Lemma 5.1
I'm going to start by proving Lemma 5.1 with just a direct calculation. I did this once before kind of brute force, so now I want to use some of the new commands and methods I'll try to use to show our paramodular version. I want to try to work them out in a case I know they work first.
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Here we create a matrix $D$ automatically rather than typing all the variables out. Not sure how I didn't bother to figure this out earlier. We start with $D$ because it is the easier matrix to find in the calculations because of the $(h,1)$.
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As our matrices will be in $\textrm{Sp}_4(F)$, we can take the parabolic element to be of the form $\begin{pmatrix} ^{t}D^{-1} & X \\ 0 & D \end{pmatrix}$.
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We can now create our general element in $P_8(F)$.
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We now have constructed all the matrices we need. We want to show that if $Q_2^{-1} p Q_2 (h,1)$ is in $\Gamma_4(\varpi)$, then $h \in \Gamma_2(\varpi)$. We look at $Q_2^{-1} p Q_2 (h,1)$ and assume this is congruent to $1_{8}$ modulo $\varpi$. We could just subtract it from $1_8$ and then say this is $0_8$ modulo $\varpi$, but I'm not sure how easily that will generalize to our case so I don't do it here. One warning is the matrix entries have SAGE labels that start at 0, so the upper left entry is the $(0,0)$ entry and not the $(1,1)$ entry so one has to be careful when picking out the entries.
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Working modulo $\varpi$ now, we have anything off the diagonal of $Z$ must be 0 modulo $\varpi$. For instance, we see below that the $(1,4)$ entry of $Z$ is $-x_{03}$, so $x_{03} = 0$.
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Modulo $\varpi$, the diagonal entries need to be 1 and the off-diagonal need to be 0.
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One now must solve the equations in each row to simplify things further. For instance, we set $z_{7,8} = 0$ and solve for $x_{2,1}$ and substitute this back into $Z$.
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Observe we have shown that if $Q_2^{-1} p Q_2 (h,1) \in \Gamma_4(\varpi)$, then modulo $\varpi$ we have $Q_2^{-1} p Q_2 (h,1) = (h,1)$. Thus, reading the this last matrix modulo $\varpi$ gives that $h \in \Gamma_2(\varpi)$ as claimed in Lemma 5.1.
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Paramodular Level
Below I will try to follow the same process for the section for the paramodular level. We want to show that if $h \in \textrm{Sp}_4(F)$ satisfies $Q_2 (h,1) = PQ_2 (h_1,k_1)$ with $P \in \textrm{Sp}_(F)$ and $h_1, k_1 \in \Gamma_2[\varpi]$, then necessarily $h \in \Gamma_2[\varpi]$. In other words, we want to use that $(h_1, k_1) = Q_2^{-1} P^{-1} Q_2 (h,1)$ to show $h \in \Gamma_2[\varpi]$. Since $P^{-1}$ is also in the parabolic subgroup, we just rewrite it as $P$.
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It seems one needs to build the paramodular matrices somewhat by hand to take into account the location of the $p$ and $p^{-1}$ terms. We can build the $P$ matrices as before though.
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We set $Z = (h_1,k_1) = Q_2^{-1}P Q_2 (h,1)$ and use that this is in $\Gamma_2[\varpi] \times \Gamma_2[\varpi]$ to try and deduce $h \in \Gamma_2[\varpi]$.
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We start by working on the third and fourth columns.
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Now the 8th column.
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Next is the 7th column.
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Pushing ahead with column 5.. Yikes!
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WARNING: Output truncated! full_output.txt WARNING: Output truncated! full_output.txt |
This is still a bit of a disaster. We may need to use that $K$ is a symplectic matrix to simplify all of these $k$ terms out.
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WARNING: Output truncated! full_output.txt WARNING: Output truncated! full_output.txt |
Section is well-defined
The following matrix is $(h_3,k_3)$ for $h_3, k_3 \in \Gamma_4[p]$.
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The following matrix is $p_1^{-1} p_2$ for the well-defined calculation, so it must be that this lies in the Siegel parabolic subgroup.
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This seems to indicate that we must have $h_3 = k_3$ in order to have an equality of the form $p_1 Q_2 (h_1, k_1) = p_2 Q_2 (h_2, k_2)$ with $p_{i}$ in the Siegel parabolic and $h_{i}, k_{i} \in \Gamma_2[p]$ and we have set $(h_3, k_3) = (h_1, k_1)(h_2^{-1},k_2^{-1})$. We use this to adjust the embedded matrices:
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Taking this into account we have the following form for $p_1^{-1} p_2$:
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This gives our matrix $A$ as:
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Note this $A$ is equal to the matrix $h_3$. That means that $A \in \textrm{Sp}_{2n}(F)$, and so it must have determinant $\pm 1$. That should be enough to give the same result as in the paper about it being well-defined. Can ignore the stuff below.
Since all of the $h_{ij}$ are integral, we see the determinant of $A$ is still integral but I am not sure how to see it is a unit.. hmmm..
We now want to show if $Q_2 (h,1)$ is in $P_4(F) Q_2 (\Gamma_4[\varpi] \times \Gamma_4[\varpi])$, then $h \in \Gamma_4[\varpi]$.
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Traceback (click to the left of this block for traceback) ... IndexError: list index out of range Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_172.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("QS5zdWJzdGl0dXRlKGQzMj0oQVsyLDBdPT0wKS5zb2x2ZShkMzIpWzBdLnJpZ2h0KCkpOw=="),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpREmFrz/___code___.py", line 3, in <module> exec compile(u'A.substitute(d32=(A[_sage_const_2 ,_sage_const_0 ]==_sage_const_0 ).solve(d32)[_sage_const_0 ].right()); File "", line 1, in <module> File "/usr/local/sage-6.10/local/lib/python2.7/site-packages/sage/structure/sequence.py", line 531, in __getitem__ return list.__getitem__(self,n) IndexError: list index out of range |
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Traceback (click to the left of this block for traceback) ... SyntaxError: keyword can't be an expression Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_138.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("ZXEuc3Vic3RpdHV0ZShBWzcsN109MCk7"),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmpsFx9GE/___code___.py", line 3 eq.substitute(A[_sage_const_7 ,_sage_const_7 ]=_sage_const_0 ); SyntaxError: keyword can't be an expression |
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Traceback (click to the left of this block for traceback) ... AttributeError: 'list' object has no attribute 'substitute' Traceback (most recent call last): File "<stdin>", line 1, in <module> File "_sage_input_387.py", line 10, in <module> exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("cy5zdWJzdGl0dXRlKHkwMD09MCk7czs="),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in <module> File "/tmp/tmp5oFAMx/___code___.py", line 3, in <module> exec compile(u's.substitute(y00==_sage_const_0 );s; File "", line 1, in <module> AttributeError: 'list' object has no attribute 'substitute' |
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