CoCalc Public Fileswww / papers / modelsX1N / modelsX1N.tex
Author: William A. Stein
1\documentclass{article}
2\include{macros}
3\title{Some genus two quotients of $X_1(N)$}
4\author{William A. Stein}
5\date{June 21, 2000}
6\bibliographystyle{amsplain}
7
8\begin{document}
9\maketitle
10\begin{abstract}
11The author has written a program to compute in $S_2(\Gamma_1(N))$.
12As an application, we use the method described clearly in
13Chapter~4 of \cite{galbraith} to compute models for
14genus two quotients of $X_1(N)$ by Atkin-Lehner involutions.
15I don't know if somebody else has already made a table like this,
16but if they did they may have made mistakes so it is good to recompute it.
17\end{abstract}
18
19\section*{Introduction}
20The following is a model for $X_1(13)$:
21    $$y^2 = x^6 + 2x^5 + x^4 + 2x^3 + 6x^2 + 4x + 1 .$$
22
23\comment{
24It was found using the \hecke{} package in \magma{}, as follows.
25\begin{verbatim}
26> G<eps>:=DirichletGroup(13,CyclotomicField(12));
27> A:=CS(MS(eps^2,2));
28> B:=CS(MS(eps^10,2));   // The Galois conjugate...
29> qEigenform(A,13);
30q + (-zeta_12^2 - 1)*q^2 + (2*zeta_12^2 - 2)*q^3 + zeta_12^2*q^4
31    + (-2*zeta_12^2 + 1)*q^5 + (-2*zeta_12^2 + 4)*q^6 +
32    (2*zeta_12^2 - 1)*q^8 - zeta_12^2*q^9 + (3*zeta_12^2 -
33    3)*q^10 - 2*q^12 + O(q^13)
34> qEigenform(B,13);
35q + (zeta_12^2 - 2)*q^2 - 2*zeta_12^2*q^3 + (-zeta_12^2 + 1)*q^4
36    + (2*zeta_12^2 - 1)*q^5 + (2*zeta_12^2 + 2)*q^6 +
37    (-2*zeta_12^2 + 1)*q^8 + (zeta_12^2 - 1)*q^9 -
38    3*zeta_12^2*q^10 - 2*q^12 + O(q^13)
39> E:=[Eltseq(Coefficient(qEigenform(A,38),n)): n in  [1..37]];
40> R<q>:=LaurentSeriesRing(Rationals());
41> f:=&+[E[n][1]*q^n : n in [1..37]]];
42> g:=-&+[E[n][3]*q^n : n in [1..37]];
43> f;
44q - q^2 - 2*q^3 + q^5 + 4*q^6 - q^8 - 3*q^10 - 2*q^12 - q^13 +
45    2*q^15 + 5*q^16 - q^18 - 4*q^19 + 2*q^20 + 6*q^23 - 2*q^24 +
46    2*q^25 - 2*q^26 - 4*q^27 - 3*q^29 - 2*q^31 - 6*q^32 + 3*q^34
47    + q^36 + 5*q^37
48> g;
49q^2 - 2*q^3 - q^4 + 2*q^5 + 2*q^6 - 2*q^8 + q^9 - 3*q^10 + 3*q^13
50    - 2*q^15 + 5*q^16 - 3*q^17 - 2*q^18 - 2*q^19 + q^20 + 6*q^23
51    + 2*q^24 - 7*q^26 - 3*q^29 + 6*q^30 - 4*q^31 - 3*q^32 +
52    6*q^34 + q^36 - 5*q^37
53> X:=f/g;
54> X;
55q^-1 + 1 + q + q^2 + q^4 - q^6 - 2*q^8 - q^11 + 2*q^12 + 2*q^14 +
56    3*q^15 - q^16 + 2*q^17 - 2*q^18 - 3*q^19 - 6*q^21 - q^22 -
57    q^23 - 2*q^24 + 7*q^25 + q^26 + 5*q^27 + 7*q^28 - 2*q^29 +
58    6*q^30 - 4*q^31 - 7*q^32 - q^33 - 14*q^34 - q^35 + O(q^36)
59> Y:=q*Derivative(X)/g;
60> Y;
61-q^-3 - 2*q^-2 - 4*q^-1 - 6 - 10*q - 10*q^2 - 12*q^3 - 11*q^4
62      - 3*q^5 - 3*q^6 + 11*q^7 + 22*q^8 + 18*q^9 + 36*q^10
63      + 12*q^11 + q^12 - 3*q^13 - 65*q^14 - 44*q^15 - 78*q^16
64      - 90*q^17 + 10*q^18 - 33*q^19 + 100*q^20 + 165*q^21
65      + 101*q^22 + 290*q^23 + 76*q^24 + 33*q^25 + 28*q^26
66      - 435*q^27 - 208*q^28 - 475*q^29 - 558*q^30 + 62*q^31
67      - 331*q^32 + 511*q^33 + O(q^34)
68> Y^2-X^6;
69-2*q^-5 - 9*q^-4 + ...
70> Y^2-X^6+2*X^5;
71q^-4 + 2*q^-3 + ...
72// etc.
73> Y^2-X^6+2*X^5-X^4+2*X^3-6*X^2+4*X-1;     // model for X_1(13)!!!
74O(q^31)
75\end{verbatim}}
76
77\section{Spaces of cusp forms}
78Observe that
79$$S_2(\Gamma_1(N)) = \bigoplus_{\chi:(\Z/N\Z)^*\ra\C^*} S_2(N,\chi),$$
80where $S_2(N,\chi)$ is the space of cusp forms on which the
81diamond operators $\langle d\rangle$ act via the character~$\chi$.
82For efficiency purposes, we compute only the spaces $S_2(N,\chi)$.
83
84Notationaly, we represent
85a character~$\chi$ is represented as follows, with appropriate modifications
86at~$2$.  Write $(\Z/N\Z)^*$ as a product of cyclic groups corresponding
87to the prime powers dividing~$N$.  Let $\eps_1,\ldots, \eps_n$ be characters
88that send minimal generators of these cyclic factors to appropriate roots
89of unity in~$\C^*$. We write characters $\chi$ in terms of the $\eps_i$.
90
91
92The dimension of $S_2(\Gamma_1(N))$ equals the genus of
93$X_1(N)$, which is a number that is recorded in
94Table~\ref{table:genus} for $N\leq 100$.
95\begin{table}
96\begin{center}
97\caption{The genus of $X_1(N)$\label{table:genus}}
98\vspace{1ex}
99
100\begin{tabular}{|lc|}\hline
101$N$ & $g$\\
102$1$ & $0$ \\
103$2$ & $0$ \\
104$3$ & $0$ \\
105$4$ & $0$ \\
106$5$ & $0$ \\
107$6$ & $0$ \\
108$7$ & $0$ \\
109$8$ & $0$ \\
110$9$ & $0$ \\
111$10$ & $0$ \\
112$11$ & $1$ \\
113$12$ & $0$ \\
114$13$ & $2$ \\
115$14$ & $1$ \\
116$15$ & $1$ \\
117$16$ & $2$ \\
118$17$ & $5$ \\
119$18$ & $2$ \\
120$19$ & $7$ \\
121$20$ & $3$ \\\hline
122\end{tabular}
124\begin{tabular}{|lc|}\hline
125$N$ & $g$\\
126$21$ & $5$ \\
127$22$ & $6$ \\
128$23$ & $12$ \\
129$24$ & $5$ \\
130$25$ & $12$ \\
131$26$ & $10$ \\
132$27$ & $13$ \\
133$28$ & $10$ \\
134$29$ & $22$ \\
135$30$ & $9$ \\
136$31$ & $26$ \\
137$32$ & $17$ \\
138$33$ & $21$ \\
139$34$ & $21$ \\
140$35$ & $25$ \\
141$36$ & $17$ \\
142$37$ & $40$ \\
143$38$ & $28$ \\
144$39$ & $33$ \\
145$40$ & $25$ \\\hline
146\end{tabular}
148\begin{tabular}{|lc|}\hline
149$N$ & $g$\\
150$41$ & $51$ \\
151$42$ & $25$ \\
152$43$ & $57$ \\
153$44$ & $36$ \\
154$45$ & $41$ \\
155$46$ & $45$ \\
156$47$ & $70$ \\
157$48$ & $37$ \\
158$49$ & $69$ \\
159$50$ & $48$ \\
160$51$ & $65$ \\
161$52$ & $55$ \\
162$53$ & $92$ \\
163$54$ & $52$ \\
164$55$ & $81$ \\
165$56$ & $61$ \\
166$57$ & $85$ \\
167$58$ & $78$ \\
168$59$ & $117$ \\
169$60$ & $57$ \\\hline
170\end{tabular}
172\begin{tabular}{|lc|}\hline
173$N$ & $g$\\
174$61$ & $126$ \\
175$62$ & $91$ \\
176$63$ & $97$ \\
177$64$ & $93$ \\
178$65$ & $121$ \\
179$66$ & $81$ \\
180$67$ & $155$ \\
181$68$ & $105$ \\
182$69$ & $133$ \\
183$70$ & $97$ \\
184$71$ & $176$ \\
185$72$ & $97$ \\
186$73$ & $187$ \\
187$74$ & $136$ \\
188$75$ & $145$ \\
189$76$ & $136$ \\
190$77$ & $181$ \\
191$78$ & $121$ \\
192$79$ & $222$ \\
193$80$ & $137$ \\\hline
194\end{tabular}
196\begin{tabular}{|lc|}\hline
197$N$ & $g$\\
198$81$ & $190$ \\
199$82$ & $171$ \\
200$83$ & $247$ \\
201$84$ & $133$ \\
202$85$ & $225$ \\
203$86$ & $190$ \\
204$87$ & $225$ \\
205$88$ & $181$ \\
206$89$ & $287$ \\
207$90$ & $153$ \\
208$91$ & $265$ \\
209$92$ & $210$ \\
210$93$ & $261$ \\
211$94$ & $231$ \\
212$95$ & $289$ \\
213$96$ & $193$ \\
214$97$ & $345$ \\
215$98$ & $235$ \\
216$99$ & $281$ \\
217$100$ & $231$ \\\hline
218\end{tabular}
219\end{center}
220\end{table}
221\section{Table of Genus 2 Curves}
222I do not know if somebody else has already made a table like this,
223but if they did they might have made mistakes so it is good that I
224have recomputed it.
225
226\setlength{\doublerulesep}{\arrayrulewidth}
227\begin{table}
228\begin{center}
229\caption{Genus 2 quotients $y^2=f(x)$ of $X_1(N)$\label{table:curves}
230that correspond to a single newform $f(q)$, {\bf
231I searched up to level $125$ and
232this is all I found!!}}
233\vspace{1ex}
234\begin{tabular}{|c|l|c|}\hline
235$X$, $\chi$ & $F(x)$, $f(q)$ & $L(J,1)\neq 0$?\\\hline\hline\hline
236&&\\
237$X_1(13)$ & $x^6 + 2x^5 + x^4 + 2x^3 + 6x^2 + 4x + 1$ & yes\\
238&&\\\hline
239&&\\
240$X_1(16)$ & $x^6 - 2x^5 - x^4 - x^2 + 2x + 1$ & yes\\
241          & $=(x - 1)(x + 1)(x^2 - 2x - 1)(x^2 + 1)$ & \\
242&&\\\hline
243&&\\
244$X_1(18)$ & $x^6 + 4x^5 + 10x^4 + 10x^3 + 5x^2 + 2x + 1$ & yes\\
245
246\comment{&&\\\hline
247&&\\
248$X_0(22)$ & $x^6 - 4x^4 + 20x^3 - 40x^2 + 48x - 32$ & yes\\
249          & $=(x^3 - 2x^2 + 4x - 4)(x^3 + 2x^2 - 4x + 8)$&\\
250
251&&\\\hline
252&&\\
253$X_0(23)$ & $x^6 - 8x^5 + 2x^4 + 2x^3 - 11x^2 + 10x - 7$ & yes\\
254          & $=(x^3 - 8x^2 + 3x - 7)(x^3 - x + 1)$ &\\
255
256&&\\\hline
257&&\\
258$X_0(26)$ & $x^6 - 8x^5 + 8x^4 - 18x^3 + 8x^2 - 8x + 1$ & yes\\
259
260
261&&\\\hline
262&&\\
263$X_0(28)$ & $x^6 + 10x^4 + 25x^2 + 28$ & \\
264          & $=(x^2 - x + 2)(x^2 + 7)(x^2 + x + 2)$ &\\
265
266&&\\\hline
267&&\\
268$X_0(29)$ & $x^6 - 4x^5 - 12x^4 + 2x^3 + 8x^2 + 8x - 7$ & \\
269
270
271&&\\\hline
272&&\\
273$X_0(30)/W_2$ & $x^6 + 2x^4 + 20x^3 - 19x^2 + 60x$ & \\
274    & $=x(x + 3)(x^2 - 2*x + 5)(x^2 - x + 4)$\\}
275
276&&\\\hline
277&&\\ % char of order 2
278$X_1(45)/W$, $\eps_2^2$ & $x^6 + 22x^3 + 125$ & yes\\
279    & $=(x^2 - 2x + 5)(x^4 + 2x^3 - x^2 + 10x + 25)$, &\\
280    & $q + \sqrt{-5}q^2 - 3q^4 - \sqrt{-5}q^5 + \cdots$ & \\
281
282&&\\\hline
283\end{tabular}
284\end{center}
285\end{table}
286
287\newpage
288\bibliography{biblio}
289
290
291\end{document}