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
|
#include "sympow.h"
#define DEBUG (FALSE || GLOBAL_DEBUG)
static void prep_moddeg_at2(int bpt)
{double ap; int c4,c6,D,M=1<<29; QDpoly v;
if ((bpt==3) || (bpt==20) || (bpt==21) || (bpt==23))
{EVEN_TOP=3; EVEN_BOTTOM=2;}
if (bpt==22) {EVEN_TOP=1; EVEN_BOTTOM=2;}
if ((bpt==1) || (bpt==3) || (bpt==8) || (bpt==9) || (bpt==24)) return;
c4=QD_modi(Ec4,M); c6=QD_modi(Ec6,M); D=QD_modi(Edisc,M);
if ((bpt==10) || (bpt==22) || (bpt==23))
{if (((c4&63)!=31) || ((c6&255)!=0))
{QD_mul1(wmax,VOLUME,2.0,VOLUME); QD_mul1(wmax,TW_EFF,2.0,TW_EFF);} return;}
if (bpt==16)
{CONDTW/=16; QD_mul1(wmax,VOLUME,4.0,VOLUME);
euler_factor_bad(2,bpt,2,&v); ap=Round(Sqrt(-v.coeff[1][0]+2.0));
QD_mul1(wmax,TW_EFF,72.0-8.0*ap*ap,TW_EFF); delQDpoly(&v); return;}
if (bpt==17)
{CONDTW/=64;
if (((c6&127)==64) && ((c4&63)==0)) QD_mul1(wmax,VOLUME,2.0,VOLUME);
else QD_mul1(wmax,VOLUME,8.0,VOLUME);
euler_factor_bad(2,bpt,2,&v); ap=Round(Sqrt(-v.coeff[1][0]+2.0));
QD_mul1(wmax,TW_EFF,9.0-ap*ap,TW_EFF); delQDpoly(&v);
if (((c6&127)==64) && ((c4&63)==0)) QD_mul1(wmax,TW_EFF,16.0,TW_EFF);
else QD_mul1(wmax,TW_EFF,64.0,TW_EFF); return;}
if (bpt==20) {CONDTW/=4; QD_mul1(wmax,TW_EFF,4.0,TW_EFF); return;}
if (bpt==21)
{QD_mul1(wmax,VOLUME,2.0,VOLUME); CONDTW/=16;
QD_mul1(wmax,TW_EFF,32.0,TW_EFF); return;}
if (bpt==25) {CONDTW/=2; QD_mul1(wmax,TW_EFF,2.0,TW_EFF); return;}
if (bpt==26)
{QD_mul1(wmax,VOLUME,2.0,VOLUME); CONDTW/=8;
QD_mul1(wmax,TW_EFF,16.0,TW_EFF); return;}
if (bpt==27)
{if (((c4&63)==0) && ((c6&255)==0))
{QD_mul1(wmax,VOLUME,2.0,VOLUME); QD_mul1(wmax,TW_EFF,2.0,TW_EFF);} return;}
if (bpt==29) /* v2(N)=6 multiplicative twist */
{CONDTW/=32; QD_mul1(wmax,VOLUME,4.0,VOLUME);
QD_mul1(wmax,TW_EFF,96.0,TW_EFF); return;}
if (bpt==30) /* v2(N)=4 multiplicative twist */
{CONDTW/=8; QD_mul1(wmax,VOLUME,4.0,VOLUME);
QD_mul1(wmax,TW_EFF,24.0,TW_EFF); return;}
errorit("bad prime type at 2 in prep_moddeg_at2");}
static llint is_square(llint x)
{llint y; if (x<0) return 0;
y=(llint) Round(Sqrt((double) x)); if (y*y==x) return y; return 0;}
static void isogcheck3(int n)
{int i,u=(n+3),v=0; QD U,V,W,R; llint p,q;
if (DEBUG) printf("isogcheck3 %i\n",n); q=(llint) n; q=q*q+9*q+27;
for (i=0;badprimes[i]!=0;i++)
{p=badprimes[i]; if (((CONDTW%p)==0) && ((n%p)==0) && ((p%6)==1)) return;
if ((q%p)==0) v++;} if (v!=1) return;
QD_copy(wmax,QD_zero,U); U[0]=(double) u; QD_powi(wmax,U,4,V);
QD_mul1(wmax,U,24.0,W); QD_sub(wmax,V,W,R); QD_round(wmax,R,R);
QD_sub(wmax,Etw4,R,R); if (R[0]!=0.0) return;
QD_powi(wmax,U,6,V); QD_powi(wmax,U,3,W); QD_mul1(wmax,W,36.0,W);
QD_sub(wmax,W,V,W); QD_copy(wmax,QD_zero,V); V[0]=-216.0; QD_add(wmax,W,V,R);
QD_sub(wmax,Etw6,R,R); if (R[0]!=0.0) return;
MANIN_TWIST=9; if (Ec6[0]==Etw6[0]) MANIN=9; return;}
static void check_manin()
{QD T,U,R; llint p,bp[32]; int i,n,tau=0; if (DEBUG) printf("check_manin\n");
MANIN=1; MANIN_TWIST=1;
if (CONDTW==11)
{if (Etw4[0]==16.0) {MANIN_TWIST=25; if (Ec4[0]==16.0) MANIN=25;} return;}
if (CONDTW==15)
{if (Etw4[0]==1.0) {MANIN_TWIST=16; if (Ec4[0]==1.0) MANIN=16;}
if (Etw4[0]==241.0) {MANIN_TWIST=4; if (Ec4[0]==241.0) MANIN=4;}
if (Etw4[0]==3841.0) {MANIN_TWIST=4; if (Ec4[0]==3841.0) MANIN=4;} return;}
if (CONDTW==17)
{if (Etw6[0]==-81.0) {MANIN_TWIST=16; if (Ec6[0]==-81.0) MANIN=16;}
if (Etw4[0]==273.0) {MANIN_TWIST=4; if (Ec4[0]==273.0) MANIN=4;}
if (Etw4[0]==4353.0) {MANIN_TWIST=4; if (Ec4[0]==4353.0) MANIN=4;} return;}
if (CONDTW==20)
{if (Etw4[0]==64.0) {MANIN_TWIST=4; if (Ec4[0]==64.0) MANIN=4;}
if (Etw4[0]==1984.0) {MANIN_TWIST=4; if (Ec4[0]==1984.0) MANIN=4;} return;}
if (CONDTW==24)
{if (Etw4[0]==-32.0) {MANIN_TWIST=4; if (Ec4[0]==-32.0) MANIN=4;} return;}
if (CONDTW==32)
{if (Ec6[0]==0.0)
{MANIN_TWIST=4; if ((COND0==32) && (QD_modi(Ec4,32)==16)) MANIN=4;}
else {MANIN_TWIST=4; if (COND0==32) MANIN=4;} return;}
if (CONDTW==40)
{if (Etw4[0]==96.0) {MANIN_TWIST=4; if (Ec4[0]==96.0) MANIN=4;} return;}
if (CONDTW==64)
{if (Ec6[0]==0.0)
{MANIN_TWIST=4; if ((COND0==64) && (QD_modi(Ec4,32)==16)) MANIN=4;} return;}
if (CONDTW==128)
{if (Etw4[0]==112.0)
{MANIN_TWIST=4; if ((COND0==128) && (QD_modi(Ec4,32)==16)) MANIN=4;} return;}
/* 3-isog case */
for (n=-10;n<=10;n++) {if (EtwD[0]==(double) (n*(n*n+9*n+27))) isogcheck3(n);}
QD_copy(wmax,QD_zero,R);
if (EtwD[0]>0.0) R[0]=Round(Root(EtwD[0],3)-3);
else R[0]=-Round(Root(-EtwD[0],3)+3);
QD_sqr(wmax,R,T); QD_mul1(wmax,R,9.0,U); QD_add(wmax,T,U,T);
QD_copy(wmax,QD_zero,U); U[0]=27.0; QD_add(wmax,T,U,T);
QD_mul(wmax,T,R,T); QD_round(wmax,T,T); QD_sub(wmax,T,EtwD,U);
if (U[0]==0.0) isogcheck3((int) R[0]);
for (i=0;badprimes[i]!=0;i++)
if ((CONDTW%badprimes[i])==0) bp[tau++]=badprimes[i];
if (tau>2) return;
if (tau==1) /* original neumann-setzer case, don't check c6 */
{if (!(is_square(CONDTW-64))) return;
if ((CONDTW-16)==(llint) Etw4[0])
{MANIN_TWIST=4; if (Ec4[0]==Etw4[0]) MANIN=4; return;} return;}
if (CONDTW&1) /* Third NS case bad for curves with big Etw6 */
{if (Etw4[0]<-48.0) return; p=(llint) Round(Sqrt(Etw4[0]+48.0));
if ((p&3)!=3) p=-p; p-=8; if ((p*p+16*p+16)!=(llint) Etw4[0]) return;
if ((p*p*p+24*p*p+120*p-64)!=(llint) Etw6[0]) return;
if (((bp[0]&3)!=3) && ((bp[1]&3)!=3)) return;
if (((p%bp[0])!=0) && ((p%bp[1])!=0)) return;
if ((((p+16)%bp[0])!=0) && (((p+16)%bp[1])!=0)) return;
MANIN_TWIST=4; if (Ec4[0]==Etw4[0]) MANIN=4; return;}
if ((CONDTW&7)!=4) return; if (!(is_square(CONDTW/4-4))) return;
if ((4*CONDTW-16)==(llint) Etw4[0]) /* second NS case, again ignore c6 */
{MANIN_TWIST=4; if (Ec4[0]==Etw4[0]) MANIN=4;}}
void prepare_moddeg(char *IN)
{int i,s,bpt,v4,v6,vd; llint p; QDpoly v; double ap,P; QD C;
QD_mul(wmax,REAL_PERIOD,IMAG_PERIOD,VOLUME);
QD_mul(wmax,VOLUME,QD_twopi,VOLUME); EVEN_TOP=1; EVEN_BOTTOM=1;
QD_copy(wmax,QD_one,TW_EFF); CONDTW=COND0;
for (i=0;badprimes[i]!=0;i++)
{p=badprimes[i]; bpt=badprimetype[i]; P=(double) p;
if (p>=3)
{if (bpt==2)
{QD_mul1(wmax,VOLUME,P,VOLUME); CONDTW/=(p*p);
euler_factor_bad(p,bpt,2,&v); ap=Round(Sqrt(-v.coeff[1][0]+P));
QD_mul1(wmax,TW_EFF,P-1.0,TW_EFF); QD_mul1(wmax,TW_EFF,P+1.0-ap,TW_EFF);
QD_mul1(wmax,TW_EFF,P+1.0+ap,TW_EFF); delQDpoly(&v);}
else if (bpt==31)
{QD_mul1(wmax,VOLUME,P,VOLUME); CONDTW/=p;
QD_mul1(wmax,TW_EFF,P+1.0,TW_EFF); QD_mul1(wmax,TW_EFF,P-1.0,TW_EFF);}
else if (bpt!=1)
{v4=QD_valuation(Ec4,P); v6=QD_valuation(Ec6,P); vd=QD_valuation(Edisc,P);
if ((v4>=2) && (v6>=3) && ((p!=3) || ((vd>=6) && (v6!=5))))
{QD_mul1(wmax,VOLUME,P,VOLUME); QD_mul1(wmax,TW_EFF,P,TW_EFF);}}
if ((bpt>=3) && (bpt<=6))
{if ((p%bpt)==1) {EVEN_TOP*=(p-1); EVEN_BOTTOM*=p;}
else {EVEN_TOP*=(p+1); EVEN_BOTTOM*=p;}}
if ((bpt==14) || (bpt==15)) {EVEN_TOP*=2; EVEN_BOTTOM*=3;}
if ((bpt==18) || (bpt==19)) {EVEN_TOP*=4; EVEN_BOTTOM*=3;}}
if (p==2) prep_moddeg_at2(bpt);}
IN[0]='2'; IN[1]='w'; IN[2]='0'; IN[3]='p';
QD_copy(wmax,QD_zero,C); C[0]=(double) CONDTW; QD_div(wmax,C,VOLUME,SCALE);
if (CM_CASE==-3)
{QD_mul(wmax,SCALE,QD_pi,SCALE); C[0]=27.0;
QD_sqrt(wmax,C,C); QD_div(wmax,SCALE,C,SCALE); HECKE=TRUE;}
if (CM_CASE==-4)
{QD_mul(wmax,SCALE,QD_pi,SCALE); QD_mul2n(wmax,SCALE,-2,SCALE); HECKE=TRUE;}
s=4+Ceil(Log10(SCALE[0])); if (s<6) s=6; if ((s>12) && (s<=16)) s=17;
if (s>28) {s=28; printf("*WARNING* reducing moddeg precision to 28");}
if (s<10) {IN[4]='0'+s; IN[5]=0;}
else {IN[4]='0'+(s/10); IN[5]='0'+(s%10); IN[6]=0;}
if (VERBOSE)
{printf("Twist effect is "); QD_intout_noplus(TW_EFF);
printf(", Even valuation gives %lli/%lli\n",EVEN_TOP,EVEN_BOTTOM);}
if (VERBOSE) printf("Twist conductor is %lli\n",CONDTW);
if (VERBOSE) printf("Using precision of %i digits\n",s);
check_manin(); if (MANIN!=1) printf("Manin effect for curve is %i\n",MANIN);
if (MANIN_TWIST!=1) printf("Manin effect for twist is %i\n",MANIN_TWIST);}
llint postpare_moddeg()
{QD C,D,W; double L; llint N=CONDTW/10;
QD_copy(wmax,QD_zero,C); QD_copy(wmax,QD_zero,W);
C[0]=(double) CONDTW; QD_div(wmax,C,VOLUME,C);
while (1)
{get_weight(N,W,0); QD_mul(wmax,C,W,D); L=Log((double) N); L*=L;
QD_mul1(wmax,D,L,D); QD_mul1(wmax,D,(double) N,D);
QD_mul1(wmax,D,(double) N,D); if (D[0]<0.5) break; N=(11*N)/10;} return(N);}
|
