Lefschetz numbers of symplectic involutions on arithmetic groups
The reduced norm-one group of a central simple algebra is an inner form of the special linear group, and an involution on the algebra induces an automorphism of . We study the action of such automorphisms in the cohomology of arithmetic subgroups of . The main result is a precise formula for Lefschetz numbers of automorphisms induced by involutions of symplectic type. Our approach is based on a careful study of the smoothness properties of group schemes associated with orders in central simple algebras. Along the way we also derive an adelic reformulation of Harder’s Gauß-Bonnet Theorem.
2010 Mathematics Subject Classification:Primary 11F75; Secondary 20H10, 20G35
Let be a semisimple linear algebraic group defined over the field of rational numbers. Given a torsion-free arithmetic subgroup it is in general a very difficult task to compute the (cohomological) Betti numbers of . However Harder’s Gauß-Bonnet Theorem  makes it possible to determine the Euler characteristic of arithmetic groups. If the Euler characteristic is non-zero, one can extract information on the Betti numbers. Moreover, whether or not the Euler characteristic vanishes only depends on the structure of the associated real Lie group (see Rem. 3.1). If the Euler characteristic vanishes, Lefschetz numbers of automorphisms of finite order of are a suitable substitute to gain insight into the cohomology of . The idea to study Lefschetz numbers in the cohomology of arithmetic groups goes back to Harder . A general method was developed by J. Rohlfs, first for Galois automorphisms  and later in a general adelic setting . Lefschetz numbers were also studied by Lee-Schwermer  and Lai . However, only very few groups have been considered in detail, most frequently Lefschetz numbers on Bianchi groups were studied (see Krämer , Rohlfs , Sengün-Türkelli , and Kionke-Schwermer ). In this article we describe a method (based on Rohlfs’ approach) to compute Lefschetz numbers of specific automorphisms on arithmetic subgroups of inner forms of the special linear group.
More precisely, let be an algebraic number field and let be a central simple -algebra. The reduced norm is a polynomial function on and the associated reduced norm-one group is a linear algebraic group defined over . Indeed, the algebraic group is an inner form of the special linear group. If has an involution of symplectic type (see Def. 2.3), then the composition of with the group inversion yields an automorphism of . We study the Lefschetz numbers of such automorphisms induced by involutions of symplectic type.
1.1. The main result
Let be an algebraic number field and let denote its ring of integers. Let be a central simple -algebra. For our purposes we may assume that for some quaternion -algebra (see 1.3).
Let be a maximal -order in , then is a maximal -order in . For a non-trivial ideal we study the cohomology of the principal congruence subgroups
of . In fact, for the groups have vanishing Euler characteristic.
The quaternion algebra is equipped with a unique involution of symplectic type , called conjugation, which induces an involution of symplectic type by , i.e. apply to every entry of the matrix and then transpose the matrix. We will call the standard involution of symplectic type on . Composition of with the group inversion yields an automorphism of order two on . Note that the congruence groups are -stable. Fix a rational representation of (defined over the algebraic closure of ) on a finite dimensional vector space. If is equipped with a compatible -action (see Def. 4.1), then we can define the Lefschetz number of in the cohomology .
Assume that is torsion-free. If is totally definite, we assume further that . The Lefschetz number is zero if is not totally real.
If is totally real, the Lefschetz number is given by the following formula
Here denotes the signed reduced discriminant of (see Def. 5.1), denotes the number of real places of ramified in , and
where denotes the set of finite places of where ramifies and denotes the Dedekind zeta-function of . If is totally real, then the Lefschetz number is zero if and only if .
We briefly give three applications of the above formula where we always assume to be a totally real number field.
1.2.1. Growth of the total Betti number
The analysis of the asymptotic behaviour of Betti numbers of arithmetic groups is an important topic. Recent results of Calegari-Emerton provide strong asymptotic upper bounds (cf. ). We can use the main theorem to obtain an asymptotic lower bound result.
Let be the reduced norm-one group associated with the central simple -algebra . For a torsion-free arithmetic subgroup we define the total Betti number . Note that this is a finite sum since torsion-free arithmetic groups are of type (FL) (see [4, Thm. 11.4.4]).
Let be an arithmetic subgroup. For any ideal we define . There is a positive real number , depending on , , , and , so that
holds for every ideal such that is torsion-free.
A proof of this corollary will be given in Section 5.5.
1.2.2. Rationality of zeta values
If is a totally real number field, then is a non-zero rational number for all integers .
Apply the main theorem with , and choose to be the trivial one-dimensional representation. We see that for every and all sufficiently small ideals , the number
is a non-zero integer. The claim follows by induction on . ∎
1.2.3. Cohomology of cocompact fuchsian groups
Let be a division quaternion algebra over such that is split at precisely one real place of . So is the number of real places ramified in .
Let be a maximal -order in . We consider the reduced norm-one group defined over . The associated real Lie group is
Note that the group is compact and so the projection onto the first factor is a proper and open homomorphism of Lie groups. In particular, every discrete torsion-free subgroup maps via isomorphically to a discrete subgroup in .
Let be a proper ideal such that is torsion-free. We will interpret as a subgroup of . Note that, since we assumed to be a division algebra, the group is a cocompact Fuchsian group (see Thm. 5.4.1 in ).
Let be the Poincaré upper half-plane.
The compact Riemann surface has genus
This implies an explicit formula for the first Betti number since
Consider the main theorem for . Note that for the automorphism is actually the identity. This means that, using the main theorem with the trivial representation, we obtain
Note that the sign of the Lefschetz number is . Since , the claim follows immediately. ∎
1.3. Reduction to quaternion algebras
Let be a central simple -algebra. If has an involution of symplectic type (see Def. 2.3), then is isomorphic to the opposed -algebra . This means that the class of has order two in the Brauer group of . Since the dimension of is even, it follows from (32.19) in  that is isomorphic to a matrix algebra over a quaternion algebra . Therefore we always assume .
Let be the standard involution of symplectic type on . Note further that in this case for an element with . Due to this observation it is only a minor restriction if we focus on the standard symplectic involution .
1.4. Structure of this article
In Section 2 give a short general treatment of smooth group schemes over Dedekind rings which are associated with orders in central simple algebras. In particular we treat integral models of inner forms of the special linear group. Further, we consider the fixed points groups attached to involutions. An important tool in the proof of the main theorem will be the pfaffian as a map in non-abelian Galois cohomology (cf. Section 2.4). In Section 3 we give an adelic reformulation of Harder’s Gauß-Bonnet Theorem which hinges on the notion of smooth group scheme. The calculation of the Lefschetz number is based on Rohlfs’ method which we summarise in Section 4. Finally the proof of the main theorem is contained in Section 5. It consists of two major steps. The first is the analysis of various non-abelian Galois cohomology sets which occur in Rohlfs’ decomposition. The second step is the calculation of the Euler characteristics of the fixed point groups using Harder’s Gauß-Bonnet Theorem.
Apart from Section 2, where we work in a more general setting, we use the following notation: is an algebraic number field and denotes its ring of integers. Let denote the set of places of . We have where (resp. ) denotes the set of archimedean (resp. finite) places of . Let be a place of , then we denote the completion of at by . The valuation ring of is denoted by and the prime ideal in is denoted by . For a non-zero ideal the ideal norm is defined by . As usual denotes the ring of adeles of and is the ring of finite adeles.
2. Group schemes associated with orders in central simple algebras
In this section we will investigate the smoothness properties of group schemes attached to orders in central simple algebras. Throughout denotes a Dedekind ring and denotes its field of fractions. For simplicity we assume . In our applications is usually the ring of integers of an algebraic number field or a complete discrete valuation ring.
The term scheme always refers to an affine scheme of finite type, the same holds for group schemes. Recall that a scheme defined over is smooth if for every commutative -algebra and every nilpotent ideal the induced map is surjective. Suppose is a complete discrete valuation ring and let denote its prime ideal. We will frequently use the following property: If is a smooth -scheme, then the induced map is surjective for every integer (cf. Cor. 19.3.11, EGA IV, ). If is a group scheme, then we denote the Lie algebra of by .
2.1. The general linear group over an order
Let be a central simple -algebra and let be an -order in . Since is a finitely generated torsion-free -module, it is a finitely generated projective -module (cf. (4.13) in ). The functor from the category of commutative -algebras to the category of rings defined by is represented by the symmetric algebra where . In fact it defines a smooth -scheme (cf. 19.3.2 in EGA IV ).
Recall that, since is finitely generated and projective, one can attach to every -linear endomorphism of its determinant . More precisely, here the determinant of is just the determinant of the -linear extension . As usual one defines the norm of an element to be the determinant of the left-multiplication with . One can check that the norm defines a morphism of schemes over
to the affine line defined over . This can be seen, for instance, by observing that the norm is a natural transformation of functors. Let be a commutative -algebra. An element is a unit if and only if . It follows from the next lemma that the associated unit group functor is a smooth group scheme over .
Let denote the affine line over . Let be an affine scheme over with a morphism . The subfunctor (from the category of commutative -algebras to the category of sets) defined by
is an affine scheme and the natural transformation is a morphism of schemes. If is smooth, then has the same property.
Let be the coordinate ring of and let be the polynomial defining . Note that is canonically isomorphic to the functor
Using this it is easily checked that the -algebra represents . Clearly, is of finite type since is of finite type.
It remains to show that is smooth, whenever is smooth. Assume to be smooth and take a commutative -algebra with an ideal such that . By assumption is surjective, so given we find projecting onto . By assumption is a unit in . In particular, we find with . However, consists entirely of units and thus . We deduce that is smooth. ∎
To stress this once more: in this article is always a functor and not a group. If we take then we call the multiplicative group (or multiplicative group scheme) defined over , and we denote it by . Note that the norm defines a homomorphism of -group schemes
We also point out that the Lie algebra of can be (and will be) identified with in a natural way.
2.2. The special linear group over an order
2.2.1. Reduced norm and trace
Let be a central simple -algebra, we consider the reduced norm and trace (for definitions see section 9 in  or IX, §2 in ). It was observed by Weil that the reduced norm and trace are polynomial functions. We reformulate this in schematic language: There is a unique element in the symmetric algebra (here ) such that for every splitting field of and every splitting the induced map maps the determinant to . Similarly there is the reduced trace with an analogous property.
Let be an -order. We show that the reduced norm and trace are defined over in the appropriate sense. For the reduced trace this is easy: Elements in are integral over , hence the reduced trace takes values in on the order and defines an -linear map . In particular we obtain a morphism of schemes over
Consider the reduced norm. From (9.7) in  one can deduce that and agree as elements in the coordinate ring . However, the coordinate ring of is integrally closed in and we conclude that the reduced norm is defined over . This means that there is a polynomial which defines the reduced norm as a morphism of -schemes
We can also restrict the reduced norm to the unit group and obtain a homomorphism of group schemes
The special linear group over the order is the group scheme over defined by the kernel of the reduced norm, this is
2.2.2. Smoothness of the special linear group
Whereas the general linear group is always smooth, independent of the chosen order, the smoothness of the special linear group depends on the underlying order. Recall the following useful result.
Proposition 2.2 (Smoothness of kernels).
Let be a morphism between two smooth group schemes over . If the derivative is surjective, then the group scheme is smooth over .
This follows from the theorem of infinitesimal points (see [7, p. 208]) and some easy diagram chasing. ∎
As a matter of fact the derivative of the reduced norm is the reduced trace. Having this in mind we make the following definition.
An -order in a central simple -algebra is called smooth if the reduced trace is surjective.
Note that smoothness of orders is a local property.
If the order is smooth then the scheme is smooth.
This follows immediately from Proposition 2.2 using the fact that the derivative of the reduced norm is the reduced trace. ∎
In fact, also the converse statement holds under the assumption . However, we shall not need this result. The next proposition shows that smooth orders exist.
Assume that is finite for every prime ideal . Then every maximal -order in a central simple -algebra is smooth.
Let be a central simple -algebra and let be a maximal -order. Since is maximal in if and only if all -adic completions are maximal orders (see (11.6) in ), and since smoothness of is a local property, we may assume that is a complete discrete valuation ring. Recall that is isomorphic to a matrix algebra over a central division algebra . Moreover, has a unique maximal -order and is (up to conjugation) the maximal order in (see (17.3) in ). It is known that the reduced trace of a matrix is given by
(cf. Cor. 2, IX.§2 in ). Hence we may assume that is a division algebra and is the unique maximal order. Let and let be the unique unramified extension of of degree . The field embeds into as a maximal subfield and the reduced trace on the elements of agrees with the field trace (cf. proof of (14.9) in ). Let denote the valuation ring of . The image of under the embedding lies in the maximal order . Finally the surjectivity of follows from the well-known surjectivity of the field trace . ∎
2.3. Involutions and fixed point groups
Let be a central simple -algebra. An involution on is an additive mapping of order two such that for all . We say that is of the first kind if is -linear. Otherwise, we say that is of the second kind. In this article all involutions are of the first kind unless the contrary is explicitly stated. We will mostly focus on involutions of symplectic type.
We say that an involution on is of symplectic type, if there is a splitting field of the algebra , a splitting
and a skew symmetric matrix satisfying for all elements . If this is the case, then every splitting (over any splitting field) has this property.
Let be an involution of the first kind. Let be an -order in and assume that is -stable. Since is -linear, we obtain a morphism of -schemes
We restrict to the unit group and compose it with the group inversion to obtain a homomorphism of group schemes
We define to be the group of fixed points of , this is, for every commutative -algebra we obtain
We analyse the smoothness properties of group schemes constructed in this way. Define and note that this -submodule of is even a direct summand of.
For every commutative -algebra , every and every we have .
We can write for certain and . The claim is linear in , hence we may assume with and . We calculate
and we see that is in since and are elements of . ∎
The order is called -smooth if the map defined by is surjective.
Clearly -smoothness is a local property.
If an -order is -smooth, then the scheme is smooth.
We set . Let be a commutative -algebra with an ideal such that . We have to show that the canonical map is surjective. Take . Since the unit group scheme is smooth (see 2.1), we find which maps to modulo . Since is in the fixed point group of , this implies that
with some .
We consider and we obtain by Lemma 2.5. Consequently, there is such that . Moreover, we have , thus there is some with . We deduce that is an element in , and thus
As last step we use once again that is -smooth and deduce that there is some with . We put , which is congruent modulo and satisfies
Therefore and maps to under the canonical map. ∎
It is possible to prove also the converse statement, however, this will not be needed in the sequel.
2.4. Involutions of symplectic type and the pfaffian
Let be a central simple -algebra with an involution of symplectic type . Let be a -stable -order in .
2.4.1. The pfaffian
Set in the notation of section 2.3. The inclusion induces a morphism of -algebras
Recall that the reduced norm is given by a polynomial function (see 2.2.1). We define . We will construct a pfaffian, i.e. a polynomial such that .
Let be any field extension. It follows from (2.9) in  that for every the reduced norm is a square in . Therefore, we may deduce that there is a polynomial such that
We normalise this polynomial such that and we call the pfaffian with respect to .
Let denote the automorphism of the symmetric -algebra which is induced by . The following assertions hold:
for all and all , we have
where is any commutative -algebra.
To prove the first claim we may work over fields. However, over fields this is the well-known statement (2.2) in .
Consider the fixed point group scheme associated with . Let for some commutative -algebra . We see from and Lemma 2.7 that
Hence the reduced norm restricts to the trivial character on .
2.4.2. The cohomological pfaffian
We study non-abelian Galois cohomology of with values in the groups and . For the definition of non-abelian cohomology we refer the reader to , [38, p. 123-126] or [19, Ch. VII]. We shall in this context often denote and by left exponents, i.e. we write for .
Let be a commutative -algebra and assume that is flat as -module. A cocycle is an element of which satisfies , or equivalently . In other words
The assumption that is flat yields that . Therefore we can apply the pfaffian associated with to cocycles in . Two cocycles and are cohomologous if there is such that . In this case it follows from Lemma 2.7 that . Therefore the pfaffian defines a morphism of pointed sets
By the same reasoning we obtain a morphism of pointed sets
For simplicity we introduce the notation and we define .
Proposition 2.8 (The cohomological diagram for symplectic involutions).
Let be an involution of symplectic type on and let be a -stable -order. For every commutative -algebra which is flat as -module, there is a commutative diagram of pointed sets with exact rows.
The map is injective and the lower row is an exact sequence of groups. Here denotes the map induced by the inclusion .
The short exact sequence of groups
is even an exact sequence of groups with -action, where acts on by inversion. Consider the initial segment of the associated long exact sequence in the cohomology (see Prop. I.38 ):
It follows from Remark 2.1 that is bijective. Thus the long exact sequence takes the form
It is easy to see that which is a subgroup of . Hence we simply replace the last term by . This yields the upper row of the diagram. It is an easy exercise to verify that the lower row is an exact sequence of groups.
It remains to verify the commutativity of the rectangles. The middle one is obviously commutative by definition of the pfaffian in the cohomology. For the last rectangle we simply use that for all by the construction of the pfaffian.
Consider the first rectangle. We recall the definition of the connecting morphism : Given , we can find an element such that , then is defined to be the class of . The pfaffian of is
(see Lemma 2.7). This proves the commutativity of the first rectangle.
Finally, note that is injective since is injective. ∎
An element lies in the image of if and only if lies in the image of the canonical map .
Let denote the canonical map. Suppose the class is in the image of , then we obtain immediately that lies in the image of .
Conversely, suppose for some . Then the diagram shows that in and therefore lies in the image of . ∎
Remark 2.2 (Twisting involutions).
Let be a central simple -algebra with an involution of symplectic type and let be a -stable -order.
Given an element , we can twist the involution with . More precisely, we define by . It is easily verified that this is again an involution on , and since , the order is -stable. Note that is again an involution of symplectic type.
Suppose is -smooth, we claim that is -smooth as well. Take an element , this is . Consequently, and by -smoothness there is an element which satisfies . The element is a unit in , hence we may write for and it follows that . We have shown that is -smooth.
Finally, for all we have on the group scheme . Since is equivalent to , such an element is a cocycle for . If we now twist with the cocycle (cf. Section 4), we obtain
2.5. Hermitian forms and non-abelian Galois cohomology
We shall also need a result due to Fainsilber and Morales from the theory of hermitian forms. Let be a central simple -algebra and let be an involution on . In this short section it is not important whether or not is of the first or of the second kind.
The notion of -smoothness is related to the theory of even hermitian forms. Let be a -stable -order in and let be a finitely generated and projective right -module. A hermitian form (or more precisely a -hermitian form) with respect to on is said to be even if there is a -sesquilinear form such that . Here is the sesquilinear form defined by
It follows immediately that is -smooth if and only if every hermitian form on (considered as right -module) is even. This is useful since even hermitian forms can be handled easier than arbitrary hermitian forms.
We consider the automorphism of defined as the composition of and the group inversion. Similarly we obtain on . Here it is not necessary to consider as a morphism of group schemes, which is a little bit more tedious if is of the second kind. We will need a Theorem of Fainsilber-Morales in the following paraphrase:
Theorem 2.10 (Fainsilber, Morales ).
Let be a field which is complete for a discrete valuation and let be its valuation ring. Let be a central simple -algebra with involution . Suppose is a -stable maximal -order in . If is -smooth, then the canonical map
Compared with  we add the assumption of -smoothness to eliminate the restriction on the residual characteristic. The proof is almost identical.
3. An adelic reformulation of Harder’s Gauß-Bonnet Theorem
We briefly describe an adelic reformulation of Harder’s Gauß-Bonnet Theorem (see ) which hinges on the notion of smooth group scheme.
Let be an algebraic number field and let denote its ring of integers. Let be a connected semisimple algebraic group defined over . We denote by the associated real Lie group, i.e.
The group is a real semisimple Lie group.
3.1. The Euler-Poincaré measure
We define what we mean by the compact dual group of , since the definition differs from author to author. Let be the real Lie algebra of and let denote its complexification. Moreover, let be a maximal compact subgroup of and consider the associated Cartan decomposition
The real vector space is a real Lie subalgebra of and is even a compact real form of (cf. [18, p. 360]). Let be the unique connected (a priori virtual) Lie subgroup of with Lie algebra . Since the real semisimple Lie algebra is a compact form, the Lie group is compact and thus closed in (see IV, Thm. 4.69 in ). Further we see that the connected component is a subgroup of . We say that is the compact dual group of containing . Note that the dual group depends on the algebraic group .
Let be a non-degenerate -bilinear form such that and are orthogonal. We extend the a -bilinear form (again denoted ) on . Note that restricted to is a non-degenerate -bilinear form . We obtain corresponding right invariant volume densities on and on which will be denoted by .
We define . Let be a torsion-free arithmetic group. Harder’s Gauß-Bonnet Theorem shows that integration over with the Euler-Poincaré measure (cf. Serre [37, §3]) yields the Euler characteristic of – even if is not cocompact. Via Hirzebruch’s proportionality principle one has the following formula for the Euler-Poincaré measure on (cf. Harder  and Serre ).
If is odd or if , then is the Euler-Poincaré measure. Otherwise, if and is even, then
Here and (resp. ) denotes the Weyl group of the complexified Lie algebra (resp. ).
3.2. The adelic reformulation
Let denote the ring of adeles of and let denote the ring of finite adeles. Let be a connected semisimple algebraic group defined over . Let be an open compact subgroup of the locally compact group . Borel showed that is the disjoint union of a finite number of double cosets, i.e.
for some representatives (see Thm. 5.1 in ). For every we obtain an arithmetic subgroup defined by
There is a -equivariant homeomorphism
Here the right hand side denotes the topologically disjoint union.
Define . Suppose that acts freely on . This is the case if and only if the groups are torsion-free for all . If is odd or if , then
This follows immediately from Harder’s Gauß-Bonnet Theorem and the homeomorphism in equation (1).
Note further that if has a complex place, then is always satisfied. Therefore we may restrict to the case where is totally real.
3.2.1. The Tamagawa measure
We derive a description of the Tamagawa measure in terms of the local volume densities. For a thorough definition of the Tamagawa measure we refer the reader to Oesterle . Let be a connected semisimple linear algebraic -group of dimension . Let be the Lie algebra of