Derived Arithmetic Fuchsian Groups of Genus Two
Derived Arithmetic Fuchsian Groups of Genus Two
We classify all cocompact torsion-free derived arithmetic Fuchsian groups of genus two by commensurability class. In particular, we show that there exist no such groups arising from quaternion algebras over number fields of degree greater than 5. We also prove some results on the existence and form of maximal orders for a class of quaternion algebras related to these groups. Using these results in conjunction with a computer program, one can determine an explicit set of generators for each derived arithmetic Fuchsian group containing a torsion-free subgroup of genus two. We show this for a number of examples.
It is a well-known result that there are finitely many conjugacy classes of arithmetic Fuchsian groups with a given signature (, ). Extensive work has been done classifying the set of -conjugacy classes of various two-generator arithmetic Fuchsian groups: triangle groups (), groups of signature (1;e) (), and groups of signature (, ). In this paper we make progress in classifying arithmetic Fuchsian groups of signature ; i.e., genus two surface groups. This is a significantly more difficult problem since these groups have much larger coarea.
An arithmetic Fuchsian group is described by the (projectivized) group of units, , in a maximal order of a quaternion algebra over a totally real number field. A derived arithmetic Fuchsian group is a subgroup of such a . Our first main result is the classification by commensurability class of derived arithmetic Fuchsian groups of genus two, and this is summarized in Theorem 4.10. There is a finite list of signatures of groups that contain a subgroup of signature . Following , we classify all commensurability classes of derived arithmetic Fuchsian groups of the form with one of these signatures by invariant quaternion algebra. Furthermore, we determine all -conjugacy classes of the groups . Sections 3 and 4 of this paper are devoted to this result and its proof.
Section 5 contains our second main result, a technique for finding all -conjugacy classes of derived arithmetic Fuchsian groups of signature . In general, the number of conjugacy classes of a subgroup of is not necessarily equal to the number of -conjugacy classes of the group . However, if has signature or , or then index of the genus two subgroup is 4, 2, or 1, respectively. In these cases, we can use a fundamental region along with our results from Theorem 4.10 to determine an explicit set of generators for (using a computer program). This ultimately pins down the -conjugacy class of the genus two subgroup as this is determined by the traces of certain products of the group generators. Although our methods are essentially computational, we also prove some general results on the structure of maximal orders for a class of quaternion algebras associated to arithmetic Fuchsian groups. In the last section, we use our results to explicitly determine a set of generators for a few examples of derived arithmetic Fuchsian groups of signature .
In order to state and prove our main results, it is necessary to give a brief overview of the theory of arithmetic Fuchsian groups. This includes a small section of number theory consisting of definitions and results that will figure prominently in our proofs.
2.1 Fuchsian Groups
In this section we collect some standard results concerning Fuchsian groups. A Fuchsian group is a discrete subgroup of that acts properly discontinuously on the hyperbolic plane . Fuchsian groups of the first kind have a presentation of the form
where the represent the conjugacy classes of maximal cyclic subgroups of order for . A Fuchsian group with the above presentation has signature
Note that is cocompact if and only if . Since we will be concerned only with cocompact groups, we will abbreviate the signature to . A finitely generated Fuchsian group of the first kind has finite coarea, i.e., has finite hyperbolic area, and its area can be computed using the Riemann-Hurwitz formula:
Furthermore, if are Fuchsian groups and then .
Also, recall that two Fuchsian groups and are commensurable if they share a finite index subgroup, i.e., and . The commensurability class of a group is the collection of groups with which is commensurable.
2.2 Arithmetic Fuchsian and Derived Arithmetic Fuchsian Groups
An arithmetic Fuchsian group has finite coarea and therefore is necessarily of the first kind. Arithmetic Fuchsian groups are defined via quaternion algebras over totally real number fields. If is a number field and a quaternion algebra over , i.e., a four-dimensional central simple algebra over , then any quaternion algebra has an associated Hilbert symbol
where for some .
The algebra is ramified at a real infinite place of if , where denotes the Hamiltonian quaternions, and unramified at if .
Similarly, if is a finite place of and the completion of corresponding to , then is ramified at if is a division algebra. Otherwise, is unramified at and .
The ramification set of will be denoted by . Furthermore, , where (resp. ) are the set of finite (resp. infinite) places at which is ramified. We will denote the product of the primes at which is ramified by .
We will use the following standard results on quaternion algebras (see ):
Let be a quaternion algebra over a number field . The number of places at which is ramified is of even cardinality.
Given a number field , a collection of real infinite places of , and a collection of finite places of such that is even, there exists a quaternion algebra defined over with and .
Let and be quaternion algebras over a number field . Then if and only if .
An order of A is a complete -lattice which is also a ring with 1, where is the ring of integers in the number field . Furthermore, an order is maximal if it is maximal with respect to inclusion.
Let be a totally real field with and a quaternion algebra over which is ramified at all but one real place. Then
If is the unique -embedding of into and a maximal order in , then the image under of the group, , of elements of norm 1 in is contained in and the group forms a finite coarea Fuchsian group. A subgroup of is an arithmetic Fuchsian group if it is commensurable with some such . In addition, is derived from a quaternion algebra or a derived arithmetic Fuchsian group if . We will denote by . The area of can be computed by the following formula ():
where is the discriminant of the number field and is the Dedekind zeta function of the field defined for by (the sum is over all ideals in ).
Throughout the remainder of the article, we will use DAFG to denote a derived arithmetic Fuchsian group.
If is an arithmetic Fuchsian group, then the corresponding quaternion algebra is uniquely determined up to isomorphism and is called the invariant quaternion algebra of . Moreover, two arithmetic Fuchsian groups are commensurable if and only if their invariant quaternion algebras are isomorphic ().
2.3 Number of Conjugacy Classes
The number of -conjugacy classes of an arithmetic Fuchsian group depends on the infinite places of the number field and the number of conjugacy classes of maximal orders of the quaternion algebra . We will be solely concerned with -conjugacy classes here, so throughout the text conjugacy class should be interpreted as -conjugacy class. Most of what follows can be found in .
For any maximal order of , will denote the arithmetic Fuchsian group
Let and be two maximal orders in quaternion algebras and , respectively. If the groups and are conjugate, then and are isomorphic and
A result of Vigneras () states that two groups and are conjugate if and only if there exists a -isomorphism such that and with .
The class number of , , is the order of the class group , where is the group of fractional ideals of and the group of nonzero principal ideals of . Let
The restricted class group, whose order we will denote by , is the group
where is the group of principal ideals with generator in . We also have that
where is the group of units of . The number of conjugacy classes of maximal orders in a quaternion algebra defined over , denoted by , is finite and is called the type number of . It is the order of the quotient of the restricted class group of by the subgroup generated by the squares of the ideals of and the prime ideals dividing the discriminant ; so we have
where is the subgroup of prime ideals dividing the discriminant . It follows that divides . In many cases, , and we will use this to show that . Also, in the case that and , from the definitions above one can deduce that .
2.4 Torsion in Arithmetic Fuchsian Groups
Throughout this section and the remainder of the text, will denote a primitive -th root of unity. Also, will denote the field , which is the unique totally real subfield of of index 2. Note that when is odd, . The existence of torsion in an arithmetic Fuchsian group defined over a number field depends primarily on the subfields of of the form and the existence of embeddings of suitable quadratic extensions of into the invariant quaternion algebra . A more detailed treatment of this topic can be found in , Ch. 12.
Let denote the elements of norm 1 in and its projectivization. Let be a maximal order in and suppose the group contains an element of order . Then contains an element of order and contains an element of order . This implies that and hence . Furthermore, is a quadratic extension of that embeds in .
Conversely, using the following theorem (cf. ), one can show that if is a quadratic extension that embeds in , then necessarily contains elements of order .
Theorem 2.1 ().
Let be a number field and a quaternion division algebra over such that there is at least one infinite place of at which is unramified. Let be a commutative -order whose field of quotients is a quadratic extension of such that . Then every maximal order in contains a conjugate of except possibly when the following conditions both hold:
(a) and are unramified at all finite places and ramified at exactly the same set of real places of ,
(b) all prime ideals dividing the relative discriminant ideal of are split in .
The order is a commutative -order whose field of quotients is . In the case of arithmetic Fuchsian groups, the field is totally real and the field is a totally imaginary extension of . Therefore, all real places of are ramified in ; however, the algebra is ramified at all real places but one. So condition (a) of Theorem 2.1 never holds. Thus, if , then every maximal order of will contain elements of order . Therefore, if contains elements of order , then so will for any maximal order of . We have just shown the following:
Theorem 2.2 ().
An arithmetic Fuchsian group contains an element of order if and only if the field embeds in .
The following theorem gives necessary and sufficient conditions for the embedding of the extension into the quaternion algebra .
Let be a quaternion algebra over a number field and let be a quadratic extension. Then embeds in if and only if is a field for each .
Let be a totally real number field such that , a quaternion algebra ramified at all but one real place over , and a maximal order in . The group will contain an element of order m if and only if does not split in for all ,
Let be a primitive -th root of unity. By Theorem 2.3, the quadratic extension of embeds in if and only if is a field for each . Since is totally imaginary, this holds automatically for all . By Theorem 2.3, of embeds in if and only if does not split in for all . Theorem 2.2 now gives the desired conclusion. ∎
In the case is a totally real field, the relative class number for the extension is defined as
where is the class number of and is the class number of , (cf.  p.38).
If a maximal order in contains elements of finite order, then we can calculate the number of conjugacy classes, , of maximal cyclic subgroups of order in , provided is a relative integral basis for the quadratic extension (). If this assumption holds, then
where is the Artin symbol (which is equal to 1, 0, or -1, according to whether splits, ramifies, or is inert, respectively, in the extension and
In some cases, we can use the following lemma to simplify the above formula (6):
Let be a totally real number field of odd degree and suppose is a relative integral basis for . If is odd, then .
Both quantities and depend only on the number field and, hence, are independent of the quaternion algebra . Since is a finite 2-group, its order is , for some non-negative integer . If is odd, then let be a quaternion algebra unramified at all finite places. Since and is odd, we must have . ∎
Since will arise frequently in our calculations, we will fix the notation . Furthermore, the following lemma can often be used to simplify formula (6):
Suppose that is a totally real number field and that 2 is unramified in , then
Likewise, if 3 is unramified in , then
Let be a totally real number field such that is not divisible by 2. Then is a relative integral basis for . Likewise, if is not divisible by 3, then is a relative integral basis for .
Let be a totally real number field and K a totally imaginary quadratic extension of . Then every unit of has the form , where is a root of unity with and is a real unit with .
Proof of Lemma 2.6.
Let us first consider the case . Since 2 does not divide the discriminant of , is a relative integral basis for the extension . Suppose that is a root of unity in . Since this is also an algebraic integer, it can be written as , where . Now, the only solutions of correspond to the units and . By Fact 2.8, any unit of is of the form , where and is a real unit. Again, using the relative integral basis, let for some . Now, any unit must satisfy the equation:
There are two possible cases to consider:
Case 1: .
Case 2: .
In Case 1, we must have and . Since is real, this implies . But since is an algebraic integer, is not a unit. Therefore, no unit corresponds to this case. In Case 2, either or . Hence, or . Therefore, the units in are of the form or . Since is a unit, this implies or and, hence, . Now, in either case, we have
This means that every unit of has norm lying in , and so
The proof for is similar. ∎
It will be necessary for us to determine which periods can arise in the various number fields . First, If , then divides and divides . With this in mind, we will require the following properties of the cyclotomic field and its proper subfield when is a prime:
Let be a prime. Then:
This first part follows from the fact that and that is a proper subfield of index two. The second and third parts can be found in , , respectively. ∎
3 Bounds on the degree of the number field
In this section, we classify commensurability classes of DAFGs of signature by invariant quaternion algebra. First, we determine all possible signatures of Fuchsian groups which can contain a subgroup of signature . Then, using arithmetic data, we show that all arithmetic Fuchsian groups with one of these signatures are defined over a number fields of degree less than or equal to 5.
The following theorem gives necessary and sufficient conditions for the existence of torsion-free subgroups of a given index in a Fuchsian group:
Theorem 3.1 ().
Let be a finitely generated Fuchsian group with the standard presentation:
Then has a torsion free subgroup of finite index if and only if is divisible by , where , and if has even type, while if has odd type. ( has odd type if , is even, but is odd for exactly an odd number of ; otherwise has even type.)
We will use this result to prove the following lemma:
Let be a cocompact Fuchsian group containing a genus two surface group. Then has one of the following signatures: , , , , , , , , , , , , , , , , ,
, , , , , , , , , , , , , , , or .
If is a genus two surface subgroup of , then , where . This implies . In particular, the genus of must be less than or equal to . Furthermore, by Theorem 3.1, depending on the signature of the group , either or divides the index , where . This gives us bounds on the possible torsion of . In particular, for fixed , this gives us an upper bound on the number of conjugacy classes of elliptic elements:
If , then has at most six conjugacy classes of elliptic elements.
If , then has at most two conjugacy classes of elliptic elements.
If , then has no elliptic elements and .
For example, suppose and has four conjugacy classes of elliptic elements of order , . By the Riemann-Hurwitz formula (2),
Therefore, if contains a torsion-free subgroup of genus two, then
This translates to the existence of integers , , satisfying the equation:
In addition, divides . Since , there exists at least one with order . Also, we can deduce the following two facts.
There cannot exist more than two distinct corresponding to the .
If for all , then .
If (i) does not hold, then , , , and , and this gives the following contradiction:
Similarly, if (ii) does not hold, then , and we arrive at the contradiction
Without loss of generality, suppose .
Case 1: and .
In this case, equation (7) becomes
The solution gives the maximal value of . The only other solutions in this case occur when .
Case 2: and .
Again, equation (7 becomes
The case gives the maximal value for and this occurs when . The only other possible solution occurs when .
Case 3: and .
Equation (7) translates to
and one can easily verify that there exist no integer solutions to this equation.
Case 4: .
In this situation,
and is the only solution. The existence of a torsion-free subgroup of index for a group of fixed signature is guaranteed by Theorem 3.1. Therefore, the only Fuchsian groups with signature containing a torsion-free subgroup of genus 2 are those with signatures , , , , , and . In this manner, we analyze torsion in groups of a fixed signature to obtain the list in the Lemma. ∎
There exist no DAFGs of signature arising from quaternion algebras over number fields of degree greater than 5.
If contains a genus two surface group of index , then
In particular, this implies
Using and in the above inequality gives
We now use Odlyzko’s lower bounds (cf. ) on the discriminant of a totally real number field to get an upper bound on the degree of :
where . Using these estimates in inequality (11) gives . However, the smallest discriminants of a totally real field of degree 7 and 8 are 20,134,393 and 282,300,416, respectively ( and ). In each case, inequality (9) is violated:
Hence, there cannot exist a DAFG of signature if .
This gives us the following upper bound on the discriminant :
According to the lists from , there are 20 number fields of degree 6 satisfying the above inequality. For each field , we investigate the behavior of small primes and, if necessary, estimate using Pari. Since , . Therefore, , where is the prime of smallest of norm in .
For example, consider the totally real field of degree 6 and discriminant . A minimal polynomial for is . By Pari, the prime of smallest norm in is the unique prime lying over 2 with . This implies that any group defined over has area at least
Hence, there exist no DAFGs of signature defined over . In this fashion, we obtain a contradiction to the inequality for each totally real field of degree 6 with discriminant satisfying (12). ∎
4 Classification by Commensurability Class
In this section, we classify DAFGs of signature by invariant quaternion algebra. All the groups in Lemma 3.2 except those with one of the three signatures , , and , have commensurability classes that have already been classified; i.e., they are all commensurable with an arithmetic Fuchsian triangle group or one having signature or So it suffices to classify the commensurability classes of the groups of these remaining three types and to extract the relevant results from , , , and .
The proof classification is exhaustive. For each fixed degree , we use equation (3) to get upper bounds on the discriminant of the number field . Then we determine the existence of all quaternion algebras whose unit groups have one of the above three signatures. Rather than go through an analysis of each number field that can correspond to such an arithmetic Fuchsian group, we give an idea of the overall approach by a few illustrative examples. Our argument will be organized by the degree of the number field.
We will make extensive use of the following lemma, which is particularly useful in the case odd (since we can have in this case):
If is a quaternion algebra defined over a totally real field ramified at all but one infinite place and unramified at all finite places, then contains elements of orders 2 and 3. Furthermore, if is a genus two surface group contained in for a maximal order in , then divides .
4.1 Quintic Number Fields
For , the only DAFGs of signature arise from quaternion algebras over the totally real fields of discriminants , and .
Suppose there exists a DAFG of genus two which is torsion-free and defined over a totally real quintic number field . Using the inequalities , in conjunction with (3) we get that
However, if , the index by Lemma 4.1. Substituting back into the area formula (3) gives . For those fields with , we analyze the behavior of small primes in to determine the possible ramification sets for each field . According to , there are 15 number fields with . In a few cases, small primes do exist, but we can eliminate these cases using torsion.
For example, let be the number field with discriminant . Using the minimal polynomial to generate the in Pari, we compute that
Furthermore, there exists a unique prime of norm 2 in . Together with the fact that is even, this implies that . So, by (3), for any maximal order in . However, by Lemma 4.1, contains elements of orders 2 and 3. Hence, is not torsion-free, and since , it neither is a genus two surface group nor does it contain a genus two subgroup.
The case yields a positive result. By Pari, using the minimal polynomial , we compute that
So if has signature , then the following equation must be satisfied:
Again the only solution to the above equation occurs when and . Since is prime, contains no proper subfields other than . Thus, the only possibilities for elements of finite order in are 2 and 3. By Lemma 4.1, any group contains elements of orders 2 and 3. So, in this case we get
We see that the only solution to this equation is . Hence, has signature in this case, and Theorem 3.1 guarantees the existence of a torsion-free subgroup of index 6.
The totally real number fields of degree 5 with discriminants 36497 and 24217 are the only other fields which yield positive existence results. ∎
4.2 Quartic Number Fields
For , . Using Proposition 2.9 to analyze the cyclotomic extensions with degree dividing 8, one can easily show that 2, 3, 4, 5, 6, 8, 10, 12, 15 are the only possible cycles for elliptic elements in this case.
There are 48 number fields with discriminants satisfying the above inequality in . Again, we eliminate all fields except those listed in Theorem 4.10 by estimating and examining the factorization of small primes using Pari.
There exist no DAFGs of signature defined over the totally real field with .
The minimal polynomial for is Using Pari, we compute that
Hence, a torsion-free genus two subgroup of index corresponds to a solution of the equation