A300201 Coefficients of non-constant terms of a Calabi-Yau modular form attached to the 4-dimensional Dwork family.

7, 54855, 1034706148, 24546181658391, 653902684588247058, 18687787944102314534628, 559904113526412529297895976, 17354691863853902532743058359703, 551873714646783457391913160851776659, 17903944334292612333529257451439091620754

Offset: 1

Author

Younes Nikdelan, Mar 28 2018

Keywords

• nonn

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 to the modular vector field R = Sum_{i=1..8} R_i d/dt_i on the enhanced moduli space arising from the 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 Section 8.3 of the Movasati & Nikdelan link.

Links

• H. Movasati, Foliation.lib.
• H. Movasati, Y. Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv:1603.09411 [math.AG], 2016-2017. See Table 2, (18/7)*t_8.

Crossrefs

Cf. A300194, A300195, A300196, A300197, A300198, A300199, A300200.

Programs

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;}
18/7*pose[8]+1/3024;