Check-GSp(4)

2996 days ago by jimlb

I use $\psi$ as $\overline{\chi}$.

var('p,chi1,chi2,psi1,psi2'); 
       
s1=Matrix(SR,[[p^3-p^2,p^2-p,p-1,1]]);s1; 
       
s2=Matrix(SR,[[p^3-p^2,p^2-p,p,0]]);s2 
       
s3=Matrix(SR,[[p^3-p^2,p^2,0,0]]);s3 
       
s4=Matrix(SR,[[p^3,0,0,0]]);s4; 
       
gamma1=Matrix(SR,[[p^(-3/2),p^(-1/2)*chi2,p^(1/2)*chi1, p^(3/2)*chi1*chi2]]);gamma1 
       
gamma1bar=Matrix(SR,[[p^(-3/2),p^(-1/2)*psi2,p^(1/2)*psi1, p^(3/2)*psi1*psi2]]);gamma1bar 
       
eq=sum([sum([p^3*s1[0,i]*s1[0,j]*gamma1[0,i]*gamma1bar[0,j] for i in [0..3]]) for j in [0..3]]) + sum([sum([p^2*s2[0,i]*s2[0,j]*gamma1[0,i]*gamma1bar[0,j] for i in [0..3]]) for j in [0..3]]) + sum([sum([p*s3[0,i]*s3[0,j]*gamma1[0,i]*gamma1bar[0,j] for i in [0..3]]) for j in [0..3]]) + sum([sum([s4[0,i]*s4[0,j]*gamma1[0,i]*gamma1bar[0,j] for i in [0..3]]) for j in [0..3]]); 
       
eq; 
       

This matches up with everything in (10.2) exactly.  I checked this by doing various substitutions such as $chi1 = 0$, etc  to isolate the coefficients.  This makes checking things quicker.

To check Theorem 10.1, we substitute $\psi_{i} = p^{t_{i}}/\chi_{i}$.

var('t1, t2') 
       
eq1=expand(eq.substitute(psi1=p^(t1)/chi1, psi2=p^(t2)/chi2)); 
       
eq2=p^2 + p^(t2+3) + p^(t1+4) + p^(t1+t2+5) + p^5 * (p-1) *(1 + chi1)*(1+p^(t1)/chi1)*(1+chi2)*(1+p^(t2)/chi2); 
       
expand(eq2); 
       
eq1-expand(eq2); 
       

This gives that the equation in Theorem 10.1 is correct.

It is easy to see case (T) of (10.4) is correct just by plugging in $t_1 = t_2 = 0$ for Theorem 10.1.

For the case (C) I get $t_1 = t_2 = -2 \beta$, and then the formula checks out in (10.4) by plugging in.

The case (SK) also checks out by plugging in.