Dihedral groups of automorphisms of compact Riemann surfaces of genus two

Transcription

1 Electronic Journal of Linear Algebra Volume 26 Volume 26 (2013) Article Dihedral groups of automorphisms of compact Riemann surfaces of genus two Qingje Yang Dan Yang Follow this and additional works at: Recommended Citation Yang, Qingje and Yang, Dan. (2013), "Dihedral groups of automorphisms of compact Riemann surfaces of genus two", Electronic Journal of Linear Algebra, Volume 26. DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact

2 DIHEDRAL GROUPS OF AUTOMORPHISMS OF COMPACT RIEMANN SURFACES OF GENUS TWO QINGJIE YANG AND DAN YANG Abstract. In this short note, the conjugacy classes of finite dihedral subgroups of the 4 4 integral symplectic group are considered. A complete list of representatives of the classes is obtained, among them six classes are realizable by analytic automorphisms of compact connected Riemann surfaces of genus two. Key words. Integral symplectic matrix, Automorphism, Riemann surface, Dihedral group. AMS subject classifications. 20H25, 57M Introduction. The problem we consider in this paper is the realizability of finite dihedral groups D n of order 2n, in SP 4 (Z), the 4 4 symplectic group over the ring of integers Z, by analytic automorphisms of compact connected Riemann surfaces of genus 2. This is a special case of a more general problem. Let S be a connected compact Riemann surface of genus g (g 2) without boundary and G be a subgroup of Aut(S), the group of analytic automorphisms of S. Then G induces a faithful group action on H 1 (S) = H 1 (S,Z), the first homology group of S, G : H 1 (S) H 1 (S). Let a 1,a 2,...,a g,b 1,b 2,...,b g be a canonical basis of H 1 (S), that is, a basis for which the intersection matrix is J = [ ] 0 Ig, I g 0 where I g is the identity matrix of degree g. For any element σ in G, let X be the matrix of σ with respect to this basis. Since σ preserves the intersection numbers, X JX = J, where X is the transpose of X. Received by the editors on May 24, Accepted for publication on July 31, Handling Editor: Roderick Gow. School of Information, Renmin University of China, Beijing, , China (yangqj@ruc.edu.cn, young.child@163.com). 553

3 554 Q. Yang and D. Yang Definition 1.1. The set of 2n 2n unimodular matrices X in M 2n (Z) such that (1.1) X JX = J is called the symplectic group of genus n over Z and is denoted by SP 2n (Z). Two symplectic matrices X, Y of SP 2n (Z) are said to be conjugate, denoted by X Y, if there is a matrix Q SP 2n (Z) such that Y = Q 1 XQ. Two subgroups G, H of SP 2n (Z) are said to be conjugate, denoted by G H, if there is a matrix Q SP 2n (Z) such that H = Q 1 GQ = {Q 1 XQ X G}. If we fix a canonical basis of H 1 (S), there is a natural group monomorphism Ψ : Aut(S) SP 2g (Z), see Farkas and Kra [4, p. 286]. Clearly, for any given subgroup G of Aut(S), the groups Ψ(G) with respect to different canonical basis are conjugate in SP 2g (Z). Definition 1.2. A subgroup H of SP 2g (Z) is said to be realizable if there is subgroup G of Aut(S) for some Riemann surface S of genus g such that Ψ(G) = H with respect to some canonical basis of H 1 (S). A question naturally arises. Which subgroups H of SP 2g (Z) can be realized? The case of a cyclic group of order p of SP p 1 (Z), where p 5 is an odd prime, was solved by Sjerve and Yang. They gave a complete list of realizable conjugacy classes of p-torsion in SP p 1 (Z), see [6, p. 583]. We have solved the problem for cyclic subgroups of SP 4 (Z), see [9, p. 2345]. In this paper we address the question of which classes of dihedral subgroups D n of SP 4 (Z) can be realized by a dihedral group action on some Riemann surface of genus 2. To explain our results we need to develop some notation. The symplectic direct sum is defined by A [ ] [ ] 1 0 B 1 0 A1 C A2 B 2 = 0 A 2 0 B 2 C 1 D 1 C 2 D 2 C 1 0 D 1 0, 0 C 2 0 D 2 where A i,b i,c i,d i are n i n i matrices, for i = 1,2. The quasi-direct sum of two matrices is 0 C [ ] [ ] 11 0 C 12 C11 C 12 D11 D 12 = D 11 0 D 12 0 C 21 C 22 D 21 D 22 0 C 21 0 C 22, D 21 0 D 22 0

4 Dihedral Groups of Automorphisms 555 where C ij M n1 n 2 (Z), D ij M n2 n 1 (Z). For properties about the symplectic direct sum and the quasi-direct sum see [8, p. 182]. We denote by D n (A,B) = A,B A n = B 2 = (AB) 2 = I any dihedral subgroup of order 2n of SP 4 (Z) generated by A,B with the relation A n = B 2 = (AB) 2 = I. Main Theorem. A complete list of realizable classes of dihedral subgroups in SP 4 (Z) is as follows: where I = I 4 or I 2, J = J 2, W = Order G Realizable Classes 2 D 1 = Z 2 I, I I 4 D 2 I I, I 6 D 3 W W,J J 8 D 4 J J,J J 12 D 6 (W W ),J J [ ] A complete list of representatives of conjugacy classes of torsion elements in SP 4 (Z) is already known, see Yang [8, p. 181], and each torsion element has order n = 1,2,3,4,5,6,8, 10,12. In Section 2, we will give a list of conjugacy classes of dihedral subgroups in SP 4 (Z). Then we can easily prove Main Theorem in Section Dihedral subgroups in SP 4 (Z). Wewanttocharacterizedihedralsubgroups of order 2n of SP 4 (Z). Note that D n (A,B) = A,B A n = B 2 = I,AB = BA 1. We see that X X 1 for any element X in D n. But for n = 5,8,10,12, any element of order n in SP 4 (Z) is not conjugate to its inverse, according to [7, p. 622] or [8, p. 182]. So, we have Lemma 2.1. There is no dihedral subgroup of order 10,16,20,24 in SP 4 (Z). We only need consider the cases of n = 2,3,4,6. Also, in [8, p. 181], a list of representative of the conjugacy classes of torsion in SP 4 (Z), which is conjugate to its

5 556 Q. Yang and D. Yang inverse, is given in following table: Order Conjugacy Classes 2 I, I ( I), I I 3 W W 4 J J, J J 6 (W W ) Note that if A A 1, then A,B A 1,B 1 for some B 1. Every dihedral subgroup in SP 4 (Z) must be conjugate to the dihedral subgroup with one generator in the table. Without loss of generality, we first choose A to be one of the matrices in the table and then determine possible involution B such that A,B, the group generated by A and B, is a dihedral group. Lemma 2.2. Let B SP 4 (Z) be a non-trivial involution. Then B is of the form (2.1) B = where ab cd+x 2 = 1. x a 0 c b x c 0 0 d x b d 0 a x Proof. We see that B J = JB from B JB = J and B 2 = I. To determine B, we need to solve the matrix equation X J = JX for X under the condition X 2 = I. Then all solutions are in the form X = where a,b,c,d,x,y are integers, with Thus, x 2 = y 2 or y = ±x. x a 0 c b y c 0 0 d x b, d 0 a y, a(x+y) = b(x+y) = c(x+y) = d(x+y) = 0, x 2 +ab cd = 1, y 2 +ab cd = 1. If x = y 0, then x + y = 2x 0, and we obtain a = b = c = d = 0 and x = y = ±1, B = ±I. If x = y, the conditions become ab cd+x 2 = 1 and y = x. We complete the proof.

6 Dihedral Groups of Automorphisms 557 Now we will determine the dihedral subgroups for each choice of A in the table of torsion classes by solving the relation AB = BA 1 or equivalent ABA = B for B, where B = I or B is of the form (2.1) in Lemma 2.2. It is not a tough work since the problem is to solve a system of linear equations. Lemma 2.3. A complete list of conjugacy classes of dihedral subgroups of order greater than or equal to 4 in SP 4 (Z) is given in following table: Order D n Conjugacy Classes 4 D 2 I ( I), I, I I, I 6 D 3 W W,J J 8 D 4 J J,J J, J J,I ( I) 12 D 6 (W W ),J J Proof. Let n = 2. Then AB = BA. Since the trivial involution I commutes every matrix and is only conjugate to itself, at least one element of D 2 (A,B) must be conjugate to I ( I) or I I. If A = I ( I), by Lemma 2.2, B = I or B = ±[I ( I)]. There is only one dihedral group for this case, D 2 (I ( I), I) = {±I,±[I ( I)]}. If A = I I, similarly, D 2 (I I, I) = {±I,±(I I)}. Let n = 3. Assume that A = W W. The matrix equation (W W )B(W W ) = B has six solutions for involution B : J J, J J, ±(V V 1 ), ±(V V 1 ), [ ] 1 0 where V = = WJ, and hence there are two different dihedral groups of order for this case: D 3 (W W,J J ) = {I,W W,W 2 W 2,J J,V V 1, (V V 1 )}, D 3 (W W,J J) = {I,W W,W 2 W 2,J J, (V V 1 ),V V 1 }. It is easy to check, by Q = J J, that D 3 (W W,J J ) D 3 (W W,J J).

7 558 Q. Yang and D. Yang Let n = 4. If A = J J, then the involutions B = ±(I I) or B = ±(J J ). There is one dihedral subgroup of order 8, D 4 (J J,J J ) = {±I,±(J J ),±(I I),±(J J )}. If A = J J, then B = ±[I ( I)] or B = ±(J J ). There is another dihedral subgroup of order 8, D 4 (J J,I ( I)) = {±I,±[J J],±[I ( I)],±(J J )}. These two subgroups are not conjugate to each other. Let n = 6. Let also A = (W W ). The same as the case of n = 3, we have D 6 ( (W W ),J J ) = {±I,±(W W ),±(W 2 W 2 ),±(J J ),±(V V 1 ),±(V V 1 )}. Therefore, we have proved the lemma. 3. Proof of Main Theorem. It is clear that the dihedral subgroups D 2 (I ( I), I) and D 4 (J J,I ( I)) are not realizable, due to the fact that I ( I) and J J are not realizable, see [9, p. 2345]. According to Bolza [2, p. 49], Kuribayashi [5, p. 25] or Breuer [3, p. 179], there are group actions of D 2,D 3,D 4,D 6 on some Riemann surfaces of genus two. Thus, every class in the table in Lemma 2.3 is realizable except D 2 (I ( I), I) and D 4 (J J,I ( I)). We complete the proof of Main Theorem. Remark 1. In fact, one can use the Riemann Surfaces represented by the normalized equations y 2 = (x 2 1)(x 2 a)(x 2 1/a) and y 2 = x 6 +1 with their fundamental octagons or dodecagons to deduce the realized dihedral subgroups D 4 or D 6 of SP 4 (Z), see [1, p. 6]. Acknowledgment. We are grateful to the referee for valuable comments and suggestion.

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