \(Y\) is a group scheme and that \(Y\to A\) is a morphism of group schemes.

By the equivalence of categories in Remark 3, it is enough to exhibit a group object structure on \(Y_0\). Indeed, once this is achieved, we also automatially know that \(Y_0\to A_0\) is a morphism of group objects (because the target is the trivial group), so Remark 3 informs us that we have also constructed a morphism of group objects in the category \(\mathrm{F\acute Et}(A)\).

Now, to put a group object structure on \(Y_0\), we note that \(Y\) is a connected object, so \(\pi_1^{\mathrm{\acute et}}(A;0)\) acts transitively on \(Y_0\). Thus, there is a (closed) finite-index subgroup \(H\subseteq\pi_1^{\mathrm{\acute et}}(A;0)\) (equal to the stabilizer of \(\overline y\)) for which there is an isomorphism \[\pi_1^{\mathrm{\acute et}}(A;0)/H\to Y_0\] of \(\pi_1^{\mathrm{\acute et}}(A;0)\)-sets given by \(g\mapsto g\overline y\). Because \(\pi_1^{\mathrm{\acute et}}(A;0)\) is an abelian group (by Lemma 9), \(H\) is normal, so the left-hand side can be given a natural group quotient group structure. Thus, \(Y_0\) is equipped with a natural quotient group structure as well (which is notably \(\pi_1^{\mathrm{\acute et}}(A;0)\)-equivariant), and we are done! \(\blacksquare\)

We are now ready to attack Theorem 1 directly, via Remark 8.

Corollary 11. Let \(A\) be an abelian variety over an algebraically closed field of characteristic zero. The finite étale covers \([n]\colon A\to A\) form a cofinal sequence.

Proof. For any finite étale cover \(\varphi\colon Y\to A\), Proposition 10 informs us that \(Y\) can automatically be upgraded into an abelian variety, and \(\varphi\colon Y\to A\) can automatically be upgraded into an isogeny.

Now, say that \(\varphi\) has degree \(n\). Then \(\ker\varphi\) is annihilated by \([n]\), so because \(\varphi\colon Y\to A\) can be viewed as an fppf quotient (by \(\ker\varphi\subseteq Y\) acting by translation), it follows that \([n]\colon A\to A\) factors through \(\varphi\). In other words, there is an isogney \(\psi\colon A\to Y\) so that the composite \[A\to Y\to A\] is \([n]\). It follows that \(Y\to A\) is covered by \([n]\colon A\to A\). \(\blacksquare\)

Proof of Theorem 1. By Remark 8 and Corollary 11, we see that \[\pi_1^{\mathrm{\acute et}}(A;0)=\lim_n\operatorname{Aut}([n]\colon A\to A).\] This latter automorphism group has been computed to be \(A[n]\), acting by translation, in Example 7. Before calling it quits, we remark that the transition maps are as follows: \([nm]\colon A\to A\) covers \([n]\colon A\to A\) via the map \([m]\colon A\to A\); on the Galois groups, this corresponds to the covering \([m]\colon A[mn]\to A[n]\) by examining the “restriction” of the translation action of \(A[mn]\) on \(A\) along the cover \([m]\colon A\to A\). \(\blacksquare\)

Remark 12. Let’s explain how to relax the hypothesis on \(\operatorname{char}k\). The same argument shows that the \(\ell\)-primary part of \(\pi_1^{\mathrm{\acute et}}(A;0)\) is isomorphic to \[T_\ell A=\lim A[\ell^\bullet]\] as long as \(\ell\) is coprime to \(\operatorname{char}k\). This equality in fact turns out to be true when \(\ell=\operatorname{char}k\), but it is more subtle to prove because \([\ell]\colon A\to A\) is no longer smooth: instead, one needs to factor out the Frobenius enough times to produce the needed cofinal sequence of finite étale covers.