EQUATIONS OF THE GENUS 6 CURVE WITH AUTOMORPHISM GROUP G = 72, 15 DAVID SWINARSKI The group G = 72, 15 is the automorphism group of a genus 6 curve X. One notable feature of this curve is that the representation of Aut(X) on H 0 (X, K) is irreducible [1, App. B]. We find equations for this curve using representation theory, algebraic geometry, and two lucky guesses. Proposition 0.1 (Swinarski, 2010) The genus 6 curve with automorphism group 72, 15 is trigonal, and its canonical ideal is generated by the following six quadrics and three cubics in the polynomial ring S = Q[ζ 9 ][a, b, c, d, e, f]: ad + be + cf a 2 + ζ9 b2 + ζ9 2 c2 d 2 + ζ 9 e 2 + ζ9 2f 2 ae ζ3 2bd bf ζ9 2ζ2 3 ce cd ζ9 2af ζ3 2ac2 + de 2 + ζ9 2 (ζ2 3 ab2 + df 2 ) a 2 b + ef 2 + ζ9 2 (ζ2 3 (bc2 + d 2 e)) b 2 c + ζ3 2d2 + ζ9 2 (a2 c + ζ3 2e2 f) 1. The action of Aut(X) on differentials We have X/ Aut(X) = P 1, and the quotient morphism is branched over three points with ramification signature (2, 4, 9). One may then use the Eichler trace formula or the Chevalley-Weil formula to compute the character of the Aut(X) action on H 0 (X, K), and then compute matrix generators for a representation with this character. Breuer gives such generators: A := 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 ζ 3 0 1 0 0 0 0 1 0 0 0 0 0 0 0 ζ3 2 0 0 0, B := 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 The matrices A, B, and C := (AB) 1 form a set of surface kernel generators for the G-action. Recall from [1] that this means given the ramification signature (2, 4, 9), we seek elements A, B, C G such that A, B, C generate G and A 2 = B 4 = C 9 = ABC = Id. In the sequel we will adopt Magma s ordering of the conjugacy classes of G and the Magma character table for G. The group G has nine conjugacy classes. Representatives of the nine classes, using Magma s ordering of the classes, are Id, B 2, A, C 3, B, C 3 B 2, C 4, C, C 2. Here is the Magma character table for G: Date: November 26, 2010. 1
2 DAVID SWINARSKI ----------------------------------------- Class 1 2 3 4 5 6 7 8 9 Size 1 3 18 2 18 6 8 8 8 Order 1 2 2 3 4 6 9 9 9 ----------------------------------------- p = 2 1 1 1 4 2 4 8 9 7 p = 3 1 2 3 1 5 2 4 4 4 ----------------------------------------- X.1 + 1 1 1 1 1 1 1 1 1 X.2 + 1 1-1 1-1 1 1 1 1 X.3 + 2 2 0 2 0 2-1 -1-1 X.4 + 2 2 0-1 0-1 Z1 Z1#2 Z1#4 X.5 + 2 2 0-1 0-1 Z1#2 Z1#4 Z1 X.6 + 2 2 0-1 0-1 Z1#4 Z1 Z1#2 X.7 + 3-1 1 3-1 -1 0 0 0 X.8 + 3-1 -1 3 1-1 0 0 0 X.9 + 6-2 0-3 0 1 0 0 0 Here, Z1 denotes ζ 4 9 + ζ5 9, where ζ 9 := e 2πi 9. The symbol # denotes algebraic conjugation ; in particular, Z1#2 = (ζ 4 9 )2 + (ζ 5 9 )2 = ζ 9 + ζ 9 1 and Z1#4 = ζ 2 + ζ 9 2. 2. The Chevalley-Weil Formula Theorem 2.1 (Chevalley Weil, 1934 [2,3]) Let V i be an irreducible G-module. If m 1 then the multiplicity of V i in H 0 (C, mk) is δ + (2m 1)(g 1) dim V i + e Q Q SKG α=0 ( (m 1) ( 1 1 ) + e Q m 1 α e Q ) n α,q,vi Here x = x x is the fractional part of x, and δ = if m = 1 and V i is the trivial representation, and 0 otherwise. The first summation is over the set of surface kernel generators, e(q) is the order of Q, and n α,q,vi eigenvalue for the surface kernel generator Q. By Noether s Theorem, the sequence is the number of times the root of unity e 2πiα e Q appears as an 0 I m Sym m H 0 (C, K) H 0 (C, mk) 0 is exact for all m 1, so we can use the Chevalley-Weil formula to determine the multiplicities of the irreducibles V i in I m for any m. From the previous section, we have all the ingredients we need (surface kernel generators plus the character table) to apply the Chevalley-Weil formula. Notation: Let G be a finite group, V a G-module, and V 1,..., V k a fixed ordering of the irreducible G-modules. We write V = (a 1,..., a k ) if V = V a 1 1 V a k k as G-modules (i.e. the vectors we write are vectors of multiplicities, not values of characters).
EQUATIONS OF THE GENUS 6 CURVE WITH AUTOMORPHISM GROUP G = 72, 15 3 Then Chevalley-Weil yields S 2 = (1, 0, 1, 1, 1, 1, 2, 0, 1) H 0 (X, 2K) = (0, 0, 1, 0, 1, 1, 1, 0, 1) I 2 = (1, 0, 0, 1, 0, 0, 1, 0, 0) S 3 = (0, 0, 1, 1, 1, 1, 3, 3, 5) H 0 (X, 3K) = (0, 0, 1, 0, 0, 1, 1, 2, 2) I 3 = (0, 0, 0, 1, 1, 0, 2, 1, 3) Gonality. The resolution of the canonical ideal of X depends on whether X is hyperelliptic, trigonal, a smooth plane quintic, or general. The center of G is trivial, so X is not hyperelliptic. We know that I 2 is generated by 6 quadrics. If X is general, then I is generated by I 2. If X is trigonal or a plane quintic, then I is generated by I 2 plus 3 cubics; in these cases, I 2 generates a scroll or the Veronese surface, respectively. Compare the multiplicities of the irreducible G-modules in I 2 S 1 and I 3 : I 2 S 1 = (0, 0, 0, 1, 1, 1, 1, 1, 4) I 3 = (0, 0, 0, 1, 1, 0, 2, 1, 3) We see that I 3 contains 2 copies of V 7, and hence I 3 I 2 S 1. Thus, I is not generated by quadrics, so X must either be trigonal or a plane quintic. (We will see below that it is trigonal.) Moreover, the cubic generators needed to define X (in addition to the quadrics) form a V 7. 3. Polynomials We decompose the action of G on S 2 and find that Span{ad + be + cf} = V 1 Span{a 2 + ζ9 + ζ9 2, d 2 + ζ 9 e 2 + ζ9f 2 2 } = V 4 Span{ae, af, bd, bf, cd, ce} = V7 2 Thus, we know that I 2 consists of ad + be + cf, a 2 + ζ9 b2 + ζ9 2 c2, and d 2 + ζ 9 e 2 + ζ9 2f 2, and a three-dimensional subspace of Span{ae, af, bd, bf, cd, ce}. How can we choose a three-dimensional subspace that will work? First, we use Magma to compute the endomorphism ring of Span{ae, af, bd, bf, cd, ce} and the primitive idempotents. This yields the decomposition Span{ae, af, bd, bf, cd, ce} = Span{ae, bf, cd} Span{bd, ce, af}. We choose the basis {ζ3 2bd, ζ2 3ce, af} of Span{bd, ce, af} so that G acts by the same matrices on Span{ae, bf, cd} and Span{ζ3 2bd, ζ2 3ce, af}:
4 DAVID SWINARSKI A action: ae bf cd ζ3 2bd ζ2 3ce af ae 1 bf ζ3 2 cd ζ 3 ζ3 2bd 1 ζ3 2ce ζ2 3 af ζ 3 B action: ae bf cd ζ3 2bd ζ2 3ce af ae bf 1 cd ζ3 2bd ζ3 2ce 1 af Thus, the remaining quadrics we seek are of the form λ 1 ae + λ 2 ζ 2 3 bd, λ 1bf + λ 2 ζ 2 3 ce, λ 1cd + λ 2 af for some λ 1, λ 2 C. Moreover, to avoid monomial generators in I (which would almost certainly give a singular scheme in P 5 ) we may assume that λ 1 and λ 2 are nonzero, and divide by λ 1 to get the quadrics ae + λζ 2 3 bd, bf + λζ2 3 ce, cd + λaf for some λ C. I experimented a little with obvious values of λ (e.g. λ = 1, z) and found that for these values, the scheme Y defined by the six quadrics ad + be + cf a 2 + ζ9 b2 + ζ 2 d 2 + ζ 9 e 2 + ζ9 2f 2 ae + λζ3 2bd bf + λζ3 2ce cd + λaf is the disjoint union of two rational curves C 1 and C 2, where C 1 is defined in Span{a, b, c} by a 2 + ζ9 b2 + ζ9 2 c2 and C 2 is defined in Span{d, e, f} by d 2 + ζ 9 e 2 + ζ9 2f 2. This led me to make my first guess: Guess 3.1 The curve X is trigonal and lies on the scroll which is the join of the two rational curves C 1 and C 2. Assume that Guess 3.1 is correct. Then to find the correct value of λ which defines a scroll, we can consider the line L between any two points p C 1 and q C 2 and solve for λ so that the quadrics in I vanish on L. I chose the points p = [1 : 0 : ζ 9 i : 0 : 0 : 0] and q = [0 : 0 : 0 : iζ 9 : 0 : 1]. Then a parametrization of L is given by [1 : 0 : ζ 9 i : iζ 9 t : 0 : t]. Then ae + λbd = 0, bf + λce = 0, and ad + be + cf = 0. But to have cd + λaf = 0, we find that we must have λ = ζ9 2. I then checked that the quadrics define a smooth scroll in P 5, as expected. 9 c2 ad + be + cf a 2 + ζ9 b2 + ζ 2 d 2 + ζ 9 e 2 + ζ9 2f 2 ae ζ3 2bd bf ζ9 2ζ2 3 ce cd ζ9 2af 9 c2
EQUATIONS OF THE GENUS 6 CURVE WITH AUTOMORPHISM GROUP G = 72, 15 5 Next I decomposed the action of G on S 3. Recall that we found above that the extra cubic generators we need form a V 7. In S 3, we have V 3 7 This splits into 3 blocks quite naturally: = Span{a 3 + d 3, b 3 + e 3, c 3 + f 3, a 2 b + ef 2, ab 2 + ζ 2 3 df 2, ζ 3 e 2 f + a 2 c, ac 2 + ζ 2 3 de2, ζ 3 d 2 f + b 2 c, bc 2 + d 2 e}. A action: a 3 + d 3 b 3 + e 3 c 3 + f 3 ζ 2 3 ac2 + de 2 a 2 b + ef 2 b 2 c + ζ 2 3 d2 f ζ 2 3 ab2 + df 2 ζ 2 3 (bc2 + d 2 e) a 2 c + ζ 2 3 e2 f a 3 + d 3 1 b 3 + e 3 1 c 3 + f 3 1 ζ 2 3 ac2 + de 2 1 a 2 b + ef 2 1 b 2 c + ζ 2 3 d2 f 1 ζ 2 3 ab2 + df 2 1 ζ 2 3 (bc2 + d 2 e) 1 a 2 c + ζ 2 3 e2 f 1 B action: a 3 + d 3 b 3 + e 3 c 3 + f 3 ζ 2 3 ac2 + de 2 a 2 b + ef 2 b 2 c + ζ 2 3 d2 f ζ 2 3 ab2 + df 2 ζ 2 3 (bc2 + d 2 e) a 2 c + ζ 2 3 e2 f a 3 + d 3 1 b 3 + e 3 c 3 + f 3 ζ 2 3 ac2 + de 2 1 a 2 b + ef 2 b 2 c + ζ 2 3 d2 f ζ 2 3 ab2 + df 2 1 ζ 2 3 (bc2 + d 2 e) a 2 c + ζ 2 3 e2 f Thus, the three cubic generators we seek are of the form for some µ 1, µ 2, µ 3 C. At this point, I made my second guess: Guess 3.2 µ 1, µ 2, µ 3 {0, ±ζ k 9 } µ 1 (a 3 + d 3 ) + µ 2 (ζ 2 3 ac2 + de 2 ) + µ 3 (ζ 2 3 ab2 + df 2 ) µ 1 (b 3 + e 3 ) + µ 2 (a 2 b + ef 2 ) + µ 3 (ζ 2 3 (bc2 + d 2 e)) µ 1 (c 3 + f 3 ) + µ 2 (b 2 c + ζ 2 3 d2 ) + µ 3 (a 2 c + ζ 2 3 e2 f) I used Magma to scan through all the schemes defined by the quadrics plus the cubics above for all possible choices of (µ 1, µ 2, µ 3 ) drawn from the set {0, ±ζ9 k }. Most such triples define schemes of dimension 0, but some such triples define schemes of dimension 1 or 2. I selected one of triples the triples that defined a curve, (µ 1, µ 2, µ 3 ) = (0, 1, ζ9 2 ), and checked that the resulting scheme was nonsingular and had the correct automorphisms. Since G is superlarge (i.e. #G > 12(g 1)) the curve cannot have any additional automorphisms; hence, we have found the curve we seek, and it is trigonal. Magma V2.16-13 Sat Nov 27 2010 21:19:43 on dopey [Seed = 1909227796] Type? for help. Type <Ctrl>-D to quit. > K<z>:=CyclotomicField(9); > w:=z^3; > GL6K:=GeneralLinearGroup(6,K); > A:=elt<GL6K 0,0,0,0,1,0, 0,0,0,1,0,0, 0,0,0,0,0,w, 0,1,0,0,0,0, 1,0,0,0,0,\ 0, 0,0,w^2,0,0,0>; > B:=elt<GL6K 0,0,0,0,0,1, 0,0,0,0,-1,0, 0,0,0,-1,0,0, 0,0,1,0,0,0, 0,-1,0,0\,0,0, -1,0,0,0,0,0>; > G:=sub<GL6K A,B>; > IdentifyGroup(G); <72, 15> > P5<a,b,c,d,e,f>:=ProjectiveSpace(K,5);
6 DAVID SWINARSKI > mu1:=0; > mu2:=1; > mu3:=-z^-2; > X:=Scheme(P5,[a*d+b*e+c*f, a^2+z^-1*b^2+z^-2*c^2, d^2+z*e^2+z^2*f^2, > a*e-w^2*z^2*b*d, b*f-w^2*z^2*c*e, c*d-z^2*a*f, mu1*(a^3+d^3) +mu2*( > w^2*a*c^2+d*e^2) + mu3*(d*f^2+w^2*a*b^2), mu1*(b^3+e^3) + > mu2*(a^2*b+e*f^2) + mu3*(w^2*b*c^2+w^2*d^2*e), mu1*(c^3+f^3)+ mu2*(b^2*c+w^2\ *d^2*f)+ mu3*(a^2*c+w^2*e^2*f)]); > Dimension(X); 1 > X:=Curve(X); > IsSingular(X); false > Genus(X); 6 > Automorphism(X,A); Mapping from: Crv: X to Crv: X with equations : e d (-z^3-1)*f b a z^3*c and inverse e d (-z^3-1)*f b a z^3*c > Automorphism(X,B); Mapping from: Crv: X to Crv: X with equations : -f -e d -c -b a and inverse f -e -d c -b -a
EQUATIONS OF THE GENUS 6 CURVE WITH AUTOMORPHISM GROUP G = 72, 15 7 References [1] Thomas Breuer, Characters and automorphism groups of compact Riemann surfaces, London Mathematical Society Lecture Note Series, vol. 280, Cambridge University Press, Cambridge, 2000. MR1796706 (2002i:14034) 1 [2] C. Chevalley and A. Weil, Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers, Abh. Math. Sem. Univ. Hamburg 10 (1934), 358 361. 2 [3] A. Weil, Über Matrizenringe auf Riemannschen Flächen und den Riemann-Rochsen Satz, Abh. Math. Sem. Univ. Hamburg 11 (1935), 110 115. 2 Software Packages Referenced [4] School of Mathematics and Statistics Computational Algebra Research Group University of Sydney, MAGMA computational algebra system (2008), available at http://magma.maths.usyd.edu.au/magma/. Version 2.15-1.