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Recent Developments in the Theory of Linear Algebraic Groups: Good Reduction and Finiteness Properties

Andrei S. Rapinchuk
Igor A. Rapinchuk
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The focus of this article is on the notion of good reduction for algebraic varieties and particularly for algebraic groups. Among the most influential results in this direction is the Finiteness Theorem of G. Faltings, which confirmed Shafarevich’s conjecture that, over a given number field, there exist only finitely many isomorphism classes of abelian varieties of a given dimension that have good reduction at all valuations of the field lying outside a fixed finite set. Until recently, however, similar questions have not been systematically considered in the context of linear algebraic groups.

Our goal is to discuss the notion of good reduction for reductive linear algebraic groups and then formulate a finiteness conjecture concerning forms with good reduction over arbitrary finitely generated fields. This conjecture has already been confirmed in a number of cases, including for algebraic tori over all fields of characteristic zero and also for some semisimple groups, but the general case remains open. What makes this conjecture even more interesting are its multiple connections with other finiteness properties of algebraic groups, first and foremost, with the conjectural properness of the global-to-local map in Galois cohomology that arises in the investigation of the Hasse principle. Moreover, it turns out that techniques based on the consideration of good reduction have applications far beyond the theory of algebraic groups: as an example, we will discuss a finiteness result arising in the analysis of length-commensurable Riemann surfaces that relies heavily on this approach. So, we hope that mathematicians working in different areas will find familiarity with good reduction quite rewarding and potentially useful. We therefore invite the reader to explore with us the fascinating symbiosis of ideas from the theory of algebraic groups and arithmetic geometry that lead to conjectures that are likely to be at the center of current efforts to develop the arithmetic theory of algebraic groups over fields more general than global.

1. Reduction Techniques in Arithmetic Geometry: A Brief Survey

It has been known since antiquity that reduction techniques can be used to show the absence of integral or rational solutions. For example, taking the equation modulo 7, one easily finds that it does not have any integral solutions. At the beginning of the 20th century, it became apparent that by reducing a given equation modulo various primes, one can not only detect the (non)existence of solutions, but in fact obtain information about the structure of the solution set. One of the earliest, and perhaps most revealing, examples arose in the study of elliptic curves. Let us consider an elliptic curve over the field of rational numbers given by its affine equation

where is a cubic polynomial with integer coefficients without multiple roots, hence having nonzero discriminant . Recall that is the projective curve obtained by adding one point at infinity to the solution set of 1, and that the chord-tangent law for addition of points makes the set of rational points into an abelian group.

In 1922, Louis Mordell showed that the group is finitely generated. One of the key steps in the argument is the so-called Weak Mordell Theorem, stating that the quotient is finite. The proof involves reducing the original equation 1 modulo rational primes and considering those primes at which our curve has bad reduction. More precisely, reducing 1 modulo a prime , we obtain the equation , where and and denote the residues of and modulo , respectively. We say that (1) has good reduction at if does not have multiple roots, and bad reduction otherwise. Geometrically, at primes of good reduction, the reduced equation still defines an elliptic curve (over ), while at primes of bad reduction it defines a singular rational curve. (Informally, the reduced curve at primes of good reduction retains the “type” of the original curve, while at primes of bad reduction it changes the type.) A somewhat subtle point here is that a change of coordinates does not essentially change an elliptic curve, i.e., results in an isomorphic curve, but may change the defining equation 1. So, we say that an elliptic curve has good reduction at if after a possible -defined change of coordinates it can be given by an equation 1 that has good reduction at . (For example, the equation has bad reduction at , but the elliptic curve it defines is isomorphic to the elliptic curve given by , which has good reduction at .) Otherwise, has bad reduction at .

For a given , the set of primes of bad reduction is contained in the set of prime divisors of the discriminant for the equation 1 defining in any coordinate system, and in particular is finite. In the proof of the Weak Mordell Theorem, plays a crucial role. More specifically, in the case where has three rational roots, the proof yields an estimate of the order of , and hence of the rank of , that depends only on the size of . We refer the interested reader to 12 for the details.

One may further wonder if the primes of bad reduction have an effect not only on the group of rational points, but on the elliptic curve itself. In his 1962 ICM talk, Shafarevich pointed out that if is a finite set of rational primes, then there are only finitely many isomorphism classes of elliptic curves over having good reduction at all .

We note that the proof of the Weak Mordell Theorem can be extended to any number field and to higher-dimensional analogues of elliptic curves, known as abelian varieties. On the other hand, while the proof of Shafarevich’s theorem can be extended to any number field and any finite set of places , it is very specific to elliptic curves. However, Shafarevich felt that his theorem was an instance of a far more general phenomenon, which prompted him to formulate the following finiteness conjecture for abelian varieties:

Let be a number field, and let be a finite set of primes of . Then for every , there exist only finitely many -isomorphism classes of abelian varieties of dimension having good reduction at all primes .

This conjecture was proved by Faltings in 1982 as a culmination of research on finiteness properties in diophantine geometry over the course of several decades. Due to its numerous implications, connections, and generalizations (cf. 3), the subject remains one of the major themes in arithmetic geometry.

2. An Overview of Algebraic Groups

To prepare for the discussion of good reduction of (linear) algebraic groups and their forms, we will now review some standard terminology. The reader who is familiar with the theory of algebraic groups may easily skip most of this section.

2.1. Basic notions and examples

An algebraic group is a subgroup of the general linear group over a “large” algebraically closed field (e.g., in characteristic 0) defined by a set of polynomial conditions; in other words, is a subgroup that is closed in the Zariski topology on . Given algebraic groups and , a morphism of algebraic groups is a group homomorphism that admits a “polynomial” representation, i.e., there exist

for , , …, , such that for all . A morphism that admits an inverse morphism is called an isomorphism.

Next, let be a subfield. When is perfect, an algebraic group is -defined or is a -group if it can be described by polynomial conditions with coefficients in . Over general fields, this notion becomes a bit more technical: one needs to require that the ideal of all functions that vanish on be generated by polynomials with coefficients in .

A morphism between two -groups is said to be -defined if the in 2 can be chosen in . It may happen that for two -groups and , there exists an isomorphism defined over a field extension of , but possibly not over . These issues are addressed in the theory of -forms (see §5) and are of central importance since the structure of algebraic groups over algebraically closed fields is well understood (see §2.2 for a brief outline and 1 for a detailed account).

While this basic approach to (linear) algebraic groups is certainly viable, a key drawback is that a given algebraic group may have a multitude of embeddings , resulting in many different sets of defining equations. This obviously creates certain inconveniences in dealing with notions that may depend on the choice of these equations (such as good reduction). The object that remains invariant for all realizations is the -algebra of regular functions in our previous notation. To keep track of the -structure when is defined over a subfield , one needs to consider the algebra of -regular functions . Without going into details, we note that the product operation on makes these algebras into commutative Hopf algebras, leading us to view as a group scheme over and , respectively. Although the perspective of the theory of group schemes is very natural for treating such topics as good reduction, in the present article we will minimize the use of this language. So, for the most part we will work with concrete defining relations associated with a given matrix realization of . One element that we will borrow from the schematic approach, though, is the possibility of avoiding the reference to a “large” algebraically closed field , allowing us to write instead of .

Example 2.1.

Standard examples include the special linear group and the symplectic group , where with and being, respectively, the zero and the identity -matrices. Both groups are defined over the prime subfield (i.e., in characteristic 0 and in characteristic ).

Furthermore, let be a nondegenerate quadratic form with coefficients in a field of characteristic , and let be its matrix. Then the orthogonal group and the special orthogonal group are -defined algebraic groups.

We will also encounter the diagonal group and the upper unitriangular group (both are again defined over the prime subfield).

2.2. Structure of algebraic groups

For any algebraic group , the connected component of the identity for the Zariski topology is a normal subgroup of finite index (e.g., is the connected component of ). So, the analysis of arbitrary algebraic groups essentially reduces to the analysis of connected ones. We will now introduce three disjoint classes of algebraic groups (unipotent groups, algebraic tori, and simple groups), and then describe how an arbitrary connected group can be constructed from these.

Unipotent groups. An algebraic group is called unipotent if the relation holds on it identically. Then there exists such that .

Algebraic tori. An algebraic group is diagonalizable if there exists such that . A connected diagonalizable group is called an algebraic torus; it is isomorphic to , where . If is defined over and an element conjugating into can be chosen in , then is said to be -split, and there is an isomorphism defined over . A -torus that does not contain any -split subtori of positive dimension is called -anisotropic.

Simple and semisimple groups. A connected algebraic group is called absolutely almost simple if it is noncommutative and does not have any proper connected normal algebraic subgroups. Then automatically has a finite center . To describe the classification of absolutely almost simple groups, we recall that a central isogeny is a surjective morphism of connected algebraic groups whose schematic kernel is finite and central (in characteristic 0, this simply means that the usual kernel is finite). An absolutely almost simple group is simply connected (resp., adjoint) if there is no nontrivial isogeny (resp., ).

Given an absolutely almost simple group , one fixes a maximal torus and then considers the corresponding root system (cf. 1, Ch. IV, §14) . The latter turns out to be reduced and irreducible, hence belongs to one of the types , , …, . The main result is that over an algebraically (and even separably) closed field, absolutely almost simple groups are classified up to isogeny by the type of the root system, and each isogeny class contains a unique (up to isomorphism) simply connected group and a unique adjoint group . While the classification of general absolutely almost simple groups over a nonclosed field is a very hard open problem (cf. §5), the classification of -split groups (i.e., those that contain a maximal -torus that splits over ) is essentially the same as over the closed fields.

A group is semisimple if there exists a central isogeny where the ’s are absolutely almost simple groups.

Example 2.2.

The groups and are absolutely almost simple simply connected groups defined and split over the prime field (such groups are also called Chevalley groups). Next, let , where is a nondegenerate -dimensional quadratic form over a field of characteristic . Then is a 1-dimensional torus for , an absolutely almost simple group for and , and a semisimple but not absolutely almost simple group for . Furthermore, it is adjoint for odd , and the corresponding simply connected group for all is the spinor group . It is -split if and only if has the maximum Witt index, and is -anisotropic (i.e., contains no -split tori) if and only if does not represent zero over .

Let us now describe the structure of an arbitrary connected algebraic group . First, possesses a maximal connected normal unipotent subgroup called the unipotent radical of . Then is trivial for the quotient , i.e., is reductive.

Now let be reductive. Then its connected center is a torus, the derived subgroup is semisimple, and the natural product map is an isogeny. Moreover, if is defined over , then and are also -defined.

3. Reduction of Linear Algebraic Groups: Examples

In this section, we will discuss several concrete examples of reduction of algebraic groups defined over the field of rational numbers at rational primes. In these examples, we will just analyze the result of reduction of a set of defining equations with integer coefficients. These considerations will motivate the formal and more general definition of good reduction in the next section.

Example 3.1.

The defining equations for the -groups , , and from §2.1 have integer coefficients, and reducing them modulo a prime we obtain the equations that define the groups of the same type over . Furthermore, let , where . Then for any prime the reduced equations define , where over .

We note that all groups considered in this example are reductive, and so are their reductions. Here is an example of a different kind of behavior.

Example 3.2.

Fix a prime and consider the binary quadratic form . Then is a 1-dimensional -anisotropic torus that consists of matrices of the form having determinant . Thus, the equations in terms of the matrix entries that define are

Reducing these equations modulo , we obtain the equations

The solutions to these equations are matrices of the form . Thus, the reduced equations define an algebraic group over which is disconnected and whose connected component is a 1-dimensional unipotent group! At the same time, reducing the equations 3 modulo any prime different from , we still get a 1-dimensional torus.

The group in this example can also be constructed as the norm torus associated with the extension of in the sense that its -points are precisely the elements of this extension having norm . We will now explore the noncommutative version of this construction.

Example 3.3.

Let be a prime, and let be the quaternion algebra over with basis and multiplication table

We recall that the reduced norm of a quaternion is given by

There exists an algebraic -group , usually denoted , whose -points correspond to the quaternions with . Using the regular representation of in the chosen basis, one can give an explicit matrix realization of and write down a set of defining equations with integer coefficients. One then proves that (1) over , the group is isomorphic to , hence is an absolutely almost simple group; and (2) the reductions modulo of the defining equations yield a group that has nontrivial unipotent radical, i.e., is not reductive (see 8, Example 3.5 for the details).

Thus, in contrast to Example 3.1, where the reductive groups had reductive reductions, in the last two examples, the reductive groups (in fact, a torus and an absolutely almost simple group) are defined by equations whose reductions yield nonreductive groups. This “anomaly” is precisely what characterizes bad reduction! (We note that in a broad sense, the situation is analogous to the reduction of an elliptic curve given by 1 at a prime dividing its discriminant in §1, where the type of the curve changes.)

This discussion motivates the following conceptual approach to the notion of good reduction for reductive -groups. First, to make our notion “local at ,” instead of , we take the localization of at the prime ideal as our base ring (concretely, is the subring of consisting of fractions in lowest terms whose denominators are prime to ). Then we say that a reductive -group has good reduction at a prime if it admits a matrix -realization where it can be defined by a system of polynomial equations with coefficients in such that the reduction of this system modulo still defines a reductive group; otherwise, we say that has bad reduction at . Thus, the groups in Example 3.1 do have good reduction at the specified primes. On the other hand, Example 3.2 suggests that the 1-dimensional torus considered therein does not have good reduction at (which can, in fact, be established with some extra work), while it does have good reduction at all odd primes .

For the group considered in Example 3.3, the situation is more delicate. We have indicated that the realization of provided by the regular representation of always results in nonreductive reduction, and one can show that indeed has bad reduction at if . For , however, is -isomorphic to , hence has a realization that results in good reduction at (cf. the discussion in §1 of why the elliptic curve actually has good reduction at ). Thus, an adequate definition of good reduction should operate with the objects independent of the matrix realization of the given reductive group and the choice of defining equations. As we explained in §2, such an object is the Hopf algebra of -regular functions, and the formal definition of good reduction in the next section is formulated precisely in terms of the existence of a -structure on the latter whose reduction modulo represents a reductive group.

4. Reductive Algebraic Groups with Good Reduction

Over general fields, one considers the notion of good reduction for reductive algebraic groups with respect to discrete valuations. So, we begin with a quick summary of the relevant definitions and some basic examples.

4.1. Brief review of valuations.

First, we recall that a (normalized) discrete valuation on a field is a surjective map such that

(a)

;

(b)

whenever .

One defines the valuation ring and the valuation ideal, respectively, by and . Then is a local ring with maximal ideal . The quotient ring is called the residue field and will be denoted . Furthermore, one considers the completion of with respect to the absolute value associated with . Then naturally extends to a discrete valuation on , which we will still denote by . The corresponding valuation ring and valuation ideal in will be denoted and , respectively; we note that the quotient ring coincides with the residue field defined above.

Example 4.1.

(a) To every rational prime , there corresponds the -adic valuation on defined as follows: if is of the form , with and relatively prime to , then . The corresponding valuation ring is the localization , and the residue field is . Furthermore, the completion is the field of -adic numbers , and the valuation ring of is the ring of -adic integers.

(b) Let be the field of rational functions in one variable over a field , and let be a (monic) irreducible polynomial. Then the same construction as in part (a) enables us to associate to a discrete valuation on . There is one additional discrete valuation on given by

Note that all of these valuations are trivial on the field of constants , and cumulatively they constitute all valuations of with this property. These valuations are often called “geometric” since they naturally correspond to the closed points of the projective line .

(c) Again let , but now assume that we are given a discrete valuation of . Then can be extended to a discrete valuation on by first extending it to the polynomial ring using the formula

for and then extending to by multiplicativity.

4.2. Definition of good reduction.

We are finally ready to introduce a formal definition of good reduction for a reductive algebraic group defined over a field with respect to a discrete valuation of . As we explained at the end of §3, one needs to formulate it in terms of the Hopf algebra of -regular functions, i.e., exercising the point of view of the theory of affine group schemes. After stating the formal definition, however, we will exhibit a number of situations where good reduction can be characterized in very concrete terms.

In our discussion, we will use the following standard notation: given an affine scheme , where is a commutative algebra over a commutative ring , and a ring extension , we will denote by the affine scheme over (usually referred to as “base change”).

Definition 4.2.

Let be a field equipped with a discrete valuation , and let be a connected reductive -group. We say that has good reduction at if there exists a reductive group scheme over with generic fiber isomorphic to .

Explicitly, this means that there should exist a Hopf -algebra such that (an -structure” on ) and the algebra over the residue field represents a connected reductive group (cf. the end of §2.2).

We conclude this section with several examples of semisimple groups where the conditions for good reduction can be described explicitly. These examples will be sufficient for understanding the rest of the article.

Example 4.3.

(a) It follows from the Chevalley construction that any absolutely almost simple simply connected -split group has good reduction at every discrete valuation of . (This generalizes the discussion of and in Example 2.1.)

(b) The construction of an absolutely almost simple group associated with the group of norm 1 elements in a quaternion algebra can be generalized to any finite-dimensional central simple algebra over a field using the reduced norm homomorphism . We will discuss this group in more detail in Example 5.2(b) below, and now only indicate that has good reduction at if and only if there exists an Azumaya -algebra such that . This means that the algebra should have an -structure such that the quotient is a central simple algebra over . Alternatively, the condition on can be expressed by saying that is unramified at .

(c) Assuming that , the spinor group of a nondegenerate quadratic form in variables over has good reduction at if and only if, over , the form is equivalent to a quadratic form with and for all , …, .

5. Forms with Good Reduction and Galois Cohomology

5.1. -forms.

The most general question in the spirit of Shafarevich’s conjecture for reductive linear algebraic groups over arbitrary fields is the following:

Let be a field equipped with a set of discrete valuations. What are the reductive algebraic groups of a given dimension that have good reduction at all ? More specifically, what are the situations in which the number of -isomorphism classes of such groups is finite?

To make this question meaningful, one needs to specialize and , which we will do in §6. However, we would first like to point out that considering reductive algebraic groups of a given dimension is far less natural than considering abelian varieties of the same dimension. Indeed, in a very coarse sense, all complex abelian varieties of dimension “look the same”: they are all analytically isomorphic to complex tori , where is a lattice of rank . At the same time, the “fine” structure and classification of these varieties are highly involved as these varieties have nontrivial moduli spaces, which leads to infinite continuous families of nonisomorphic varieties.

On the contrary, reductive algebraic groups of the same dimension may look completely different; however, the structure theory (cf. §2.2) implies that over an algebraically (or separably) closed field, for each integer , there are only finitely many isomorphism classes of (connected) reductive groups of dimension (thus, there are no nontrivial moduli spaces for such groups). So, in the analysis of finiteness phenomena for reductive groups, it makes sense to focus on those classes of groups that become isomorphic over a separable closure of the base field; the main issue then becomes the passage from an isomorphism over the separable closure to an isomorphism over the base field (so-called Galois descent). This brings us to the following.

Definition 5.1.

Let be a linear algebraic group over a field , and let be a field extension. An algebraic -group is called an -form of if there exists an -isomorphism .

We will be interested mostly in -forms of a linear algebraic group , where is a fixed separable closure of ; these will often be referred to simply as -forms of . Here are some examples of forms of algebraic groups that we will encounter in the article.

Example 5.2.

(a) Let be an -dimensional -split torus. Any other -dimensional -torus splits over , hence over . This means that all -dimensional -tori are -forms of .

Similarly, let be an absolutely almost simple simply connected split algebraic group over , and let be any absolutely almost simple simply connected -group of the same type as . Then becomes split over , entailing the existence of a -isomorphism . Thus, is a -form of , and hence every absolutely almost simple simply connected -group is a -form of an absolutely almost simple simply connected split -group.

(b) Let be a central simple -algebra of degree , and let be the algebraic -group associated with the group of elements of reduced norm 1 in . There is an isomorphism of -algebras and in terms of this isomorphism, the reduced norm of is given by . It follows that the group is -isomorphic to ; in other words, is a -form of . In fact, it is an inner form, and all inner forms are obtained this way; see 5, Ch. II, §2.3.4 for the details.

(c) Assume that . Let be a nondegenerate quadratic form in variables over , and let be the corresponding spinor group. Then any other nondegenerate quadratic form over in variables is equivalent to over , implying that is a -form of . If is odd, then the groups account for all -forms of . However, for even, there may be other -forms of defined in terms of the (universal covers of) unitary groups of (skew)-hermitian forms over noncommutative central division -algebras with an involution of the first kind (such as, for example, quaternion algebras).

We can now give a more precise version of the above question concerning reductive groups with good reduction.

Question 5.3.

Let be a field equipped with a set of discrete valuations. What are the -forms (or even just inner -forms; see Example 5.4 below) of a given reductive algebraic -group that have good reduction at all ? More specifically, in what situations is the number of -isomorphism classes of such -forms finite?

We will return to this question in the next section and formulate precise conjectures in several situations. First, however, we would like to review how the forms of algebraic groups (and other objects) are classified in terms of Galois cohomology.

5.2. Classification of forms and Galois cohomology.

Let be a group, and let be a -module, i.e., an abelian group equipped with a -action. In homological algebra, one defines the cohomology groups for all . This definition can be extended to a not necessarily abelian -group for , in which case does not have a natural group structure and is treated as a set with a distinguished element—the equivalence class of the trivial cocycle. We refer the reader to 11 for the description of this construction and its basic properties, and will focus here on the applications to the classification of forms.

Given a finite Galois extension with Galois group , there are two sources of -groups. First, for any algebraic -group , the group naturally acts on the group of -points . In this case, instead of