rcruz.tex

Note that in view of \eqref{mappingconelift} the lift of classes
of $H_2({X}, \partial {X})$ or $H^2(X)$ is the sum of two in general
non-holomorphic modular forms (see below).

In \cite{FMres} we systematically study for $\Orth(p,q)$ the
restriction of the classes $\theta_{\varphi^V_{q}}$ (also with
non-trivial coefficients) to the Borel-Serre boundary. Whenever the
restriction vanishes cohomologically, we can expect that a similar
analysis to the one given in this paper will give analogous extensions
of the geometric theta correspondence. In fact, aside from this
paper we have at present managed to do this for several other cases,
namely for modular curves with non-trivial coefficients \cite{FMspec}
generalizing work of Shintani \cite{Shintani} and for Picard modular
surfaces \cite{FM-Cogdell} generalizing work of Cogdell \cite{Cogdell}.



\subsubsection*{Linking numbers in $3$-manifolds of type Sol}

The theta series $\theta_{\phi_1^W}$ at the boundary is of independent
interest and has geometric meaning in its own right. Recall that
for two disjoint (rationally) homological trivial $1$-cycles $a$
and $b$ in a $3$-manifold $M$ we can define the {\it linking number}
of $a$ and $b$ as the intersection number
\[
\Lk(a,b) = A \cdot b
\]
of (rational) chains in $M$. Here $A$ is a $2$-chain in $M$ with boundary $a$. We show 

\begin{theorem}\label{FM-linking} (Theorem~\ref{xi'-integralP})
Let $c$ be \bf{homologically} trivial $1$-cycle in $\partial \overline{X}$ which
is disjoint from the torus fibers containing components of $\partial
C_n$. Then the holomorphic part of the weight $2$ non-holomorphic
modular form $\int_c \theta_{\phi_1^W}$ is given by the generating
series of the linking numbers $\sum_{n>0}\Lk(\partial C_n,c) q^n$.
\end{theorem}