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Update ConstantInvariant.tex
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asilverb authored Dec 12, 2019
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Expand Up @@ -930,7 +930,7 @@ \section{Proof of Theorem~\ref{thm:curves-dense}}
By Lemma~\ref{lem:rggt-dense}, there exists an element of $C$ of the form $rGG^t$ with $r \in \R^+$ and $G \in \Sl_g(\Z[1/\ell])$. Since $C$ is a cone, we may scale this element to obtain a matrix $A \in \detl \cap C$. % If $A$ does not satisfy~\eqref{eq:q}, then by Proposition~\ref{prop:q-expansion}, we may replace $A$ with $\ell^m A$ for some positive integer $m$ until the latter formula applies. Observe that $\ell^m A \in C$ also.

Let $n = \min \{\tr(AQ) \colon Q \in S\}$. By the definition of $C$, there is a unique $Q_0 \in S$ such that $\tr(AQ_0) = n$.
Thus, the coefficient of $q^n$ in the $q$-expansion of $\psimod^*(f)$ is $c_{Q_0}$. Since $Q_0 \in S$, we have $c_{Q_0} \ne 0$. Hence $\psimod^*(f) \neq 0$, as desired.
Thus, the coefficient of $q^n$ in the $q$-expansion of $\psimod^*(f)$ is $c(Q_0)$. Since $Q_0 \in S$, we have $c(Q_0) \ne 0$. Hence $\psimod^*(f) \neq 0$, as desired.
\end{proof}


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