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\begin{document}

\section{The conjecture}
Let $z_1, z_2 \in \HH$ be two different complex multiplication points. Let $Q_1$, $Q_2$ be the corresponding positive definite primitive quadratic forms. Let $D_1$, $D_2$ be their discriminants. Let $k=2,3,4,5$. The conjecture is
\begin{conjecture}
There exists an algebraic number $f$, such that the lifted value of the Green's function equals
\[
    \wt{G}_k^{\HH/PSL_2\Z}(z_1, z_2) = (D_1 D_2)^{\frac{1-k}2} \log f.
\]
\end{conjecture}
Our aim here is to make the conjecture more precise. In fact we would like to describe the conjectural prime decomposition of the fractional ideal generated by $f$.
\end{document}@