OFFSET
1,2
COMMENTS
The 8-tuple (1/36 + 20*A300194, -1 + 216*A300195, -1/36 + 14*A300196, -1/6 + 24*A300197, -1/72 + 2*A300198, -1/46656 * A300199, 1/36 - 2*A300200, -1/7776 + 7/18 * A300201) gives a solution of the modular vector field R = Sum_{i=1..8} R_i d/dt_i on the enhanced moduli space arising from 4-dimensional Dwork family, where d/dt_i's give the standard basis of the tangent space in the chart (t_1,t_2,...,t_8) and
R_1 = -t_1*t_2+t_3;
R_2 = (-t_1^6*t_2^2+1/36*t_3^2*t_4*t_8+t_2^2*t_6)/(t_1^6-t_6);
R_3 = (-3*t_1^6*t_2*t_3+1/36*t_3^2*t_5*t_8+3*t_2*t_3*t_6)/(t_1^6-t_6);
R_4 = (-t_1^6*t_2*t_4-1/36*t_3^2*t_7*t_8+t_2*t_4*t_6)/(t_1^6-t_6);
R_5 = (-2*t_1^6*t_3*t_4-4*t_1^6*t_2*t_5+5*t_1^4*t_3*t_8+1/36*t_3*t_5^2*t_8+ 2*t_3*t_4*t_6+4*t_2*t_5*t_6)/(2*(t_1^6-t_6));
R_6 = -6*t_2*t_6;
R_7 = -18*t_1^2+1/2*t_4^2;
R_8 = (-3*t_1^6*t_2*t_8+3*t_1^5*t_3*t_8+3*t_2*t_6*t_8)/(t_1^6-t_6);
For more details see the Movasati & Nikdelan link Section 8.3.
LINKS
H. Movasati, Y. Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv:1603.09411 [math.AG], 2016-2017. See Table 2, (-6^6)*t_6.
H. Movasati, Foliation.lib.
PROG
(SINGULAR)
// This program has to be compiled in SINGULAR. By changing "int iter" you can
// calculate more coefficients. Note that this program is using a library calling
// "foliation.lib" written by H. Movasati, which is available in the link given in
// LINKS section as Foliation.lib.
LIB "linalg.lib"; LIB "foliation.lib";
ring r=0, (t_1, t_2, t_3, t_4, t_5, t_6, t_7, t_8, q), dp;
int pm=1; number t10=1/36; number ko=1/216; number c4=ko^2; number t20=-1; number t81=49/18; number a=-6*t20;
poly dis=t_1^6-t_6;
poly dt1=dis*(-t_1*t_2+t_3);
poly dt2=(1296*c4*t_3^2*t_4*t_8-t_1^6*t_2^2+t_2^2*t_6);
poly dt3=(1296*c4*t_3^2*t_5*t_8-3*t_1^6*t_2*t_3+3*t_2*t_3*t_6);
poly dt4=(-1296*c4*t_3^2*t_7*t_8-t_1^6*t_2*t_4+t_2*t_4*t_6);
poly dt5=(1296*c4*t_3*t_5^2*t_8-4*t_1^6*t_2*t_5-2*t_1^6*t_3*t_4+5*t_1^4*t_3*t_8+4*t_2*t_5*t_6+2*t_3*t_4*t_6)/(2);
poly dt6=dis*(-6*t_2*t_6);
poly dt7=dis*((1296*c4*t_4^2-t_1^2)/(2592*c4));
poly dt8=(-3*t_1^6*t_2*t_8+3*t_1^5*t_3*t_8+3*t_2*t_6*t_8);
list pose;
pose=(60*ko)/(49*t10^2)*t81*q+(t10), (-162*t20*ko)/(49*t10^3)*t81*q+(t20), (-66*t20*ko)/(7*t10^2)*t81*q+(t10*t20), 16/(147*t10^2)*t81*q+(-t10)/(36*ko), 45/(49*t10)*t81*q+(-t10^2)/(12*ko), (3888*t10^3*ko)/49*t81*q, 1/(1512*t10*t20*ko)*t81*q+(-t10^2)/(1296*t20*ko^2), t81*q+(-t10^3)/(36*ko);
list vecfield=dt1, dt2, dt3, dt4, dt5, dt6, dt7, dt8;
list denomv=dis, dis, dis, dis, dis, dis, dis, dis;
intvec upto=1, 1, 1, 1, 1, 1, 1, 1; intvec whichpow;
int iter=20;
int n;
for (n=2; n<=iter; n=n+1){upto=n, n, n, n, n, n, n, n; whichpow=upto; pose=qexpansion(vecfield, denomv, pose, upto, upto, a); n; }
-6^6*pose[6];
CROSSREFS
KEYWORD
sign
AUTHOR
Younes Nikdelan, Mar 22 2018
STATUS
approved