Hoshino, Kenji

Nakamura, Hiroaki

Okayama University Scientific Achievement Repository

Mathematical Journal of OkayamaUniversityVolume 52, Issue 1 2010 Article 4JANUARY 2010BELYI FUNCTION ON X0(49) OF DEGREE7Kenji Hoshino∗ Hiroaki Nakamura†∗Department of Mathematics, Faculty of Science, Okayama University,khoshino@math.okayama-u.ac.jp†Department of Mathematics, Faculty of Science, Okayama University, h-naka@math.okayama-u.ac.jpCopyright c©2010 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouMath. J. Okayama Univ. 52 (2010), 61–63BELYI FUNCTION ON X0(49) OF DEGREE 7Appendix to: “The Belyi functions and dessin d’enfants corresponding tothe non-normal inclusions of triangle groups” by K.HoshinoKenji HOSHINO and Hiroaki NAKAMURAIn [H] §4, we identified two types of the non-normal inclusions of trianglegroups (type A and type C from Singerman’s list [S]) as those correspondingto subcovers of the Klein quartic. N. D. Elkies [E] closely studied thosesubcovers in view of modular curves, i.e., as subcovers under the ellipticmodular curveX7 = X(7) identified with the Klein quartic defined byX3Y+Y 3Z + Z3Y = 0. Let C(X7) be the function field of X7 generated byy := Y/X and z := Z/X with a relation y + y3z + z3 = 0, and consider twoautomorphisms of C(X7) defined by(A1) σ :{y 7→ ζ3y,z 7→ ζz,τ :{y 7→ 1/z,z 7→ y/z,where ζ := e2pii/7. Then, σ and τ are automorphisms respectively of order7 and 3, and they form an automorphism group H of C(X7) of order 21,a subgroup of the full automorphism group G of order 168 of the Kleinquartic. In this respect, the genus zero covers of type A and type C of [S]are respectivelyX1(7)→ X(1) and X0(7)→ X(1) arising from the inclusionrelations of 〈σ〉 ⊂ H ⊂ G. Remarkably, Elkies [E] studied deep arithmeticproperties of a genus one subcover E fixed by 〈τ〉 with showing that E isQ-isomorphic to X0(49). Especially, he explicitly presented its function fieldC(E) as C(E) = C(u, v) with v2 = 4u3 + 21u2 + 28u, where(A2) u = −(y + z + yz)2(1 + y + z)yz, v = −(2− y − z + 2y2 − yz + 2z2)(y + z + yz)yz(1 + y + z)Recalling also from [E] that standard coordinates ofX1(7) ∼= P1t andX0(7)∼=P1s may be given ast := −y2z, s := t+11− t+t− 1t,we would like to interpret the degree 7 cover E → X0(7) arising from 〈τ〉 ⊂H by expressing s by u, v explicitly. In this note, we showProposition A. Notations being as above, the covering of P1s by the ellipticcurve E : v2 = 4u3+21u2+28u is ramified only above s = 3ρ, 3ρ−1,∞ (whereρ = e2pii/6), and the equation is given bys =12{(u2 + 7u+ 7)v + (7u3 + 35u2 + 49u+ 16)}.611Hoshino and Nakamura: BELYI FUNCTION ON X<sub>0</sub>(49) OF DEGREE 7Produced by The Berkeley Electronic Press, 201062 KENJI HOSHINO AND HIROAKI NAKAMURAThus β = s−3ρ3ρ−1−3ρ gives a Belyi function (i.e., unramified outside β =0, 1,∞) of degree 7 on E with valency list [331, 331, 7].In effect, one can induce an isomorphism of coversX0(49)∼−−−−→ Ey yX0(7)∼−−−−→ X0(7)from the conjugacy by the ‘Fricke involution’± 1√7(07−10 ) between the modulargroup Γ0(49) and {±(acbd) ∈ PSL2(Z) : b ≡ c ≡ 0 mod 7} in PSL2(R) (cf.[E] p.90). Therefore, the computation of E → X0(7) may be reduced tocombining classically well known equations that relate X0(7), X0(49) withthe J-line X(1) found in, e.g., [F] pp.395–403. Here, however, we shallemploy an alternative enjoyable discussion following the Elkies scheme:X1(7) = P1tf←−−−−〈σ〉X7gy 〈τ〉yqX0(7) = P1sp←−−−− E.First of all, since dsdt =(t2−t+1)2(t−1)2t2 , the 3-cyclic cover g is ramified only overs = 3ρ, 3ρ−1 at t = ρ, ρ−1, and the fiber over s = ∞ is formed by thethree points t = 0, 1,∞. We next chase the fibers of f over these pointsand their images in E by q. Noticing that C(X7) = C(y, z) = C(y, t) withy7 = t31−t , we see that f is totally ramified over t = 0, 1,∞, and their imagesby q coincide at the infinity point on E : v2 = 4u3 + 21u2 + 28u. Fromthis follows that p : E → P1s is totally ramified at the infinity point of Eover s = ∞, hence s is of the form s = F (u) + G(u)v with F,G ∈ C[u],deg(F ) = 3, deg(G) = 2.The fiber of f over t = ρ forms one orbit under the action of σ (y 7→ζy) whose points are represented by the set of their y-coordinates Sρ :={ξ2, ξ5, ξ8, ξ11, ξ14, ξ17, ξ20}, where ξ := e2pii/21. The action of τ preservest = ρ and transforms those y-coordinates (over t = ρ) as y 7→ y2ξ−14, hencedecomposes Sρ into the three τ -orbits S1ρ := {ξ2, ξ11, ξ8}, S2ρ := {ξ5, ξ17, ξ20},and S3ρ := {ξ14 = ρ4} (This also explains the branch type of p over s = 3ρ is‘331’). Then we compute the u-coordinates u1, u2, u3 of the images of theseorbits by q after the formula (A2); it turns out that u1 = −ξ2− ξ11− ξ8−1,u2 = −ξ7 + ξ2 + ξ11 + ξ8 − 2 and u3 = 3ρ− 1. Eliminating v from F (u) +2Mathematical Journal of Okayama University, Vol. 52 [2010], Iss. 1, Art. 4http://escholarship.lib.okayama-u.ac.jp/mjou/vol52/iss1/4BELYI FUNCTION ON X0(49) OF DEGREE 7 63G(u)v = 3ρ, we should then have an equation of the form(4u3 + 21u2 + 28u)G(u)2 − (F (u)− 3ρ)2 = (u− u1)3(u− u2)3(u− u3).Using a symbolic computation software such as MAPLE to compare thecoefficients of the both sides above (plus a slight more consideration of signs),we conclude F (u) = 12(7u3 + 35u2 + 49u + 16), G(u) = 12(u2 + 7u + 7) asstated in Proposition A. Remark. The monodromy representation associated with the aboveBelyi function β is given by x = (142)(356)(7), y = (175)(346)(2), z =(1234567). The following picture illustrates the uniformization of this Groth-endieck dessin.References[E] N. D. Elkies, The Klein Quartic in Number Theory, in “The Eightfold Way”, MSRIPubl. 35 (1998), 51–101.[F] R. Fricke, Die Elliptischen Funktionen und Ihre Anwendungen, Zweiter Teil, Teub-ner 1922.[H] K. Hoshino, The Belyi functions and dessin d’enfants corresponding to the non-normal inclusions of triangle groups, Preprint 2008.[S] D. Singerman, Finitely maximal Fuchsian group, J. London Math. Soc. (2) 6 (1972),29–38.Department of Mathematics, Faculty of Science, Okayama University,Okayama 700-8530, Japane-mail address: khoshino@math.okayama-u.ac.jp, h-naka@math.okayama-u.ac.jp(Received May 12, 2008 )3Hoshino and Nakamura: BELYI FUNCTION ON X<sub>0</sub>(49) OF DEGREE 7Produced by The Berkeley Electronic Press, 2010

BELYI FUNCTION ON X<sub>0</sub>(49) OF DEGREE 7

Abstract

Similar works

Full text

Available Versions

Okayama University Scientific Achievement Repository