ldanielson.tex

We also give a simple formula in Theorem~\ref{linkSol} for the
linking number of two circles contained in the fiber of a $3$-manifold
$M$ of type Sol in terms of the glueing homeomorphism for the bundle.

One can reformulate the previous theorem stating that
$\sum_{n>0}\Lk(\partial C_n,c) q^n$ is a "mixed Mock modular form"
of weight $2$; it is the product of a Mock modular form of weight
$3/2$ with a unary theta series. Such forms, which originate with
the famous Ramanujan Mock theta functions, have recently generated
great interest.

Theorem~\ref{FM-linking} (and its analogues for the Borel-Serre
boundary of modular curves with non-trivial coefficients and Picard
modular surfaces) suggest that there is a more general connection
between modular forms and linking numbers of nilmanifold subbundles
over special cycles in nilmanifold  bundles over locally symmetric
spaces.









\subsubsection*{Relation to the work of Hirzebruch and Zagier}

In their seminal paper \cite{HZ}, Hirzebruch-Zagier provided a map
from the second homology of the smooth compactification of certain
Hilbert modular surfaces $j:X \hookrightarrow \tilde{X}$ to modular
forms. They introduced the Hirzebruch-Zagier curves $T_n$ in $X$,
which are given by the closure of the cycles $C_n$ in $\tilde{X}$.
They then defined "truncated" cycles $T_n^c$ as the
projections of $T_n$ orthogonal to the subspace of $H_2(\tilde{X},\Q)$
spanned by the
compactifying divisors of $\tilde{X}$. The principal result of
\cite{HZ} was that $\sum_{n \geq 0} [T_n^c] q^n$ defines a holomorphic
modular form of weight $2$ with values in $H_2(\tilde{X},\Q)$.
We show $j_{\ast} C_n^c = T_n^c$ (Proposition~\ref{CnTn}), and hence
the Hirzebruch-Zagier theorem follows easily from Theorem~\ref{FMHZ-main}
above, see Theorem~\ref{HZTheorem}.

The main work in \cite{HZ} was to show that the generating function
\[\vspace{-.1cm}
F(\tau) = \sum_{n=0}^{\infty} (T_n^c \cdot T_m) q^n \vspace{-.1cm}
\]
for the intersection numbers in $\tilde{X}$ of $T^c_n$ with a fixed
$T_m$ is a modular form of weight $2$. The Hirzebruch-Zagier proof
of the modularity of $F$ was a remarkable synthesis of algebraic
geometry, combinatorics, and modular forms. They explicitly computed
the intersection numbers $T_n^c \cdot T_m$ as the sum of two terms,
$T_n^c \cdot T_m = (T_n \cdot T_m)_X  + ({T}_n \cdot {T}_m)_{\infty}$,
where $(T_n \cdot T_m)_X $ is the geometric intersection number of
$T_n$ and $T_m$ in the interior of $X$ and $({T}_n \cdot {T}_m)_{\infty}$
which they called the "contribution from infinity". They then proved
both generating functions $\sum_{n=0}^{\infty} (T_n \cdot T_m)_X
q^n$ and $\sum_{n=0}^{\infty}  (T_n \cdot T_m)_{\infty} q^n$ are
the holomorphic parts of two non-holomorphic forms $F_X$ and
$F_{\infty}$ with the {\it same} non-holomorphic part (with opposite
signs). Hence combining these two forms gives $F(\tau)$.