Some genus two quotients of X_{1}(N)

William A. Stein

June 21, 2000

Abstract The author has written a program to compute in S_{2}(Γ_{1}(N)). As an application, we use the method described clearly in Chapter 4 of [1] to compute models for genus two quotients of X_{1}(N) by Atkin-Lehner involutions. I don’t know if somebody else has already made a table like this, but if they did they may have made mistakes so it is good to recompute it.

Introduction

The following is a model for X_{1}(13):

y^{2} = x^{6} + 2x^{5} + x^{4} + 2x^{3} + 6x^{2} + 4x + 1.

1 Spaces of cusp forms

Observe that

S_{2}(Γ_{1}(N)) = ⨁_{χ:(Z/NZ)^{∗}→C^{∗}} S_{2}(N, χ),

where S_{2}(N, χ) is the space of cusp forms on which the diamond operators ⟨d⟩ act via the character χ. For efficiency purposes, we compute only the spaces S_{2}(N, χ).

Notationaly, we represent a character χ is represented as follows, with appropriate modifications at 2. Write (Z/NZ)^{∗} as a product of cyclic groups corresponding to the prime powers dividing N. Let ε_{1}, …, ε_{n} be characters that send minimal generators of these cyclic factors to appropriate roots of unity in C^{∗}. We write characters χ in terms of the ε_{i}.

The dimension of S_{2}(Γ_{1}(N)) equals the genus of X_{1}(N), which is a number that is recorded in Table 1 for N ≤ 100.

2 Table of Genus 2 Curves

I do not know if somebody else has already made a table like this, but if they did they might have made mistakes so it is good that I have recomputed it.

Table 1: The genus of X_{1}(N)

\begin{matrix}N & g \\ 1 & 0 \\ 2 & 0 \\ 3 & 0 \\ 4 & 0 \\ 5 & 0 \\ 6 & 0 \\ 7 & 0 \\ 8 & 0 \\ 9 & 0 \\ 10 & 0 \\ 11 & 1 \\ 12 & 0 \\ 13 & 2 \\ 14 & 1 \\ 15 & 1 \\ 16 & 2 \\ 17 & 5 \\ 18 & 2 \\ 19 & 7 \\ 20 & 3 \end{matrix}  \begin{matrix}N & g \\ 21 & 5 \\ 22 & 6 \\ 23 & 12 \\ 24 & 5 \\ 25 & 12 \\ 26 & 10 \\ 27 & 13 \\ 28 & 10 \\ 29 & 22 \\ 30 & 9 \\ 31 & 26 \\ 32 & 17 \\ 33 & 21 \\ 34 & 21 \\ 35 & 25 \\ 36 & 17 \\ 37 & 40 \\ 38 & 28 \\ 39 & 33 \\ 40 & 25 \end{matrix}  \begin{matrix}N & g \\ 41 & 51 \\ 42 & 25 \\ 43 & 57 \\ 44 & 36 \\ 45 & 41 \\ 46 & 45 \\ 47 & 70 \\ 48 & 37 \\ 49 & 69 \\ 50 & 48 \\ 51 & 65 \\ 52 & 55 \\ 53 & 92 \\ 54 & 52 \\ 55 & 81 \\ 56 & 61 \\ 57 & 85 \\ 58 & 78 \\ 59 & 117 \\ 60 & 57 \end{matrix}  \begin{matrix}N & g \\ 61 & 126 \\ 62 & 91 \\ 63 & 97 \\ 64 & 93 \\ 65 & 121 \\ 66 & 81 \\ 67 & 155 \\ 68 & 105 \\ 69 & 133 \\ 70 & 97 \\ 71 & 176 \\ 72 & 97 \\ 73 & 187 \\ 74 & 136 \\ 75 & 145 \\ 76 & 136 \\ 77 & 181 \\ 78 & 121 \\ 79 & 222 \\ 80 & 137 \end{matrix}  \begin{matrix}N & g \\ 81 & 190 \\ 82 & 171 \\ 83 & 247 \\ 84 & 133 \\ 85 & 225 \\ 86 & 190 \\ 87 & 225 \\ 88 & 181 \\ 89 & 287 \\ 90 & 153 \\ 91 & 265 \\ 92 & 210 \\ 93 & 261 \\ 94 & 231 \\ 95 & 289 \\ 96 & 193 \\ 97 & 345 \\ 98 & 235 \\ 99 & 281 \\ 100 & 231 \end{matrix}

Table 2: Genus 2 quotients y^{2} = f(x) of X_{1}(N) that correspond to a single newform f(q), I searched up to level 125 and this is all I found!!

X, χ  F (x), f(q)  L(J, 1) ≠ 0?
X_{1}(13)  x^{6} + 2x^{5} + x^{4} + 2x^{3} + 6x^{2} + 4x + 1  yes

X_{1}(16)  x^{6} − 2x^{5} − x^{4} − x^{2} + 2x + 1  yes
= (x − 1)(x + 1)(x^{2} − 2x − 1)(x^{2} + 1)

X_{1}(18)  x^{6} + 4x^{5} + 10x^{4} + 10x^{3} + 5x^{2} + 2x + 1  yes

X_{1}(45)/W, ε_{2}^{2} x^{6} + 22x^{3} + 125  yes
= (x^{2} − 2x + 5)(x^{4} + 2x^{3} − x^{2} + 10x + 25),
q + \sqrt{−5}q^{2} − 3q^{4} − \sqrt{−5}q^{5} + ⋯

References

[1] S. D. Galbraith, Equations for modular curves, Oxford Ph.D. thesis (1996), 91 pp.