## On FOCs

FOCs are frequently occuring congruences (Ramanujan’s 691s). These should be made usual suspects when searching for mirror duals of polarization structures (“degrees”). For instance, the fact that there is a del Pezzo of deg 5 signals that 5 must be a FOC for elliptic curves, i.e. a_p=p+1 mod 5 for all p’s for many curves. In the same manner, the existence of a V_22 means that 11 is a FOC in a suitable problem on 3-dim’l Galois reps.

I attach a PARI script that produces FOCs for elliptic curves. When run from within the dir with elliptic curves data in the standard distribution, it inputs the vec of ell curves with small conductors and outputs the congruences. Adjust SLOWDOWN according to the speed of your computer.

Do report on your findings.

default(parisize,120000000);

SLOWDOWN=10^4;

e=read(“ell0″);

congruences(E)=

{

res=0;

forprime(p=10^5,10^5+SLOWDOWN,res=gcd(res,1+p-ellap(E,p)));

if(7==7,print(res,”\t”,”\t”));

}

cong(ii)=

{

num_curves=matsize(e[ii])[2];

for(jj=2,num_curves,EE=ellinit(e[ii][jj][2]);congruences(EE));

}

for(ii=1,matsize(e)[2],cong(ii));

Example: del Pezzo of deg 6 and the 6-division point. Incidentally, what is the shortest way to find phi1^-1(div_point)? Cannot do that literally…

Q:=Rationals();

R

:=RationalFunctionField(Q);:=ProjectiveSpace(R,2); P2

C := Curve(P2, x^2*y*z+y^2*x*z+x*z^3+y*z^2*z+x^2*y^2+z^4-tt*x*y*z^2);pt:=C![0,1,0];

E1, phi1 := EllipticCurve(C, pt);

div_point:=E1![1/(2+tt),0,1];

Order(div_point);