# Check-GSp(4)

## 2996 days ago by jimlb

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

var('p,chi1,chi2,psi1,psi2');
 $\newcommand{\Bold}{\mathbf{#1}}\left(p, \chi_{1}, \chi_{2}, \psi_{1}, \psi_{2}\right)$
s1=Matrix(SR,[[p^3-p^2,p^2-p,p-1,1]]);s1;
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} p^{3} - p^{2} & p^{2} - p & p - 1 & 1 \end{array}\right)$
s2=Matrix(SR,[[p^3-p^2,p^2-p,p,0]]);s2
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} p^{3} - p^{2} & p^{2} - p & p & 0 \end{array}\right)$
s3=Matrix(SR,[[p^3-p^2,p^2,0,0]]);s3
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} p^{3} - p^{2} & p^{2} & 0 & 0 \end{array}\right)$
s4=Matrix(SR,[[p^3,0,0,0]]);s4;
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} p^{3} & 0 & 0 & 0 \end{array}\right)$
gamma1=Matrix(SR,[[p^(-3/2),p^(-1/2)*chi2,p^(1/2)*chi1, p^(3/2)*chi1*chi2]]);gamma1
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} \frac{1}{p^{\frac{3}{2}}} & \frac{\chi_{2}}{\sqrt{p}} & \chi_{1} \sqrt{p} & \chi_{1} \chi_{2} p^{\frac{3}{2}} \end{array}\right)$
gamma1bar=Matrix(SR,[[p^(-3/2),p^(-1/2)*psi2,p^(1/2)*psi1, p^(3/2)*psi1*psi2]]);gamma1bar
 $\newcommand{\Bold}{\mathbf{#1}}\left(\begin{array}{rrrr} \frac{1}{p^{\frac{3}{2}}} & \frac{\psi_{2}}{\sqrt{p}} & \sqrt{p} \psi_{1} & p^{\frac{3}{2}} \psi_{1} \psi_{2} \end{array}\right)$
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;
 $\newcommand{\Bold}{\mathbf{#1}}\chi_{1} \chi_{2} p^{6} \psi_{1} \psi_{2} + \chi_{1} \chi_{2} {\left(p - 1\right)} p^{5} \psi_{1} + \chi_{1} {\left(p - 1\right)} p^{5} \psi_{1} \psi_{2} + \chi_{1} {\left(p - 1\right)}^{2} p^{4} \psi_{1} + {\left(p^{2} - p\right)} \chi_{1} \chi_{2} p^{4} \psi_{2} + {\left(p^{2} - p\right)} \chi_{2} p^{4} \psi_{1} \psi_{2} + {\left(p^{2} - p\right)} \chi_{2} {\left(p - 1\right)} p^{3} \psi_{1} + \chi_{1} p^{5} \psi_{1} + {\left(p^{2} - p\right)} \chi_{1} {\left(p - 1\right)} p^{3} \psi_{2} + {\left(p^{3} - p^{2}\right)} \chi_{1} \chi_{2} p^{3} + {\left(p^{2} - p\right)} \chi_{2} p^{3} \psi_{1} + {\left(p^{2} - p\right)}^{2} \chi_{2} p^{2} \psi_{2} + {\left(p^{2} - p\right)} \chi_{1} p^{3} \psi_{2} + \chi_{2} p^{4} \psi_{2} + {\left(p^{3} - p^{2}\right)} p^{3} \psi_{1} \psi_{2} + {\left(p^{3} - p^{2}\right)} \chi_{1} {\left(p - 1\right)} p^{2} + {\left(p^{3} - p^{2}\right)} {\left(p - 1\right)} p^{2} \psi_{1} + {\left(p^{2} - p\right)}^{2} \chi_{2} p \psi_{2} + {\left(p^{3} - p^{2}\right)} {\left(p^{2} - p\right)} \chi_{2} p + {\left(p^{3} - p^{2}\right)} \chi_{1} p^{2} + {\left(p^{3} - p^{2}\right)} p^{2} \psi_{1} + {\left(p^{3} - p^{2}\right)} {\left(p^{2} - p\right)} p \psi_{2} + {\left(p^{3} - p^{2}\right)} {\left(p^{2} - p\right)} \chi_{2} + {\left(p^{3} - p^{2}\right)} \chi_{2} p + p^{3} + {\left(p^{3} - p^{2}\right)} {\left(p^{2} - p\right)} \psi_{2} + {\left(p^{3} - p^{2}\right)} p \psi_{2} + {\left(p^{3} - p^{2}\right)}^{2} + \frac{{\left(p^{3} - p^{2}\right)}^{2}}{p} + \frac{{\left(p^{3} - p^{2}\right)}^{2}}{p^{2}}$

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')
 $\newcommand{\Bold}{\mathbf{#1}}\left(t_{1}, t_{2}\right)$
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);
 $\newcommand{\Bold}{\mathbf{#1}}\chi_{2} p^{6} p^{t_{1}} + p^{6} p^{t_{1}} p^{t_{2}} + \chi_{1} p^{6} p^{t_{2}} + \chi_{1} \chi_{2} p^{6} - \chi_{2} p^{5} p^{t_{1}} + \frac{\chi_{2} p^{6} p^{t_{1}}}{\chi_{1}} + p^{6} p^{t_{1}} + \frac{p^{6} p^{t_{1}} p^{t_{2}}}{\chi_{1}} + \frac{p^{6} p^{t_{1}} p^{t_{2}}}{\chi_{2}} - \chi_{1} p^{5} p^{t_{2}} + \frac{\chi_{1} p^{6} p^{t_{2}}}{\chi_{2}} + p^{6} p^{t_{2}} - \chi_{1} \chi_{2} p^{5} + \chi_{1} p^{6} + \chi_{2} p^{6} - \frac{\chi_{2} p^{5} p^{t_{1}}}{\chi_{1}} - p^{5} p^{t_{1}} + \frac{p^{6} p^{t_{1}}}{\chi_{1}} - \frac{p^{5} p^{t_{1}} p^{t_{2}}}{\chi_{1}} - \frac{p^{5} p^{t_{1}} p^{t_{2}}}{\chi_{2}} + \frac{p^{6} p^{t_{1}} p^{t_{2}}}{\chi_{1} \chi_{2}} - \frac{\chi_{1} p^{5} p^{t_{2}}}{\chi_{2}} - p^{5} p^{t_{2}} + \frac{p^{6} p^{t_{2}}}{\chi_{2}} - \chi_{1} p^{5} - \chi_{2} p^{5} + p^{6} + p^{4} p^{t_{1}} - \frac{p^{5} p^{t_{1}}}{\chi_{1}} - \frac{p^{5} p^{t_{1}} p^{t_{2}}}{\chi_{1} \chi_{2}} - \frac{p^{5} p^{t_{2}}}{\chi_{2}} - p^{5} + p^{3} p^{t_{2}} + p^{2}$
eq1-expand(eq2);
 $\newcommand{\Bold}{\mathbf{#1}}0$

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.