1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
|
// FILE PCURVE.CC
//////////////////////////////////////////////////////////////////////////
//
// Copyright 1990-2012 John Cremona
//
// This file is part of the eclib package.
//
// eclib is free software; you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by the
// Free Software Foundation; either version 2 of the License, or (at your
// option) any later version.
//
// eclib is distributed in the hope that it will be useful, but WITHOUT
// ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
// FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
// for more details.
//
// You should have received a copy of the GNU General Public License
// along with eclib; if not, write to the Free Software Foundation,
// Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
//
//////////////////////////////////////////////////////////////////////////
//
// computes a possible equation from periods for newforms
// ONLY needs eigs data, not full intdata
// BUT only finds curves "up to isogeny" as it does not
// determine the full Gamma_0(N) period lattice.
#include <iostream>
#include <eclib/marith.h>
#include <eclib/moddata.h>
#include <eclib/symb.h>
#include <eclib/oldforms.h>
#include <eclib/homspace.h>
#include <eclib/cperiods.h> //from qcurves, for computing conductors
#include <eclib/newforms.h>
#include <eclib/periods.h>
#include <eclib/pcprocs.h>
//#define SINGLE
#ifndef SINGLE
#define AUTOLOOP
#endif
#define DMAX 50 // Upper bound for d = (2,2) matrix entry used
#define LMAX 200 // Upper bound twisting prime l-
int main(void)
{
initprimes("PRIMES",0);
set_precision(20);
long limit,n=1;
int dump=1, detail;
long maxn, dmax=DMAX;
//#ifdef SINGLE
detail=2;
//#else
cout << "See details? "; cin>>detail;
//#endif
cout << "Enter max d: "; cin>>dmax;
cout << "Enter max scaling factor for periods: "; cin>>maxn;
#ifdef AUTOLOOP
cout<<"Enter first and last N: ";cin>>n>>limit;
n--; cout<<"\n";
while (n<limit) { n++;
#else
while (n>0) { cout<<"Enter level: "; cin>>n;
#endif
if (n>0)
{
cout << "N = " << n << endl;
newforms nf(n,detail);
int noldap=25;
nf.createfromdata(1,noldap);
int squarelevel=nf.squarelevel;
long fac=nf.sqfac;
long nnf = nf.n1ds;
long inf = 1;
#ifdef SINGLE
cout << "Enter form number: "; cin>>inf;
nnf=inf;
#endif
primevar pr; long p0; // First "good" prime
while (p0=(long)pr, ::divides(p0,n)) pr++;
int anyfound=0;
for(long i=inf-1; i<nnf; i++)
{
if(detail) cout<<"\n"<<"Form number "<<i+1<<"\n";
else cout<<(i+1)<<" ";
newform* nfi = &((nf.nflist)[i]);
if(detail>1) {cout<<"\n Newform details:\n";nfi->display();cout<<endl;}
bigfloat x0=to_bigfloat(10), y0=to_bigfloat(10), ratio;
// flags to say that we have real/imag period
// and that we have a cycle giving both together
int rp_known=0, have_both=0;
long lplus=nfi->lplus, mplus=nfi->mplus;
long s = nfi->sfe;
//nfi->dotplus=1;
nfi->dotminus=1; // In case wrong values were on the file!
// STEP 1: find the real period using L(f,1) is L/P nonzero,
// or via L(f,chi,1)
// Now done within get_matrix_periods()
// STEP 2: compute periods of lots of matriced in Gamma_0(N) over all
// symbols {0,b/d} for d<dmax. We are looking for (i) a symbol whose
// imaginary part is nonzero; (ii) a symbol whose real and imaginary
// parts are both non-zero, which we store.
have_both = nf.find_matrix( i, dmax, rp_known, x0, y0);
if(!have_both)
cout<<"Problem! find_matrix() returns 0!"<<endl;
if(detail)
{
cout << "Minimal periods found: x0 = "<<x0<<", y0 = "<<y0<<"\n";
cout << "Matrix ("<<nfi->a<<","<<nfi->b<<";"<<n*nfi->c<<","<<nfi->d<<"):\t";
cout << "dotplus = "<< nfi->dotplus << ", dotminus = "<< nfi->dotminus<< "\n";
cout << "Searching for scaling factors.\n";
}
// STEP 3: Now we have nonzero integer multiples of real and imag
// periods, we search for a sub-multiple of both which gives a genuine
// curve.
long nx, ny; int type;
long maxnx=maxn;
if(rp_known) maxnx=1;
int found = get_curve(n, fac, maxnx, maxn, x0, y0, nx, ny, type, detail);
if(found)
{
anyfound=1;
// cout<<"before rescaling, dotplus="<<nfi->dotplus<<", dotminus="<<nfi->dotminus<<endl;
// cout<<"rescaling by nx = "<<nx<<", ny="<<ny<<endl;
nfi->dotplus *= nx;
nfi->dotminus *= ny;
nfi->type=type;
cout << "[(" <<nfi->a<<","<<nfi->b<<";"<<nfi->c
<<","<<nfi->d<<"),"<<nfi->dotplus<<","<<nfi->dotminus
<<";"<<nfi->type<<"]"<<endl;
// STEP 4: We find a suitable twisting prime for the imaginary period
if(!squarelevel)
{
bigfloat y1 = y0/to_bigfloat(ny);
int ok = nf.find_lminus(i, 0, y1);
if(!ok)
{
cout<<"No suitable twisting prime "
<<" found for imaginary period!"<<endl;
}
}
if(detail)
{
cout<<"Updated newform data:\n"; nfi->display();
cout<<"test reconstruction of curve from updated data:"<<endl;
}
// test reconstruction of the curve
bigfloat rperiod;
Curve C = nf.getcurve(i, -1, rperiod, detail);
Curvedata CD(C,1); // The 1 causes minimalization
cout << "Curve = \t";
cout << (Curve)CD << "\t";
CurveRed CR(CD);
bigint nc = getconductor(CR);
cout << "N = " << getconductor(CR) << endl;
}
else cout<<"No curve found\n";
// dump complete data file into NF_DIR/x$N.
} // end of forms loop
if (anyfound)
{
cout<<"Store newform data? "; cin>>dump;
if(dump)
{
nf.output_to_file();
}
}
} // end of if(n)
} // end of while()
} // end of main()
|
