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

Theorem 1 Let be an elliptic curve with .
1. If the endomorphism algebra is an imaginary quadratic field , then splits in and is an order in of conductor prime to . Conversely, all such orders occur as endomorphism rings.
2. If is supersingular, then is a maximal order in the unique rational quaternion algebra ramified only at and . 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 and , using an approach different from Deuring's original proof. Our argument of the maximality comprises two major parts. On the -part, can be identified with the endomorphisms of the -adic Tate module. On the -part, can be identified with the endomorphisms of the formal group .

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.

## Endomorphism rings and heights

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

Theorem 2 Let be an elliptic curve over with . Denote the -th Frobenius map by . Then the following are equivalent:
1. is supersingular.
2. is purely inseparable for one (all) .
3. is purely inseparable and .
4. for one (all) .
Proof (b)(d) Since -Frobenius maps are purely inseparable ([2, II 2.11]), we know that Hence the equivalence of (b) and (d) follows.

(a)(b) Suppose is separable for all , then . Consider the natural map , then is injective in this case. That is because if for some , then , hence for all , which implies since nonconstant isogeny has finite kernel. Now is commutative and the map is injective, so must be commutative.

(b)(c) Let be the -Frobenius map. Since and is purely inseparable, if is purely inseparable, then so is .

Because is purely inseparable, it factors through the -Frobenius map on , namely where is an isomorphism. Therefore , which means .

(c)(a) Under condition (c), we claim that there are only finitely many elliptic curves that are isogenous to . Suppose is an isogeny. Since the isogeny on is purely inseparable and , we know that the isogeny on is also purely inseparable. So . There are only finitely many -invariants for , hence the claim follows.

Now suppose is not supersingular, then is either or an imaginary quadratic extension of . Since there only finitely many that is isogenous to , we can find a prime such that remains a prime in each . Since by Theorem 2, we can find a sequence of cyclic groups satisfying . Using [2, III 4.12], we get a separable isogeny with kernel . By the claim above again, there exist some positive integers such that . Now the natural map has degree and has kernel . Since is a prime in , comparing degrees we obtain where is an isomorphism. But the kernel of is not cyclic, a contradiction. ¡õ

Corollary 1 Up to isomorphism, there are only finitely many supersingular elliptic curves over .
Proof By Theorem 2, . So there are only finitely many such -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 be two elliptic curves over with . Let be an isogeny and be the associated homomorphism of formal groups. Then the inseparable degree , where is the height of the homomorphism .
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 is the -Frobenius map, then . On the other hand, the associated homomorphism , so .

Suppose is a separable map, then and , where is the invariant differential. So as a differential on formal group , we have . But ([2, IV 4.3]), it follows that , therefore . ¡õ

Corollary 2 Let be an elliptic curve over with . Then is either 1 or 2.
Proof By definition, . Since has degree , is either 1, or , therefore is either 0, 1, or 2. But and , where is the normalized invariant differential on , hence we conclude that . It follows that , so the possible height is either 1 or 2. ¡õ
Theorem 4 Let be an elliptic curve over with . Then
1. is supersingular if and only if .
2. if and only if and is transcendental over .
3. is an order in an imaginary quadratic extension of if and only if and .

In particular, when is defined over a finite field.

Proof For the first part, by Theorem 2, is supersingular if and only if is purely inseparable, if and only if . By Theorem 3, the last statement is true if and only if .

For the remaining two parts, by Corollary 2, we know that . Let us show that if , then is transcendental over . Otherwise, suppose . We may assume is defined over some finite field . Thus the Frobenius map , and comparing degrees we obtain that . Hence is purely inseparable, a contradiction to by Theorem 3.

Conversely, if is transcendental, then is a generic elliptic curve and every elliptic curve over can be realized by specializing the value of . By Theorem 1, every imaginary quadratic field in which splits can occur as an endomorphism algebra and these endomorphism algebras have intersection . Thus . ¡õ

## Quaternion 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 -algebra is called a quaternion algebra over .
Remark 1 When , can be written as an algebra with basis satisfying , and . We denote it by . Therefore this more general Definition 1 is compatible with the earlier Definition 5.

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

The quaternion algebra is ramified if and only if has no solution in . By the Hasse-Minkowski Principle, the latter can be determined locally. Moreover, when , we can use Hilbert symbol as a calculation tool to determine whether has a solution in . Then the Hilbert reciprocity law ensures that the number of places where ramifies must be even.

Two quaternion algebras are isomorphic if and only if they are isomorphic locally everywhere. Hence the places where ramifies uniquely determine . Moreover, an order in is maximal if and only if is maximal for all finite places.

Theorem 5 Let be a supersingular elliptic curve with . Then is an order in the unique rational quaternion algebra ramified only at and .
Proof Let . By Definition 5 and Theorem 8, we know that for some negative numbers and . Hence has no solution in , which implies that is ramified at . For a prime , we have an injection into by Theorem 5 and Theorem 1. So is isomorphic to , namely is split at all finite places away from . But the total number of ramified places is even, hence we conclude that must ramify at . ¡õ
Remark 2 Let be a quaternion algebra over . is called definite if is ramified at , otherwise is called indefinite. Therefore Theorem 6 justifies this terminology in Definition 5.

## Supersingular 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 is a maximal order in for all primes .

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 be an elliptic curve over a finite field , , . Then for , there are isomorphisms where the subscript denotes the endomorphisms which commute with the -action.

If is in addition supersingular, then

Theorem 7 Let be a formal group of height over a separably closed field , . Then is a free -module of rank and is the maximal order in the local division algebra .

Using these two strong and useful results, we are now ready to prove the maximality of .

Theorem 8 Let be a supersingular elliptic curve over a finite field with . Then is a maximal order in the rational quaternion algebra ramified only at and .
Proof By Theorem 6, it suffices to show the maximality. Consider . Since by Theorem 2, can be actually defined over . So is generated by the -th Frobenius map . Since , we know that where 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 of , acts as scalars on . So . By Theorem 6, we know which is a maximal order in . But every endomorphism of is defined over some finite field, so is a maximal order in .

Since is separably closed, using Theorem 7 we know that is isomorphic to the maximal order in the local quaternion algebra. The theorem follows. ¡õ

## Abelian 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 is a complete connected group variety over , or equivalently, a projective connected group variety over ([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- map is an isogeny of degree , where . Moreover, it is an étale map if , where ([5, I 7.2]). Similarly, for a prime , the -adic Tate module of an abelian variety over is the inverse limit of the -torsion part (over ) of ; it is a -module along with a -action. So is a free -module of rank ([6, 19]). For any two abelian varieties and , we have a natural homomorphism of -modules 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 is a free -module of rank at most .

The next corollary follows easily from Theorem 9.

Corollary 3 Let be an abelian variety. Then is a semisimple -algebra.
Proof Suppose is a simple abelian variety, then every endomorphism is automatically an isogeny, hence is a division algebra . Therefore the endomorphism algebra of any abelian variety is a product of matrix algebras . Now, by Theorem 9, the 's are finite dimensional, hence is semisimple by the Artin-Wedderburn theorem. ¡õ

Taking -invariants on both sides of (1) we get an injective homomorphism Tate [7] conjectured that in many cases it is also an isomorphism.

Conjecture 1 (Tate) The natural map (2) is an isomorphism if 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 -analog of the -adic Tate module called the Dieudonné module , constructed from the -divisible group . Tate conjectured that

Conjecture 2 (Tate) The natural map is an isomorphism if 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 -divisible groups and formal groups [16]. In particular, when is a supersingular elliptic curve (hence without -torsion), the -divisible group can be identified with the formal group .

## Finiteness 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 be an abelian variety and be a line bundle on , then the theorem of square ([5, I 5.5]) ensures that is a homomorphism where is the translation-by- isogeny. Let be the isomorphism classes of all line bundles satisfying for any . When is an elliptic curve and is a divisor on , then if and only if , so this definition is compatible with the earlier . One can show actually maps onto ([6, 8]).

There is a pair consisting of a dual abelian variety and a Poincaré bundle which is the solution to the moduli problem of parametrizing ([6, 13]). A polarization of is an isogeny such that for some ample line bundle on . If has degree one, then is called a principal polarization.

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

Theorem 10 Let be a line bundle on an abelian variety . Denote its Euler characteristic by . Then
1. . If , then where is the -fold self-intersection number of ([6, 16]).
2. if is ample, then is very ample for every ([6, 17]).

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

Theorem 11 Let be a finite field and be two fixed positive integers. Then up to isomorphism, there are only finitely many abelian varieties over of dimension possessing a polarization of degree .
Proof By Theorem 10, an abelian variety possessing a polarization of degree can be realized as a subvariety of degree in a dimensional projective space, in other words, a rational point on the corresponding Chow variety. But 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 be an abelian variety over a number field , then up to isomorphism there are only finitely many abelian varieties over which are isogenous to .

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

Theorem 12 Let be an abelian variety over a finite field , be a prime, . Then the natural map is an isomorphism.
Proof (Sketch of Proof) Step 1 Let . Reduce to showing that the natural map is an isomorphism. Since this is an injection and the dimension of is independent of , it suffices to show that it is an isomorphism for one and that all 's have the same dimension.

Step 2 Let be the subalgebra of generated by the automorphisms of defined by elements of . To prove the isomorphism, it is equivalent to show that is the commutant of in . Since is semisimple, it suffices to show that is the commutant of in by the von Neumann bicommutant theorem.

Step 3 Let be a polarization of . Define a pairing on by setting where is the pairing of and induced by the Weil pairing. Let be a subspace of which is maximally isotropic with respect to the pairing and stable under -action. Use the Finiteness Theorem 11 to construct an endomorphism such that .

Step 4 Let be the subalgebra of generated by the Frobenius endomorphisms of over . Show that for any which splits completely in , we have and use Step 3 to show that is the commutant of in .

Step 5 Use the semisimplicity of the Frobenius map on to show that all '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.