[-] Contents
Endomorphism rings and heights
Quaternion algebras
Supersingular elliptic curves and maximal orders
Abelian varieties and Tate conjecture
Finiteness theorem and Tate's isogeny theorem

In his classical paper [1], Deuring proved the following refined result concerning the endomorphism rings of elliptic curves (and much more).

Theorem 1 Let $E/K$ be an elliptic curve with $p=\Char(K)>0$.
  1. If the endomorphism algebra $\End(E)\otimes\mathbb{Q}$ is an imaginary quadratic field $F$, then $p$ splits in $F$ and $\End(E)$ is an order in $F$ of conductor prime to $p$. Conversely, all such orders occur as endomorphism rings.
  2. If $E$ is supersingular, then $\End(E)$ is a maximal order in the unique rational quaternion algebra ramified only at $\infty$ and $p$. Conversely, all such orders occur as endomorphism rings.

In this chapter, we will prove part of Deuring's theorem. Namely, we will prove that the endomorphism ring of a supersingular elliptic curve over a finite field is an maximal order in the rational quaternion algebra ramified only at $\infty$ and $p$, using an approach different from Deuring's original proof. Our argument of the maximality comprises two major parts. On the $\ell$-part, $\End(E)\otimes \mathbb{Z}_{\ell}$ can be identified with the endomorphisms of the $\ell$-adic Tate module. On the $p$-part, $\End(E)\otimes \mathbb{Z}_p$ can be identified with the endomorphisms of the formal group $\hat E$.

We will begin with giving a criterion for endomorphism rings in terms of the heights of the formal groups. Next, we will discuss several interesting properties of quaternion algebras. By utilizing the nontrivial results of Tate and Dieudonné-Lubin, we will be able to achieve our goal mentioned above. Finally, two sections will be devoted to a short discussion of abelian varieties and Tate's isogeny theorem.

TopEndomorphism rings and heights

As promised in the first chapter, we will first connect the supersingularity with the $p$-torsion part of an elliptic curve. We mainly follow [2] in this section.

Theorem 2 Let $E$ be an elliptic curve over $K$ with $p=\Char(K)>0$. Denote the $p^r$-th Frobenius map by $\phi_r$. Then the following are equivalent:
  1. $E$ is supersingular.
  2. $\hat\phi_r$ is purely inseparable for one (all) $r\ge1$.
  3. $[p]$ is purely inseparable and $j(E)\in \mathbb{F}_{p^2}$.
  4. $E[p^r]=0$ for one (all) $r\ge1$.
Proof (b)$\Leftrightarrow$(d) Since $p^r$-Frobenius maps are purely inseparable ([2, II 2.11]), we know that $$\# E[p^r]=\# \ker [p^r]=\deg_s[p^r]=\deg_s \hat\phi_r, \quad r\ge1.$$ Hence the equivalence of (b) and (d) follows.

(a)$\Rightarrow$(b) Suppose $\hat\phi_r$ is separable for all $r\ge1$, then $\# E[p^r]=p^r$. Consider the natural map $T_p:\End(E)\rightarrow \End(T_p(E))$, then $T_p$ is injective in this case. That is because if $T_p(\psi)=0$ for some $\psi\in\End(E)$, then $\psi(E[p^r])=0$, hence $\#\ker\psi\ge p^r$ for all $r\ge1$, which implies $\psi=0$ since nonconstant isogeny has finite kernel. Now $\End(T_p(E))=\mathbb{Z}_p$ is commutative and the map $T_p$ is injective, so $\End(E)$ must be commutative.

(b)$\Rightarrow$(c) Let $\phi=\phi_1$ be the $p$-Frobenius map. Since $[p]=\hat\phi\circ\phi$ and $\phi$ is purely inseparable, if $\hat\phi$ is purely inseparable, then so is $[p]$.

Because $\hat\phi: E^{(p)}\rightarrow E$ is purely inseparable, it factors through the $p$-Frobenius map $\phi'$ on $E^{(p)}$, namely $\hat\phi=\psi\circ\phi'$ where $\psi:E^{(p^2)}\rightarrow E$ is an isomorphism. Therefore $j(E)^{p^2}=j(E)$, which means $j(E)\in\mathbb{F}_{p^2}$.

(c)$\Rightarrow$(a) Under condition (c), we claim that there are only finitely many elliptic curves that are isogenous to $E$. Suppose $\psi:E\rightarrow E'$ is an isogeny. Since the isogeny $[p]$ on $E$ is purely inseparable and $[p]\circ\psi=\psi\circ[p]$, we know that the isogeny $[p]$ on $E'$ is also purely inseparable. So $j(E')\in \mathbb{F}_{p^2}$. There are only finitely many $j$-invariants for $E'$, hence the claim follows.

Now suppose $E$ is not supersingular, then $\End(E)\otimes\mathbb{Q}$ is either $\mathbb{Q}$ or an imaginary quadratic extension of $\mathbb{Q}$. Since there only finitely many $E'$ that is isogenous to $E$, we can find a prime $\ell\ne p$ such that $\ell$ remains a prime in each $\End(E')$. Since $E[\ell^n]=\mathbb{Z}/\ell^n\mathbb{Z}\times\mathbb{Z}/\ell^n\mathbb{Z}$ by Theorem 2, we can find a sequence of cyclic groups $\Phi_1\subseteq\Phi_2\cdots\subseteq E$ satisfying $\Phi_n\cong\mathbb{Z}/\ell^n\mathbb{Z}$. Using [2, III 4.12], we get a separable isogeny $E\rightarrow E/\Phi_n$ with kernel $\Phi_n$. By the claim above again, there exist some positive integers $n,m$ such that $E/\Phi_{n+m}\cong E/\Phi_{m}$. Now the natural map $\lambda: E/\Phi_{m}\rightarrow E/\Phi_{n+m}\cong E/\Phi_{m}$ has degree $\ell^n$ and has kernel $\mathbb{Z}/\ell^n\mathbb{Z}$. Since $\ell$ is a prime in $\End(E/\Phi_{m})$, comparing degrees we obtain $\lambda=\psi\circ[\ell^{n/2}]$ where $\psi$ is an isomorphism. But the kernel of $[\ell^{n/2}]$ is not cyclic, a contradiction. ¡õ

Corollary 1 Up to isomorphism, there are only finitely many supersingular elliptic curves over $K$.
Proof By Theorem 2, $j(E)\in\mathbb{F}_{p^2}$. So there are only finitely many such $j$-invariants. ¡õ

The following theorem connects the inseparable degree of an isogeny with the height of the associated map of formal groups in positive characteristic. Combining it with Theorem 2, we will be able to give a characterization of endomorphism rings in terms of the height of the formal group (Theorem 4).

Theorem 3 Let $E_1, E_2$ be two elliptic curves over $K$ with $p=\Char(K)>0$. Let $\phi:E_1\rightarrow E_2$ be an isogeny and $f:\hat E_1\rightarrow\hat E_2$ be the associated homomorphism of formal groups. Then the inseparable degree $\deg_i(\phi)=p^{\Ht(f)}$, where $\Ht(f)$ is the height of the homomorphism $f$.
Proof Every isogeny can be decomposed as a Frobenius map and a separable map. Since the inseparable degrees multiply and the heights add under composition, it suffices to show for the theorem of Frobenius maps and separable maps.

Suppose $\phi$ is the $p^r$-Frobenius map, then $\deg_i(\phi)=p^r$. On the other hand, the associated homomorphism $f(T)=T^{p^r}$, so $\Ht(f)=r$.

Suppose $\phi$ is a separable map, then $\deg_i(\phi)=1$ and $\phi^*\omega\ne0$, where $\omega$ is the invariant differential. So as a differential on formal group $\hat E_2$, we have $(\omega\circ f)(T)\ne0$. But $(\omega\circ f)(T)=f'(0)\omega(T)$ ([2, IV 4.3]), it follows that $f'(0)\ne0$, therefore $\Ht(f)=0$. ¡õ

Corollary 2 Let $E$ be an elliptic curve over $K$ with $p=\Char(K)>0$. Then $\Ht(\hat E)$ is either 1 or 2.
Proof By definition, $\Ht(\hat E)=\Ht([p])$. Since $[p]$ has degree $p^2$, $\deg_i([p])$ is either 1, $p$ or $p^2$, therefore $\Ht([p])$ is either 0, 1, or 2. But $[p]'(T)\omega(T)=p\omega(T)=0$ and $\omega(T)=(1+\cdots)dT$, where $\omega(T)$ is the normalized invariant differential on $\hat E$, hence we conclude that $[p]'(T)=0$. It follows that $\Ht([p])\ge1$, so the possible height is either 1 or 2. ¡õ
Theorem 4 Let $E$ be an elliptic curve over $K$ with $p=\Char(K)>0$. Then
  1. $E$ is supersingular if and only if $\Ht(\hat E)=2$.
  2. $\End(E)=\mathbb{Z}$ if and only if $\Ht(\hat E)=1$ and $j(E)$ is transcendental over $\mathbb{F}_p$.
  3. $\End(E)$ is an order in an imaginary quadratic extension of $\mathbb{Q}$ if and only if $\Ht(\hat E)=1$ and $j(E)\in\overline{\mathbb{F}_p}$.

In particular, $\End(E)\ne\mathbb{Z}$ when $E$ is defined over a finite field.

Proof For the first part, by Theorem 2, $E$ is supersingular if and only if $[p]$ is purely inseparable, if and only if $\deg_i[p]=p^2$. By Theorem 3, the last statement is true if and only if $\Ht(\hat E)=2$.

For the remaining two parts, by Corollary 2, we know that $\Ht(\hat E)=1$. Let us show that if $\End(E)=\mathbb{Z}$, then $j(E)$ is transcendental over $\mathbb{F}_p$. Otherwise, suppose $j(E)\in \overline{\mathbb{F}_p}$. We may assume $E$ is defined over some finite field $\mathbb{F}_{p^r}$. Thus the Frobenius map $\phi_r\in\End(E)=\mathbb{Z}$, and comparing degrees we obtain that $\phi_r=[\pm p^{r/2}]$. Hence $[p]$ is purely inseparable, a contradiction to $\Ht(\hat E)=1$ by Theorem 3.

Conversely, if $j(E)$ is transcendental, then $j(E)$ is a generic elliptic curve and every elliptic curve over $K$ can be realized by specializing the value of $j(E)$. By Theorem 1, every imaginary quadratic field in which $p$ splits can occur as an endomorphism algebra and these endomorphism algebras have intersection $\mathbb{Q}$. Thus $\End(E)=\mathbb{Z}$. ¡õ

TopQuaternion algebras

We will do a quick summary of the basic properties of quaternion algebras in this section. See [3] for more proofs.

Definition 1 A four-dimensional central simple $K$-algebra ${\cal K}$ is called a quaternion algebra over $K$.
Remark 1 When $\Char(K)\ne 2$, ${\cal K}$ can be written as an algebra with basis $\{1,\alpha,\beta,\alpha\beta\}$ satisfying $\alpha^2=a\in K, \beta^2=b\in K$, and $\alpha\beta=-\beta\alpha$. We denote it by ${\cal K}=(a,b)_K$. Therefore this more general Definition 1 is compatible with the earlier Definition 5.

Let $K$ be a global field, $v$ be a place of $K$, then there are only two isomorphism classes of quaternion algebras over the local field $K_v$. One is the split case, i.e., the matrix algebra $M_2(K_v)$. Another is the ramified case, i.e., the unique division algebra $D_v$ of dimension 4.

The quaternion algebra ${\cal K}=(a,b)_K$ is ramified if and only if $ax^2+by^2=1$ has no solution $(x,y)$ in $K$. By the Hasse-Minkowski Principle, the latter can be determined locally. Moreover, when $K=\mathbb{Q}$, we can use Hilbert symbol $(a,b)_v$ as a calculation tool to determine whether $ax^2+by^2=1$ has a solution in $\mathbb{Q}_v$. Then the Hilbert reciprocity law $\prod_v (a,b)_v=1$ ensures that the number of places where ${\cal K}$ ramifies must be even.

Two quaternion algebras are isomorphic if and only if they are isomorphic locally everywhere. Hence the places where ${\cal K}$ ramifies uniquely determine ${\cal K}$. Moreover, an order $\mathcal{R}$ in ${\cal K}$ is maximal if and only if $\mathcal{R}$ is maximal for all finite places.

Theorem 5 Let $E/K$ be a supersingular elliptic curve with $p=\Char(K)>0$. Then $\End(E)$ is an order in the unique rational quaternion algebra ramified only at $\infty$ and $p$.
Proof Let ${\cal K}=\End(E)\otimes\mathbb{Q}$. By Definition 5 and Theorem 8, we know that ${\cal K}=(a,b)_\mathbb{Q}$ for some negative numbers $a$ and $b$. Hence $ax^2+by^2=1$ has no solution in $\mathbb{R}=\mathbb{Q}_\infty$, which implies that ${\cal K}$ is ramified at $\infty$. For a prime $\ell\ne p$, we have an injection ${\cal K}\otimes\mathbb{Q}_\ell$ into $M_2(\mathbb{Q}_\ell)$ by Theorem 5 and Theorem 1. So ${\cal K}\otimes\mathbb{Q}_\ell$ is isomorphic to $M_2(\mathbb{Q}_\ell)$, namely ${\cal K}$ is split at all finite places away from $p$. But the total number of ramified places is even, hence we conclude that ${\cal K}$ must ramify at $p$. ¡õ
Remark 2 Let ${\cal K}$ be a quaternion algebra over $\mathbb{Q}$. ${\cal K}$ is called definite if ${\cal K}$ is ramified at $v=\infty$, otherwise ${\cal K}$ is called indefinite. Therefore Theorem 6 justifies this terminology in Definition 5.

TopSupersingular elliptic curves and maximal orders

Now we hope to show that the endomorphism ring of a supersingular elliptic curve over a finite field is actually a maximal order. It suffices to prove the maximality for all primes, that is, that $\End(E)\otimes\mathbb{Z}_\ell$ is a maximal order in ${\cal K}\otimes\mathbb{Q}_\ell$ for all primes $\ell$.

This could be hard. But it will be fairly easy if we quote the following two theorems. The first theorem is a special case of Tate's isogeny theorem about abelian varieties (see the next two sections). The second theorem is due to Dieudonné-Lubin ([4, Page 72]).

Theorem 6 Let $E$ be an elliptic curve over a finite field $K$, $p=\Char(K)$, $G=\Gal(\overline{K}/K)$. Then for $\ell\ne p$, there are isomorphisms $$\End_K(E)\otimes\mathbb{Z}_\ell\cong \End_G(T_\ell(E)),$$ where the subscript $G$ denotes the endomorphisms which commute with the $G$-action.

If $E$ is in addition supersingular, then $$\End_K(E)\otimes\mathbb{Z}_p\cong \End_K(\hat E).$$

Theorem 7 Let ${\cal F}$ be a formal group of height $h$ over a separably closed field $K$, $p=\Char(K)$. Then $\End({\cal F})$ is a free $\mathbb{Z}_p$-module of rank $h^2$ and is the maximal order in the local division algebra $\End({\cal F})\otimes\mathbb{Q}_p$.

Using these two strong and useful results, we are now ready to prove the maximality of $\End(E)$.

Theorem 8 Let $E$ be a supersingular elliptic curve over a finite field with $p=\Char(K)$. Then $\End(E)$ is a maximal order in the rational quaternion algebra ramified only at $\infty$ and $p$.
Proof By Theorem 6, it suffices to show the maximality. Consider $\ell\ne p$. Since $j(E)\in \mathbb{F}_p$ by Theorem 2, $E$ can be actually defined over $\mathbb{F}_{p^2}$. So $G$ is generated by the $p^2$-th Frobenius map $\phi_2$. Since $\deg\phi_2=p^2$, we know that $\psi\circ[p]=\phi_2$ where $\psi$ is an isomorphism ([2, II 2.12]). But the automorphism group of an elliptic curve has finite order ([2, III 10.1]), hence after passing to a finite extension $L$ of $K$, $G'{}=\Gal(\overline{K}/L)$ acts as scalars on $T_\ell(E)$. So $\End_{G'}(T_\ell E)\cong M_2(\mathbb{Z}_\ell)$. By Theorem 6, we know $\End_L(E)\otimes \mathbb{Z}_{\ell} \cong M_2(\mathbb{Z}_{\ell})$ which is a maximal order in $M_2(\mathbb{Q}_\ell)$. But every endomorphism of $E$ is defined over some finite field, so $\End(E)\otimes \mathbb{Z}_{\ell}\cong M_2(\mathbb{Z}_\ell)$ is a maximal order in $\End(E)\otimes\mathbb{Q}_\ell$.

Since $\overline{K}$ is separably closed, using Theorem 7 we know that $\End(E)\otimes \mathbb{Z}_p\cong\End(\hat E)$ is isomorphic to the maximal order in the local quaternion algebra. The theorem follows. ¡õ

TopAbelian varieties and Tate conjecture

Abelian varieties can be viewed as a higher dimensional generalization of elliptic curves. In this section we introduce some basic notions about abelian varieties and state the Tate conjecture and list the results about it. The main references for this and the next sections are [5] and [6].

An abelian variety over $K$ is a complete connected group variety over $K$, or equivalently, a projective connected group variety over $K$ ([5, I 6.4]). As we expected, its group structure is automatically commutative ([5, I 1.4]).

Analogously to elliptic curves, an isogeny between two abelian varieties is an homomorphism of abelian varieties which is surjective and has finite kernel. An abelian variety is called simple if it does not contain any nontrivial abelian variety. Every abelian variety is isogenous to a product of simple abelian varieties ([5, I 10.1]).

The multiplication-by-$m$ map is an isogeny of degree $m^{2g}$, where $g=\dim A$. Moreover, it is an étale map if $p\nmid m$, where $p=\Char(K)$ ([5, I 7.2]). Similarly, for a prime $\ell\ne p$, the $\ell$-adic Tate module $T_\ell(A)$ of an abelian variety $A$ over $K$ is the inverse limit of the $\ell^n$-torsion part (over $\overline{K}$) of $A$; it is a $\mathbb{Z}_{\ell}$-module along with a $G=\Gal(\overline{K}/K)$-action. So $T_\ell(A)$ is a free $\mathbb{Z}_{\ell}$-module of rank $2g$ ([6, 19]). For any two abelian varieties $A$ and $B$, we have a natural homomorphism of $\mathbb{Z}_{\ell}$-modules \begin{equation*} T_\ell: \Hom(A, B)\otimes \mathbb{Z}_{\ell} \rightarrow \Hom(T_\ell(A), T_\ell(B)). \tag{1} \end{equation*} The injectivity of Theorem 5 also holds for abelian varieties using similar argument ([5, I 10.5]).

Theorem 9 The natural map (1) is injective. Moreover the cokernel is torsion-free, hence $\Hom(A,B)$ is a free $\mathbb{Z}$-module of rank at most $4\dim A\cdot\dim B$.

The next corollary follows easily from Theorem 9.

Corollary 3 Let $A$ be an abelian variety. Then $\End(A)\otimes\mathbb{Q}$ is a semisimple $\mathbb{Q}$-algebra.
Proof Suppose $B$ is a simple abelian variety, then every endomorphism $\phi\in\End(B)$ is automatically an isogeny, hence $\End(B)\otimes\mathbb{Q}$ is a division algebra $D$. Therefore the endomorphism algebra of any abelian variety $A$ is a product of matrix algebras $M_{n_i}(D_i)$. Now, by Theorem 9, the $D_i$'s are finite dimensional, hence $\End(A)\otimes\mathbb{Q}$ is semisimple by the Artin-Wedderburn theorem. ¡õ

Taking $G$-invariants on both sides of (1) we get an injective homomorphism \begin{equation*} T_\ell: \Hom_K(A, B)\otimes \mathbb{Z}_{\ell} \rightarrow \Hom_G(T_\ell(A), T_\ell(B)). \tag{2} \end{equation*} Tate [7] conjectured that in many cases it is also an isomorphism.

Conjecture 1 (Tate) The natural map (2) is an isomorphism if $K$ is finitely generated over its prime field (e.g. finite fields, number fields or global function fields).

This conjecture for finite fields was proved by Tate himself [8], now called Tate's isogeny theorem. Zarhin [9] proved the Tate conjecture for function fields of positive characteristic. The number field case was proved by Faltings [10] as one of the steps in proving the Mordell conjecture. Faltings [11] proved the case for function fields of characteristic zero.

Remark 3 These deep results give an amazing connection between the endomorphism ring of an abelian variety over certain fields (a geometric object) to the endomorphism ring of the related Tate module (an algebraic object). As an application of Tate's isogeny theorem, Tate [8] characterized the endomorphism algebra of an abelian variety over a finite field. Using Tate's isogeny theorem, Tate and Honda were able to develop a full classification of isogeny classes of abelian varieties over finite fields — they are in a bijection with the conjugacy classes of Weil numbers. See [12], [13] and [14] for more.

There is also a $p$-analog of the $\ell$-adic Tate module $T_\ell(A)$ called the Dieudonné module $T_p(A)$, constructed from the $p$-divisible group $A[p^{\infty}]$. Tate conjectured that

Conjecture 2 (Tate) The natural map $$\Hom_K(A,B)\otimes\mathbb{Z}_p\rightarrow \Hom_G(T_p(B), T_p(A))$$ is an isomorphism if $K$ is finitely generated over its prime field (e.g., finite fields, number fields or global function fields).

This conjecture was proved for finite fields by Tate himself (see [14]), and for function fields over finite fields by de Jong [15]. Under certain conditions, there is an equivalence of categories between $p$-divisible groups and formal groups [16]. In particular, when $A=E$ is a supersingular elliptic curve (hence without $p$-torsion), the $p$-divisible group can be identified with the formal group $\hat E$.

TopFiniteness theorem and Tate's isogeny theorem

In this section, we will discuss some basic properties of abelian varieties. We will sketch the main steps of the proof of Tate's isogeny theorem. Before doing that, we will first prove a finiteness theorem on abelian varieties over a finite field, which turns out to be one of the key steps in the proof.

Let $A$ be an abelian variety and $\mathcal{L}$ be a line bundle on $A$, then the theorem of square ([5, I 5.5]) ensures that $$\lambda_\mathcal{L}: A\rightarrow \Pic(A),\qquad x\mapsto t_x^*\mathcal{L}\otimes \mathcal{L}^{-1}$$ is a homomorphism where $t_x$ is the translation-by-$x$ isogeny. Let $\Pic^0(A)$ be the isomorphism classes of all line bundles ${\cal L}$ satisfying $\lambda_{\cal L}(x)=0$ for any $x\in A$. When $A$ is an elliptic curve and $D$ is a divisor on $A$, then $\deg D=0$ if and only if $\lambda_D=0$, so this definition is compatible with the earlier $\Pic^0(E)$. One can show $\lambda_{\cal L}$ actually maps onto $\Pic^0(A)$ ([6, 8]).

There is a pair consisting of a dual abelian variety and a Poincaré bundle $(A^\vee, {\cal P})$ which is the solution to the moduli problem of parametrizing $\Pic^0(A)$ ([6, 13]). A polarization of $A$ is an isogeny $\lambda: A\rightarrow A^\vee$ such that $\lambda_{\overline{K}}=\lambda_{\cal L}$ for some ample line bundle ${\cal L}$ on $A_{\overline{K}}$. If $\lambda$ has degree one, then $\lambda$ is called a principal polarization.

There are several interesting results about line bundles on abelian varieties.

Theorem 10 Let ${\cal L}$ be a line bundle on an abelian variety $A$. Denote its Euler characteristic by $\chi({\cal L})$. Then
  1. $\deg\lambda_{\cal L}=\chi({\cal L})^2$. If ${\cal L}={\cal O}(D)$, then $\chi({\cal L})=(D^g)/g!$ where $(D^g)$ is the $g$-fold self-intersection number of $D$ ([6, 16]).
  2. if ${\cal L}$ is ample, then ${\cal L}^n$ is very ample for every $n\ge3$ ([6, 17]).

Over a finite field, there are only finitely many isomorphism classes of elliptic curves since there are only finitely many $j$-invariants. This finiteness result can be extended to abelian varieties ([5, I 11.2]).

Theorem 11 Let $K$ be a finite field and $g, d$ be two fixed positive integers. Then up to isomorphism, there are only finitely many abelian varieties over $K$ of dimension $g$ possessing a polarization of degree $d^2$.
Proof By Theorem 10, an abelian variety possessing a polarization of degree $d^2$ can be realized as a subvariety of degree $3^gd(g!)$ in a $(3^gd-1)$ dimensional projective space, in other words, a rational point on the corresponding Chow variety. But $K$ is a finite field, so there are only finite many rational points on this Chow variety. The result follows. ¡õ
Remark 4 Over a number field, Faltings showed a finiteness theorem as a step in proving the Tate conjecture for the number fields case: Let $A$ be an abelian variety over a number field $K$, then up to isomorphism there are only finitely many abelian varieties over $K$ which are isogenous to $A$.

Let us sketch the major steps of the proof of Tate's isogeny theorem to end this chapter.

Theorem 12 Let $A$ be an abelian variety over a finite field $K$, $\ell\ne p=\Char(K)$ be a prime, $G=\Gal(\overline{K}/K)$. Then the natural map $$T_\ell: \Hom_K(A, B)\otimes \mathbb{Z}_{\ell} \rightarrow \Hom_G(T_\ell(A), T_\ell(B))$$ is an isomorphism.
Proof (Sketch of Proof) Step 1 Let $V_\ell(A)=T_\ell(A)\otimes\mathbb{Q}$. Reduce to showing that the natural map $$\End(A)\otimes\mathbb{Q}_\ell\rightarrow\End_G(V_\ell(A))$$ is an isomorphism. Since this is an injection and the dimension of $\End(A)\otimes\mathbb{Q}_\ell$ is independent of $\ell$, it suffices to show that it is an isomorphism for one $\ell$ and that all $\End_G(V_\ell(A))$'s have the same dimension.

Step 2 Let $F_\ell$ be the subalgebra of $\End(V_\ell(A))$ generated by the automorphisms of $V_\ell(A)$ defined by elements of $G$. To prove the isomorphism, it is equivalent to show that $\End_K(A)\otimes\mathbb{Q}_\ell$ is the commutant of $F_\ell$ in $\End(V_\ell(A))$. Since $\End_K(A)\otimes\mathbb{Q}_\ell$ is semisimple, it suffices to show that $F_\ell$ is the commutant of $\End_K(A)\otimes\mathbb{Q}_\ell$ in $\End(V_\ell(A))$ by the von Neumann bicommutant theorem.

Step 3 Let $\lambda$ be a polarization of $A$. Define a pairing $E^{\cal L}$ on $V_\ell(A)$ by setting $E^{\cal L}(x, y)=e(x, \lambda y)$ where $e$ is the pairing of $V_\ell(A)$ and $V_\ell(A^\vee)$ induced by the Weil pairing. Let $W$ be a subspace of $V_\ell(A)$ which is maximally isotropic with respect to the pairing $E^{\cal L}$ and stable under $G$-action. Use the Finiteness Theorem 11 to construct an endomorphism $u\in \End_G(V_\ell(A))$ such that $u(V_\ell(A))=W$.

Step 4 Let $F$ be the subalgebra of $\End_K(A)\otimes\mathbb{Q}$ generated by the Frobenius endomorphisms of $A$ over $K$. Show that for any $\ell$ which splits completely in $F$, we have $F_\ell\cong F\otimes\mathbb{Q_\ell}$ and use Step 3 to show that $F_\ell$ is the commutant of $\End_K(A)\otimes\mathbb{Q}_\ell$ in $\End(V_\ell(A))$.

Step 5 Use the semisimplicity of the Frobenius map on $V_\ell(A)$ to show that all $\End_G(V_\ell(A))$'s have the same dimension and complete the proof by Step 1. ¡õ

References

[1]Deuring, M., Die typen der multiplikatorenringe elliptischer funktionenkörper, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, 14 1941, 197--272.

[2]Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer, 2010.

[3]T. Y. Lam, Introduction To Quadratic Forms Over Fields, Amer Mathematical Society, 2004.

[4]Frohlich, A, Formal groups, Springer-Verlag, Berlin, New York, 1968.

[5]J. S. Milne, Abelian Varieties, 2008, http://www.jmilne.org/math/CourseNotes/AV.pdf.

[6]David Mumford, Abelian Varieties, Oxford University, 1970.

[7]John Tate, Arithmetical Algebraic Geometry, Algebraic Cycles and Poles of Zeta Functions, Harper and Row, 1965.

[8]John Tate, Endomorphisms of abelian varieties over finite fields, Inventiones mathematicae 2 (1966), no.2, 134--144.

[9]Zarhin, J. G., Endomorphisms of Abelian varieties over fields of finite characteristic, Mathematics of the USSR-Izvestiya 9 (1975), 255.

[10]Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Inventiones mathematicae 73 (1983), no.3, 349--366.

[11]Gerd Faltings and Gisbert Wuestholz, Rational Points, Friedrich Vieweg \& Sohn Verlag, 1992.

[12]Frans Oort, Abelian varieties over finite fields, Higher-Dimensional Geometry over Finite Fields, IOS Press, 2008.

[13]Waterhouse, WC, Abelian varieties over finite fields, Ann. Sci. Ecole Norm. Sup 4 (1969), no.2, 521--560.

[14]Waterhouse, WC and Milne, JS, Abelian varieties over finite fields, 1969 Number Theory Institute 4 (1971), 53 -- 64.

[15]de Jong, A.J., Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic, Inventiones Mathematicae 134 (1998), no.2, 301--333.

[16]John Tate, p-Divisible groups, Proc Conf. Local Fields, Driebergen, 1966, 158--183.

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