The Geometric Theta Correspondence for Hilbert Modular Surfaces

THE GEOMETRIC THETA CORRESPONDENCE FOR HILBERT MODULAR SURFACES

JENS FUNKE* AND JOHN MILLSON**

Date: August 30, 2011.

* Partially supported by NSF grant DMS-0710228.

** Partially supported by NSF grant DMS-0907446, NSF FRG grant DMS-0554254, and the Simons Foundation.

1. Introduction

In a series of papers [11, 12, 13, 14] we have been studying the geometric theta correspondence (see below) for non-compact arithmetic quotients of symmetric spaces associated to orthogonal groups. It is our overall goal to develop a general theory of geometric theta liftings in the context of the real diﬀerential geometry/topology of non-compact locally symmetric spaces of orthogonal and unitary groups which generalizes the theory of Kudla-Millson in the compact case, see [24].

In this paper we study in detail the geometric theta lift for Hilbert modular surfaces. In particular, we will give a new proof and an extension (to all ﬁnite index subgroups of the Hilbert modular group) of the celebrated theorem of Hirzebruch and Zagier [16] that the generating function for the intersection numbers of the Hirzebruch-Zagier cycles is a classical modular form of weight $2$ .1 In our approach we replace Hirzebuch’s smooth complex analytic compactiﬁcation $\tilde{X}$ of the Hilbert modular surface $X$ with the (real) Borel-Serre compactiﬁcation $\bar{X}$ . The various algebro-geometric quantities that occur in [16] are then replaced by topological quantities associated to $4$ -manifolds with boundary. In particular, the “boundary contribution” in [16] is replaced by sums of linking numbers of circles (the boundaries of the cycles) in the $3$ -manifolds of type Sol (torus bundle over a circle) which comprise the Borel-Serre boundary.

The geometric theta correspondence. We ﬁrst explain the term “geometric theta correspondence”. The Weil (or oscillator) representation gives us a method to construct closed diﬀerential forms on locally symmetric spaces associated to groups which belong to dual pairs. Let $V$ be a rational quadratic space of signature $(p, q)$ with for simplicity even dimension. Then the Weil representation induces an action of ${SL}_{2} (ℝ) \times O (V_{ℝ})$ on $𝒮 (V_{ℝ})$ , the Schwartz functions on $V_{ℝ}$ . Let $G = {SO}_{0} (V_{ℝ})$ and let $K$ be a maximal compact subgroup. We let $𝔤$ and $𝔨$ be their respective Lie algebras and let $𝔤 = 𝔭 \oplus 𝔨$ be the associated Cartan decomposition. Suppose

φ \in {(𝒮 (V_{ℝ}) \otimes \land^{r} 𝔭^{*})}^{K}

is a cocycle in the relative Lie algebra complex for $G$ with values in $𝒮 (V)$ . Then $φ$ corresponds to a closed diﬀerential $r$ -form $\tilde{φ}$ on the symmetric space $D = G ∕ K$ of dimension $p q$ with values in $𝒮 (V)$ . For a coset of a lattice $ℒ$ in $V$ , we deﬁne the theta distribution $Θ = Θ_{ℒ}$ by $Θ = \sum_{ℓ \in ℒ} δ_{ℓ}$ , where $δ_{ℓ}$ is the delta measure concentrated at $ℓ$ . It is obvious that $Θ$ is invariant under $Γ = Stab (ℒ) \subset G$ . There is also a congruence subgroup $Γ^{'}$ of $SL (2, ℤ)$ ) such that $Θ$ is also invariant under $Γ^{'}$ . Hence we can apply the theta distribution to $\tilde{φ}$ to obtain a closed $r$ -form $𝜃_{φ}$ on $X = Γ ∖ D$ given by

𝜃_{φ} (ℒ) = ⟨ Θ_{ℒ}, \tilde{φ} ⟩ .

Assume now in addition that $φ$ has weight $k$ under the maximal compact subgroup $SO (2) \subset {SL}_{2} (ℝ)$ . Then $𝜃_{φ}$ also gives rise to a (in general) non-holomorphic function on the upper half place $ℍ$ which is modular of weight $k$ for $Γ^{'}$ . We may then use $𝜃_{φ}$ as the kernel of a pairing of modular forms $f$ with (closed) diﬀerential $(p q - r)$ -forms $η$ or $r$ -chains (cycles) $C$ in $X$ . The resulting pairing in $f$ , $η$ (or $C$ ), and $φ$ as these objects vary, we call the geometric theta correspondence.

The cocycle of Kudla-Millson. The key point of the work of Kudla and Millson [21, 22] is that they found (in greater generality) a family of cocycles $φ_{q}^{V}$ in ${(𝒮 (V) \otimes \land^{q} 𝔭^{*})}^{K}$ with weight $(p + q) ∕ 2$ for ${SL}_{2}$ . Moreover, these cocycles give rise to Poincaré dual forms for certain totally geodesic, “special” cycles in $X$ . Recently, it has now been shown, ﬁrst [17] for $SO (3, 2)$ , and then [1] for all $SO (p, q)$ and $p + q > 6$ (with $p \geq q$ ) in the cocompact (standard arithmetic) case that the geometric theta correspondence specialized to $φ_{q}^{V}$ induces on the adelic level an isomorphism from the appropriate space of classical modular forms to $H^{q} (X)$ . In particular, for any congruence quotient, the dual homology groups are spanned by special cycles. This gives further justiﬁcation to the term geometric theta correspondence and highlights the signiﬁcance of these cocycles. In [12] we generalize $φ_{q}^{V}$ to allow suitable non-trivial coeﬃcient systems (and one has an analogous isomorphism in [1]).

The main results. In the present paper, we consider the case when $V$ has signature $(2, 2)$ with $ℚ$ -rank $1$ . Then $D ≃ ℍ \times ℍ$ , and $X$ is a Hilbert modular surface. We let $\bar{X}$ be the Borel-Serre compactiﬁcation of $X$ which is obtained by replacing each isolated cusp associated to a rational parabolic $P$ with a boundary face $e^{'} (P)$ which turns out to be a torus bundle over a circle, a $3$ -manifold of type Sol. This makes $\bar{X}$ a $4$ -manifold with boundary. For simplicity, we assume that $X$ has only one cusp so that $\partial \bar{X} = e^{'} (P)$ , and we write $k : \partial \bar{X} ↪ \bar{X}$ for the inclusion. The special cycles $C_{n}$ 2 in question are now embedded modular and Shimura curves, and are parameterized by $n \in ℕ$ . They deﬁne relative homology classes in $H_{2} (X, \partial X, ℚ)$ .

The geometric theta correspondence of Kudla-Millson [24] for the cocycle $φ_{2}^{V}$ in this situation takes the following shape. For a compact cycle $C$ in $X$ , we have that

⟨ 𝜃_{φ_{2}^{V}}, C ⟩ = \int_{C} 𝜃_{φ_{2}^{V}} = \sum_{n \geq 0} (C_{n} \cdot C) q^{n}

(1.1)

is a holomorphic modular form of weight $2$ and is equal to the generating series of the intersection numbers with $C_{n}$ . Here $q = e^{2 π i τ}$ with $τ \in ℍ$ . (There is a similar statement for the pairing of $𝜃_{φ_{2}^{V}}$ with a closed compactly supported diﬀerential $2$ -form on $X$ representing a class in $H_{c}^{2} (X)$ , see Theorem 7.1). Our ﬁrst result is

Theorem 1.1. (Theorem 7.3) The diﬀerential form $𝜃_{φ_{2}^{V}}$ on $X$ extends to a form on $\bar{X}$ , and the restriction $k^{*}$ of $𝜃_{φ_{2}^{V}}$ to $\partial \bar{X}$ gives an exact diﬀerential form on $\partial \bar{X}$ . Moreover, there exists a theta series $𝜃_{ϕ_{1}^{W}}$ for a space $W$ of signature $(1, 1)$ of weight $2$ with values in the $1$ -forms on $\partial \bar{X}$ such that $𝜃_{ϕ_{1}^{W}}$ is a primitive for $k^{*} 𝜃_{φ_{2}^{V}}$ :

d (𝜃_{ϕ_{1}^{W}}) = k^{*} 𝜃_{φ_{2}^{V}} .

Considering the mapping cone for the inclusion $k : \partial \bar{X} ↪ \bar{X}$ (see Section 3.3) we then view the pair $[𝜃_{φ_{2}^{V}}, 𝜃_{ϕ_{1}^{W}}]$ as an element of the compactly supported cohomology $H_{c}^{2} (X)$ . Explictly, let $C$ be a relative cycle in $\bar{X}$ representing a class in $H_{2} (X, \partial X, ℤ)$ . Then the Kronecker pairing between $[𝜃_{φ_{2}^{V}}, 𝜃_{ϕ_{1}^{W}}]$ and $C$ is given by

⟨ [𝜃_{φ_{2}^{V}}, 𝜃_{ϕ_{1}^{W}}], C ⟩ = \int_{C} 𝜃_{φ_{2}^{V}} - \int_{\partial C} 𝜃_{ϕ_{1}^{W}} .

(1.2)

In this way, we obtain an extension of the geometric theta lift which captures the non-compact situation.

To describe the geometric interpretation of this extension, we study the cycle $C_{n}$ at the boundary $\partial \bar{X}$ (Section 4). The intersection of $C_{n}$ with $\partial \bar{X}$ is a union of circles contained in the torus ﬁbers of Sol. But rationally such circles are homologically trivial. Hence we can ﬁnd a (suitably normalized) rational $2$ -chain $A_{n}$ in $\partial \bar{X}$ whose boundary is the boundary of $C_{n}$ in $\partial \bar{X}$ . “Capping” oﬀ $C_{n}$ by $A_{n}$ , we obtain a closed cycle $C_{n}^{c}$ in $\bar{X}$ deﬁning a class in $H_{2} (X, ℚ)$ . Our main result is the extension of (1.1):

Theorem 1.2. (Theorem 7.7) Let $C$ be a relative cycle in $\bar{X}$ . Then

⟨ [𝜃_{φ_{2}^{V}}, 𝜃_{ϕ_{1}^{W}}], C ⟩ = \sum_{n \geq 0} (C_{n}^{c} \cdot C) q^{n}

is a holomorphic modular form of weight $2$ and is equal to the generating series of the intersection numbers with the capped cycles $C_{n}^{c}$ . (Similarly for the pairing with an arbitrary closed $2$ -form on $\bar{X}$ representing a class in $H^{2} (X)$ ).

Note that in view of (1.2) 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 [13] we systematically study for $O (p, q)$ the restriction of the classes $𝜃_{φ_{q}^{V}}$ (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 [14] generalizing work of Shintani [27] and for Picard modular surfaces [15] generalizing work of Cogdell [6].

Linking numbers in $3$ -manifolds of type Sol. The theta series $𝜃_{ϕ_{1}^{W}}$ 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 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

Theorem 1.3. (Theorem 6.3) Let $c$ be homologically trivial $1$ -cycle in $\partial \bar{X}$ 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 $\int_{c} 𝜃_{ϕ_{1}^{W}}$ is given by the generating series of the linking numbers $\sum_{n > 0} Lk (\partial C_{n}, c) q^{n}$ .

We also give a simple formula in Theorem 4.10 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 1.3 (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.

Relation to the work of Hirzebruch and Zagier. In their seminal paper [16], Hirzebruch-Zagier provided a map from the second homology of the smooth compactiﬁcation of certain Hilbert modular surfaces $j : X ↪ \tilde{X}$ 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 [16] 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 $j_{*} C_{n}^{c} = T_{n}^{c}$ (Proposition 4.7), and hence the Hirzebruch-Zagier theorem follows easily from Theorem 1.2 above, see Theorem 7.9.

The main work in [16] was to show that the generating function

F (τ) = \sum_{n = 0}^{\infty} (T_{n}^{c} \cdot T_{m}) q^{n}

for the intersection numbers in $\tilde{X}$ of $T_{n}^{c}$ 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 same non-holomorphic part (with opposite signs). Hence combining these two forms gives $F (τ)$ .

We recover this feature of the original Hirzebruch-Zagier proof via (1.2) with $C = C_{m}$ . The ﬁrst term on the right hand side of (1.2) was studied in the thesis of the ﬁrst author of this paper [9] and gives the interior intersections ${(T_{n} \cdot T_{m})}_{X}$ encoded in $F_{X}$ . So via Theorem 1.2 the second term on the right hand side of (1.2) must match the boundary contribution $F_{\infty}$ in [16], that is, we obtain

Theorem 1.4.

{(T_{n} \cdot T_{m})}_{\infty} = Lk (\partial C_{n}, \partial C_{m}) .

Hence we give an interpretation for the boundary contribution in [16] in terms of linking numbers in $\partial \bar{X}$ . In fact, the construction of $𝜃_{ϕ_{1}^{W}}$ owes a great deal to Section 2.3 in [16], where a scalar-valued version of $𝜃_{ϕ_{1}^{W}}$ is introduced, see also Example 6.4. Using Theorem 4.11 one can also make the connection between our linking numbers and the formulas of the boundary contribution in [16] explicit.

To summarize, we start with the diﬀerence of theta integrals (1.2) (which we know a priori is a holomorphic modular form), then by functorial diﬀerential topological computations we relate its Fourier coeﬃcients to intersection/linking numbers, and by direct computation of the integrals involved we obtain the explicit formulas of Hirzebruch-Zagier and a “closed form” for their generating function.

Note that Bruinier [4] and Oda [26] use related theta series to consider [16], but their overall approach is diﬀerent.

Currents. One of the key properties of the cocycle $φ_{2}^{V}$ is that the $n$ -th Fourier coeﬃcients of $𝜃_{φ_{2}^{V}}$ represents the Poincaré dual class for the cycle $C_{n}$ . Kudla-Millson establish this by showing that $φ_{2}^{V}$ gives rise to a Thom form for the normal bundle of each of the components of $C_{n}$ . To prove our main result, Theorem 1.2, we follow a diﬀerent approach using currents which is implicit in [5] and is closely related to the Green’s function $Ξ (n)$ for the divisors $C_{n}$ constructed by Kudla [18, 19]. This function plays an important role in the Kudla program (see eg [20]) which considers the analogous generating series for the special cycles in arithmetic geometry. In the non-compact situation however, one needs to modify $Ξ (n)$ to obtain a Green’s function for the cycle $T_{n}^{c}$ in $\tilde{X}$ . Discussions with U. Kühn suggest that the constructions in this paper indeed give rise to such a modiﬁcation of $Ξ (n)$ , see Remark 8.5.

We would like to thank Rolf Berndt, Jan Bruinier, Jose Burgos, Misha Kapovich, and Ulf Kühn for fruitful and extensive discussions on the constructions and results of this paper. As always it is a great pleasure to thank Steve Kudla for his interest and encouragement. Each of us began the work of relating theta lifts and special cycles with him.

We dedicate this paper to the memory of Gretchen Taylor Millson, beloved wife of the second author.

2. The Hilbert modular surface and its Borel-Serre compactiﬁcation

2.1. The symmetric space and its arithmetic quotient.

2.1.1. The orthogonal group and its symmetric space. Let $V$ be a rational vector space of dimension $4$ with a non-degenerate symmetric bilinear form $(,)$ of signature $(2, 2)$ . We let $\underset{̲}{G} = SO (V)$ , viewed as an algebraic group over $ℚ$ . We let $G = {\underset{̲}{G}}_{0} (ℝ) ≃ {SO}_{0} (2, 2)$ be the connected component of the identity of the real points of $\underset{̲}{G}$ . It is most convenient to identify the associated symmetric space $D = D_{V}$ with the space of negative $2$ -planes in $V (ℝ)$ on which the bilinear form $(,)$ is negative deﬁnite:

D = {z \subset V_{ℝ}; dim z = 2 and (,) |_{z} < 0} .

We pick an orthogonal basis ${e_{1}, e_{2}, e_{3}, e_{4}}$ of $V_{ℝ}$ with $(e_{1}, e_{1}) = (e_{2}, e_{2}) = 1$ and $(e_{3}, e_{3}) = (e_{4}, e_{4}) = - 1$ . We denote the coordinates of a vector $x$ with respect to this basis by $x_{i}$ . We pick as base point of $D$ the plane $z_{0} = [e_{3}, e_{4}]$ spanned by $e_{3}$ and $e_{4}$ , and we let $K ≃ SO (2) \times SO (2)$ be the maximal compact subgroup of $G$ stabilizing $z_{0}$ . Thus $D ≃ G ∕ K$ . Of course, $D ≃ ℍ \times ℍ$ , the product of two upper half planes.

We let $\underset{̲}{P}$ be a rational parabolic subgroup stabilizing a rational isotropic line $ℓ$ and deﬁne $P = {\underset{̲}{P}}_{0} (ℝ)$ as before. We let $\underset{̲}{N}$ be its unipotent subgroup and $N = \underset{̲}{N} (ℝ)$ . We let $u = (e_{1} + e_{4}) ∕ \sqrt{2}$ and $u^{'} = (e_{1} - e_{4}) ∕ \sqrt{2}$ be two isotropic vectors so that $(u, u^{'}) = 1$ . We assume that $u, u^{'}$ are deﬁned over $ℚ$ and obtain a rational Witt decomposition

V = ℓ \oplus W \oplus ℓ^{'}

with $ℓ = ℚ u$ , $ℓ^{'} = ℚ u^{'}$ , and a subspace $W = ℓ^{⊥} \cap {ℓ^{'}}^{⊥}$ such that $W_{ℝ} = {span}_{ℝ} (e_{2}, e_{3})$ . The choice of $u^{'}$ gives a Levi splitting of $\underset{̲}{P}$ , and we write

P = N A M

for the Langlands decomposition. Here, with respect to the basis $u, e_{2}, e_{3}, u^{'}$ , we have

\begin{array}{l} N & = \{n (w) = (\begin{matrix} 1 & (\cdot, w) & - (w, w) ∕ 2 \\ 1_{W} & - w \\ 1 \end{matrix}); w \in W_{ℝ}\}, \\ A & = \{a (t) = (\begin{matrix} t \\ 1_{W} \\ t^{- 1} \end{matrix}); t \in ℝ_{+}\}, \\ M & = \{m (s) = (\begin{matrix} 1 \\ \begin{matrix} cosh (s) & sinh (s) \\ sinh (s) & cosh (s) \end{matrix} \\ 1 \end{matrix}); s \in ℝ\} . \end{array}

Note $N ≃ W_{ℝ}$ . We obtain coordinates for $D$ by $z = z (t, s, w)$ where $z$ is the negative two-plane in $V_{ℝ}$ with $z = [n (w) a (t) m (s) e_{3}, n (w) a (t) m (s) e_{4}]$ .

2.1.2. Arithmetic Quotient. We let $L$ be an even lattice in $V$ of level $N$ , that is $L \subseteq L^{#}$ , the dual lattice, $(x, x) \in 2 ℤ$ for $x \in L$ , and $q (L^{#}) ℤ = \frac{1}{N} ℤ$ . We ﬁx $h \in L^{#}$ and let $Γ \subseteq Stab L$ be a subgroup of ﬁnite index of the stabilizer of $ℒ : = L + h$ in $G$ . For each isotropic line $ℓ = ℚ u$ , we assume that $u$ is primitive in the lattice $L$ in $V$ . We will throughout assume that the $ℚ$ -rank of $\underset{̲}{G}$ is $1$ , that is, $V$ splits exactly one hyperbolic plane over $ℚ$ . Then we deﬁne the Hilbert modular surface

X = Γ ∖ D .

Example 2.1.

An important example is the following. Let $d > 0$ be the discriminant of the real quadratic ﬁeld $K = ℚ (\sqrt{d})$ over $ℚ$ , $𝒪_{K}$ its ring of integers. We denote by $x \mapsto x^{'}$ the Galois involution on $K$ . We let $V \subset M_{2} (K)$ be the space of skew-hermitian matrices in $M_{2} (K)$ , i.e., which satisfy $^{t} x^{'} = - x$ . Then the determinant on $M_{2} (K)$ gives $V$ the structure of a non-degenerate rational quadratic space of signature $(2, 2)$ and $ℚ$ -rank $1$ . We deﬁne the integral skew-hermitian matrices by

L = \{x = (\begin{matrix} a \sqrt{d} & λ \\ - λ^{'} & b \sqrt{d} \end{matrix}) : a, b \in ℤ, λ \in 𝒪_{K}\} .

Then $L$ is a lattice of level $d$ . We embed ${SL}_{2} (K)$ into ${SL}_{2} ℝ \times {SL}_{2} (ℝ)$ by $g \mapsto (g, g^{'})$ so that ${SL}_{2} (𝒪_{K})$ acts on $L$ by $γ . x = γ x^{t} γ^{'}$ as isometries. Hirzebruch and Zagier actually considered this case for $d \equiv 1 (mod 4)$ a prime.

The quotient space $X$ is in general an oriented uniformizable orbifold with isolated singularities. We will treat $X$ as a manifold - we will use Stokes’ Theorem and Poincaré duality over $ℚ$ on $X$ . This is justiﬁed because in each instance we can pass to a ﬁnite normal cover $Y$ of $X$ with $Y$ a manifold. Hence, the formulas we want hold on $Y$ . We then then go back to the quotient by taking invariants or summing over the group $Φ$ of covering transformations. The point is that the de Rham complex of $X$ is the algebra of $Φ$ -invariants in the one of $Y$ and the rational homology (cohomology) groups of $X$ are the groups of $Φ$ -coinvariants (invariants) of those of $Y$ .

2.2. Compactiﬁcations.

2.2.1. Admissible Levi decompositions of $P$ . We let $Γ_{P} = Γ \cap P$ and $Γ_{N} = Γ_{P} \cap N$ . Then the quotient $Γ_{P} ∕ Γ_{N}$ is a non-trivial arithmetic subgroup of $\underset{̲}{P} ∕ \underset{̲}{N}$ and lies inside the connected component of the identity of the real points of $\underset{̲}{P} ∕ \underset{̲}{N}$ . Furthermore, $Γ_{P} ∕ Γ_{N}$ acts as isometries of spinor norm $1$ on the anisotropic quadratic space $ℓ^{⊥} ∕ ℓ$ of signature $(1, 1)$ . Hence $Γ_{P} ∕ Γ_{N} ≃ ℤ$ is inﬁnite cyclic. Therefore the exact sequence

1 \to Γ_{N} \to Γ_{P} \to Γ_{P} ∕ Γ_{N} \to 1

splits. We ﬁx $g \in Γ_{P}$ such that its image $\bar{g}$ generates $Γ_{P} ∕ Γ_{N}$ . Then $g$ deﬁnes a Levi subgroup $M$ . In fact, the element $g$ generates $Γ_{M} : = Γ_{P} \cap M$ . Hence

Γ_{P} = Γ_{M} ⋉ Γ_{N} .

We will say a Levi decomposition $P = N A M$ is admissible if $Γ_{P} = (M \cap Γ_{P}) ⋉ Γ_{N}$ . In the following we assume that we have picked an admissible Levi decomposition for each rational parabolic.

2.2.2. Borel-Serre compactiﬁcation. We let $\bar{D}$ be the (rational) Borel-Serre enlargement of $D$ , see [3] or [2], III.9. For any parabolic $\underset{̲}{P}$ as before with admissible Levi decomposition $P = N A M$ , we deﬁne the boundary component

e (P) = M N ≃ D_{W} \times W .

Here $D_{W} ≃ M ≃ ℝ$ is the symmetric space associated to the orthogonal group of $W$ . Then $\bar{D}$ is given by

\bar{D} = D \cup \underset{\underset{̲}{P}}{∐} e (P),

where $\underset{̲}{P}$ varies over all rational parabolics. The action of $Γ$ on $D$ extends to $\bar{D}$ in a natural way, and we let

\bar{X} : = Γ ∖ \bar{D}

be the Borel-Serre compactiﬁcation of $X = Γ ∖ D$ . This makes $\bar{X}$ a manifold with boundary such that

\partial \bar{X} = \underset{[\underset{̲}{P}]}{∐} e^{'} (P),

where for each cusp, the corresponding boundary component is given by

e^{'} (P) = Γ_{P} ∖ e (P) .

Here $[\underset{̲}{P}]$ runs over all $Γ$ -conjugacy classes. The space $X_{W} : = Γ_{M} ∖ D_{W}$ is a circle. Hence $e^{'} (P)$ is a torus bundle over the circle, where the torus $T^{2}$ is given by $Γ_{N} ∖ N$ . That is, $e^{'} (P) = X_{W} \times T^{2}$ , and we have the natural map $κ : e^{'} (P) \to X_{W}$ . We have a natural product neighborhood of $e (P)$ in $\bar{D}$ and hence for $e^{'} (P)$ in $\bar{X}$ given by $[(T, \infty] \times e^{'} (P)]$ for $T$ suﬃciently large given by $z (t, s, w)$ with $t > T$ . We let $i : X ↪ \bar{X}$ and $i_{P} : e^{'} (P) ↪ \bar{X}$ be the natural inclusions.

It is one of the fundamental properties of the Borel-Serre compactiﬁcation $\bar{X}$ that it is homotopic equivalent to $X$ itself. Hence their (co)homology groups coincide.

2.2.3. Hirzebruch’s smooth compactiﬁcation. We let $X^{'}$ be the Baily-Borel compactifciation of $X$ , which is obtained by collapsing in $\bar{X}$ each boundary component $e^{'} (P)$ to a single point or topologically by taking a cone on each component of the Borel-Serre boundary. It is well known that $X^{'}$ is a projective algebraic variety. We let $\tilde{X}$ be Hirzebruch’s smooth resolution of the cusp singularities and $π : \tilde{X} \to X^{'}$ be the natural map collapsing the compactifying divisors for each cusp. We let $j : X ↪ \tilde{X}$ be the natural embedding. Note that the Borel-Serre boundary separates $\tilde{X}$ into two pieces, the (connected) inside $X^{i n}$ , which is isomorphic to $X$ and the (disconnected) outside $X^{o u t}$ , which for each cusp is a neighborhood of the compactifying divisors. Note that we can view $e^{'} (P)$ as lying in both $X^{i n}$ and $X^{o u t}$ since the intersection $X^{i n} \cap X^{o u t}$ is equal to $∐_{\underset{̲}{P}} e (P)$ .

3. (Co)homology

In this section we describe the relationship between the (co)homology of the various compactiﬁcations.

3.1. The homology of the boundary components.

Every element of $Γ_{N} = π_{1} (T^{2})$ is a rational multiple of a commutator in $Γ_{P}$ and accordingly the image of $H_{1} (T^{2}, ℚ)$ in $H_{1} (e^{'} (P), ℚ)$ is trivial. Let $a_{P} \in H_{1} (e^{'} (P), ℤ)$ be the class of the identity section of $κ : e^{'} (P) \to X_{W}$ and $b_{P} \in H_{2} (e^{'} (P), ℤ)$ be the class of the torus ﬁber of $κ$ . It is clear that the intersection number of $a_{P}$ and $b_{P}$ is $1$ (up to sign) whence $a_{P}$ and $b_{P}$ are nontrivial primitive classes. Furthermore, $a_{P}$ generates $H_{1} (e^{'} (P), ℚ)$ and $H_{2} (e^{'} (P), ℤ) ≅ ℤ$ , generated by $b_{P}$ . So

Lemma 3.1.

The ﬁrst rational homology group of $e^{'} (P)$ is generated by $a_{P}$ .
The second homology group of $e^{'} (P)$ is generated by $b_{P}$ .

Remark 3.2. To compute the homology over $ℤ$ one has only to use the Wang sequence for a ﬁber bundle over a circle, see [25], page 67.

Let $Ω_{P}$ be the unique $P$ -invariant $2$ -form on $e^{'} (P)$ such that

\int_{b_{P}} Ω_{P} = 1 .

(3.1)

Since $b_{P}$ is the image of the fundamental class of $T^{2}$ inside $H_{2} (e^{'} (P), ℤ)$ , we see that that the restriction of $Ω_{P}$ to $T^{2}$ lifts to the area form on $W_{ℝ} ≃ N$ normalized such that $T^{2} = Γ_{N} ∖ N$ has area $1$ .

3.2. Homology and cohomology of $X$ and $\tilde{X}$ .

Accordingly to the discussion in Section 2.2.3 we have the Mayer-Vietoris sequence

0 \to \oplus_{P} H_{2} (e^{'} (P)) \to H_{2} (X) \oplus (\oplus_{P} S_{P}) \to H_{2} (\tilde{X}) \to 0 .

Here $S_{P}$ denotes the span of the classes deﬁned by compactifying divisors at the cusp associated to $P$ . The zero on the left comes from $H_{3} (\tilde{X}) = 0$ and the zero on the right comes from the fact that for each $P$ the class $a_{P}$ injects into $H_{1} (X^{o u t})$ , see [28], II.3. Since the generator $b_{P}$ has trivial intersection with each of the compactifying divisors, $b_{P}$ bounds on the outside so a fortiori it bounds in $\tilde{X}$ . Thus the above short exact sequence is the sum of the two short exact sequences $\oplus_{P} H_{2} (e^{'} (P)) \to H_{2} (X) \to j_{*} H_{2} (X)$ and $0 \to \oplus_{P} S_{P} \to \oplus_{P} S_{P}$ . By adding the third terms of the two sequences and equating them to $H_{2} (\tilde{X})$ we obtain the orthogonal splittings (for the intersection pairing) - see also [28], p.123,

\begin{array}{l} H_{2} (\tilde{X}) = j_{*} H_{2} (X) \oplus (\oplus_{[P]} S_{P}), H^{2} (\tilde{X}) = j_{#} H_{c}^{2} (X) \oplus (\oplus_{[P]} S_{P}^{\lor}) . \end{array}

Here $j_{#}$ is the push-forward map. Furthermore, the pairings on each summand are non-degenerate. Considering $\oplus_{P} H_{2} (e^{'} (P)) \to H_{2} (X) \to j_{*} H_{2} (X)$ we also obtain

Proposition 3.3. $H_{2} (\partial \bar{X})$ is the kernel of $j_{*}$ so that

j_{*} H_{2} (X) ≃ H_{2} (X) ∕ \sum_{[P]} H_{2} (e^{'} (P)) .

3.3. Compactly supported cohomology and the cohomology of the mapping cone.

We brieﬂy review the mapping-cone-complex realization of the cohomology of compact supports of $X$ . For a more detailed discussion, see [14], section 5.

We let $A_{c}^{∙} (X)$ be the complex of compactly supported diﬀerential forms on $X$ which gives rise to $H_{c}^{∙} (X)$ , the cohomology of compact supports. We now represent the compactly-supported cohomology of $X$ by the cohomology of the mapping cone $C^{∙}$ of $i^{*}$ , see [29], p.19, where as before $i : X ↪ \bar{X}$ . However, we will change the sign of the diﬀerential on $C^{∙}$ and shift the grading down by one. Thus we have

C^{i} = {(a, b), a \in A^{i} (\bar{X}), b \in A^{i - 1} (\partial \bar{X})}

with $d (a, b) = (d a, i^{*} a - d b)$ . If $(a, b)$ is a cocycle in $C^{∙}$ we will use $[[a, b]]$ to denote its cohomology class. We have

Proposition 3.4. The cochain map $A_{c}^{∙} (X) \to C^{∙}$ given by $c \mapsto (c, 0)$ is a quasi-isomorphism.

We now give a cochain map from $C^{∙}$ to $A_{c}^{∙} (X)$ which induces the inverse to the above isomorphism. We let $V$ be a product neighborhood of $\partial \bar{X}$ as in Section 2.2.2, and we let $π : V \to \partial \bar{X}$ be the projection. If $b$ is a form on $\partial \bar{X}$ we obtain a form $π^{*} b$ on V. Let $f$ be a smooth function of the geodesic ﬂow coordinate $t$ which is $1$ near $t = \infty$ and zero for $t \leq T$ for some suﬃciently large $T$ . We may regard $f$ as a function on $V$ by making it constant on the $\partial \bar{X}$ factor. We extend $f$ to all of $\bar{X}$ by making it zero oﬀ of $V$ . Let $(a, b)$ be a cocycle in $C^{i}$ . Then there exist a compactly supported closed form $α$ and a form $μ$ which vanishes on $\partial \bar{X}$ such that

a - d (f π^{*} b) = α + d μ .

We deﬁne the cohomology class $[a, b]$ in the compactly supported cohomology $H_{c}^{i} (X)$ to be the class of $α$ , and the assignment $[[a, b]] \mapsto [a, b]$ gives the desired inverse. From this we obtain the following integral formulas for the Kronecker pairings with $[a, b]$ .

Lemma 3.5. Let $η$ be a closed form on $\bar{X}$ and $C$ a relative cycle in $\bar{X}$ of appropriate degree. Then

⟨ [a, b], [η] ⟩ = \int_{\bar{X}} a \land η - \int_{\partial \bar{X}} b \land i^{*} η, and ⟨ [a, b], C ⟩ = \int_{C} a - \int_{\partial C} b .

4. Capped special cycles and linking numbers in Sol

For $x \in V$ such that $(x, x) > 0$ , we deﬁne

D_{x} = {z \in D; z ⊥ x} .

Then $D_{x}$ is an embedded upper half plane in $D$ . We let $Γ_{x} \subset Γ$ be the stabilizer of $x$ and deﬁne the special or Hirzebruch-Zagier cycle by

C_{x} = Γ_{x} ∖ D_{x},

and by slight abuse identify $C_{x}$ with its image in $X$ . These are modular or Shimura curves. For positive $n \in ℚ$ , we write $ℒ_{n} = {x \in ℒ; \frac{1}{2} (x, x) = n}$ . Then the composite cycles $C_{n}$ are given by

C_{n} = \sum_{x \in Γ ∖ ℒ_{n}} C_{x} .

Since the divisors deﬁne in general relative cycles, we take the sum in $H_{2} (X, \partial X, ℤ)$ .

4.1. The closure of special cycles in the Borel-Serre boundary and the capped cycle $C_{x}^{c}$ .

We now study the closure of $C_{x}$ in $\partial \bar{X}$ , which is the same as the intersection of ${\bar{C}}_{x}$ or $\partial C_{x}$ with the union of the hypersurfaces $e^{'} (P)$ . A straightforward calculation gives

Proposition 4.1. If $(x, u) \neq 0$ then there exists a neighborhood $U_{\infty}$ of $e (P)$ such that

D_{x} \cap U_{\infty} = \emptyset .

If $(x, u) = 0$ , then ${\bar{D}}_{x} \cap e (P)$ is contained in the ﬁber of $p$ over $s (x)$ , where $s (x)$ is the unique element of $ℝ$ satifying

(x, m (s (x)) e_{3}) = 0 .

At $s (x)$ the intersection ${\bar{D}}_{x} \cap e (P)$ is the aﬃne line in $W$ given by

{w \in W : (x, w) = (u^{'}, x)} .

We deﬁne $c_{x} \subset \partial C_{x}$ to be the closed geodesic in the ﬁber over $s (x)$ which is the image of ${\bar{D}}_{x} \cap e (P)$ under the covering $e (P) \to e^{'} (P)$ . We have

Proposition 4.2.

The $1$ -cycle $\partial C_{x}$ is a ﬁnite union of circles.
At a cusp associated to $P$ , each circle is contained in a ﬁber of the map $κ : e^{'} (P) \to X_{W}$ and hence is a rational boundary (by Lemma 3.1).
Two boundary circles $c_{x}$ and $c_{y}$ are parallel if they are contained in the same ﬁber. In particular, $c_{x} \cap c_{y} \neq \emptyset \Leftrightarrow c_{x} = c_{y} .$

We now describe the intersection of ${\bar{C}}_{n}$ or $\partial C_{n}$ with $e^{'} (P)$ . For $ℒ_{V} = ℒ = L + h$ we can write

ℒ_{W} = ℒ_{W_{P}} = (ℒ \cap u^{⊥}) ∕ (ℒ \cap u) ≃ \underset{k}{∐} (L_{W, k} + h_{W, k})

for some lattices $L_{W, k} \subset W$ and vectors $h_{W, k} \in L_{W, k}^{#}$ .

Via the isomorphism $W ≃ N$ , we can identify $Γ_{N} = N \cap Γ$ with a lattice $Λ_{W}$ in $W$ . Since $u$ is primitive in $L$ and $n (w) x = x + (w, x) u$ for a vector $x \in u^{⊥}$ we see that $ℒ_{W}$ is contained in the dual lattice of $Λ_{W}$ .

Lemma 4.3. The intersection $\partial C_{n} \cap e^{'} (P)$ is given by

\underset{\begin{array}{c} x \in Γ_{M} ∖ ℒ_{W} \\ (x, x) = 2 n \end{array}}{∐} \underset{0 \leq k < {min}_{λ \in Λ_{W}}^{'} | (λ, x) |}{∐} c_{x + k u} .

Here ${min}^{'}$ denotes that we take the minimum over all nonzero values of $| (λ, x) |$ .

Proof.

We will ﬁrst prove $\partial C_{n, P} : = \partial C_{n} \cap e^{'} (P)$ is a disjoint union

\partial C_{n, P} = \underset{y \in Γ_{P} ∖ ℒ_{n, u}}{∐} c_{y},

(4.1)

where $ℒ_{n, u} = {x \in ℒ \cap u^{⊥}; (x, x) = 2 n}$ . Indeed, ﬁrst note that by Proposition 4.1 only vectors in $ℒ_{n, u}$ can contribute to $\partial C_{n, P}$ . The action of $Γ$ on $V$ induces an equivalence relation $\sim_{Γ}$ on the set $Γ_{p} ∖ ℒ_{n, u} \subset V$ which is consequently a union of equivalence classes $[x_{i}] = {[x_{i}]}_{P}, 1 \leq i \leq k$ . We may accordingly organize the union $R$ on the right-hand side of (4.1) as $R = ∐_{i = 1}^{k} ∐_{y \in [x_{i}]} c_{y} .$ But it is clear that ${(\partial C_{x_{i}})}_{P} = ∐_{y \in [x_{i}]} c_{y}$ and hence we have the equality of $1$ -cycles in $e^{'} (P)$ and $\partial X$

{(\partial C_{x_{i}})}_{P} = \sum_{y \in {[x_{i}]}_{P}} c_{y} and \partial C_{x_{i}} = \sum_{[P]} \sum_{y \in {[x_{i}]}_{P}} c_{y},

(4.2)

since an element $y \in [x_{i}]$ gives rise to the lift $D_{y}$ of $C_{x_{i}}$ to $D$ that intersects $e (P)$ and this intersection projects to $c_{y}$ . Thus we may rewrite the right-hand side of(4.1) as $R = ∐_{\sim_{Γ} ∖ ℒ_{n, u}} \partial C_{x_{i}} .$ But it is clear that this latter union is $\partial C_{n, P}$ and (4.1) follows. Finally, we easily see that $∐_{\begin{array}{c} x \in Γ_{M} ∖ ℒ_{W} \\ (x, x) = 2 n \end{array}} ∐_{0 \leq k < {min}_{λ \in Λ_{W}}^{'} | (λ, x) |} x + k u$ is a complete set of representatives of $Γ_{P}$ -equivalence classes in $ℒ_{n, u}$ . These give the circles $c_{x + k u}$ above. □

Proposition 4.4. Let $x \in ℒ_{n, u}$ with $n > 0$ . Then there exists a rational $2$ -chain $a_{x}$ in $e^{'} (P)$ such that

$\partial a_{x} = c_{x}$
$\int_{a_{x}} Ω_{P} = 0$ , here $Ω_{P}$ is the area form for the ﬁbers (see (3.1))

Proof.

Except for the rationality of the cap this follows immediately from Proposition 4.2. The problem is to ﬁnd a cap $a_{x}$ such that $\int_{a_{x}} Ω_{P} \in ℚ$ . We will prove this in Section 4.3 below. □

We will deﬁne ${(A_{x})}_{P}$ by ${(A_{x})}_{P} = \sum_{y \in [x]} a_{x}$ . Then sum over the components $e^{'} (P)$ to obtain $A_{x}$ a rational $2$ -chain in $\partial X$ . Then we have (noting that ${(\partial C_{x})}_{P} = \sum_{y \in [x]} c_{y}$ )

\partial A_{x} = \partial C_{x} .

Deﬁnition 4.5. We deﬁne the rational absolute $2$ -cycle in $\bar{X}$ by

C_{x}^{c} = C_{x} \cup (- A_{x})

with the $2$ -chain $A_{x}$ in $\partial \bar{X}$ as in Proposition 4.4. In particular, $C_{x}^{c}$ deﬁnes a class in $H_{2} (\bar{X}) = H_{2} (X)$ . In the same way we obtain $C_{n}^{c}$ .

4.2. The closure of the special cycles in $\tilde{X}$ and the cycle $T_{n}^{c}$ .

Following Hirzebruch-Zagier we let $T_{n}$ be the cycle in $\tilde{X}$ given by the closure of the cycle $C_{n}$ in $\tilde{X}$ . Hence $T_{n}$ deﬁnes a class in $H_{2} (\tilde{X})$ .

Deﬁnition 4.6. Consider the decomposition $H_{2} (\tilde{X}) = j_{*} H_{2} (X) \oplus (\oplus_{[P]} S_{p})$ , which is orthogonal with respect to the intersection pairing on $\tilde{X}$ . We let $T_{n}^{c}$ be the image of $T_{n}$ under orthogonal projection onto the summand $j_{*} H_{2} (X)$ .

Proposition 4.7. We have

j_{*} C_{n}^{c} = T_{n}^{c} .

Proof. For simplicity, we assume that $X$ has only one cusp. The $3$ -manifold $e^{'} (P)$ separates $T_{n}$ and we can write $T_{n} = T_{n} \cap X^{i n} + T_{n} \cap X^{o u t}$ as (appropriately oriented) $2$ -chains in $\tilde{X}$ . It is obvious that we have $j_{*} {\bar{C}}_{n} = T_{n} \cap X^{i n}$ as $2$ -chains. We write $B_{n} = T_{n} \cap X^{o u t}$ . We have $\partial C_{n} = - \partial B_{n}$ . Hence we can write $T_{n} = j_{*} C_{n}^{c} + B_{n}^{c}$ , the sum of two $2$ -cycles in $\tilde{X}$ . Here $B_{n}^{c}$ is obtained by ‘capping’ $B_{n}$ in $e^{'} (P)$ with the negative of the cap $A_{n}$ of $C_{n}^{c}$ . Since $j_{*} C_{n}^{c}$ is clearly orthogonal to $S_{P}$ (since it lies in $X^{i n}$ ) and $B_{n}^{c} \in S_{P}$ (since it lies in $X^{o u t}$ ) the decomposition $T_{n} = j_{*} C_{n}^{c} + B_{n}^{c}$ is just the decomposition of $T_{n}$ relative to the splitting $H_{2} (\tilde{X}) = j_{*} H_{2} (X) \oplus S_{P}$ . Hence $T_{n}^{c} = j_{*} C_{n}^{c}$ , as claimed. □

4.3. Rationality of the cap.

We will now prove Proposition 4.4. In fact we will show that it holds for any circle $α$ contained in a torus ﬁber of $e (P)$ and passing through a rational point. We would like to thank Misha Kapovich for simplifying our original argument. The idea is to construct, for each component of $\partial C_{x}$ , a $2$ -chain $A$ with that component as boundary so that $A$ is a sum $P + T + ℳ (γ_{0})$ of three simplicial $2$ -chains in $M$ . We then verify that the “parallelogram” $P$ and the “triangle” $T$ have rational area and the period of $Ω$ over the “monodromy chain” $ℳ (γ_{0})$ is zero.

In what follows we will pass from pictures in the plane involving directed line segments, triangles and parallelograms to identities in the space of simplicial $1$ -chains $C_{1} (T^{2})$ on $T^{2}$ . The principal behind this is that any $k$ -dimensional subcomplex $S$ of a simplicial complex $Y$ which is the fundamental cycle of an oriented $k$ -submanifold $| S |$ (possibly with boundary) of $Y$ corresponds in a unique way to a sum of oriented $k$ -simplices in $C_{k} (Y)$ .

In this subsection we will work with a general $3$ -manifold $M$ with Sol geometry. Of course this includes all the manifolds $e^{'} (P)$ that occur in this paper. Let $f \in SL (2, ℤ)$ be a hyperbolic element. We will then consider the $3$ -manifold $M$ obtained from $ℝ \times T^{2}$ (with the $2$ -torus $T^{2} = W ∕ ℤ^{2}$ ) given by the relation

(s, w) \sim (s + 1, f (w)) .

(4.3)

We let $π : ℝ \times T^{2} \to M$ be the resulting inﬁnite cyclic covering.

We now deﬁne notation we will use below. We will use Greek letters to denote closed geodesics on $T^{2}$ , a subscript $c$ will indicate that the geodesic starts at the point $c$ on $T^{2}$ . We will use the analogous notation for geodesic arcs on $W$ . We will use $[α]$ to denote the corresponding homology class of a closed geodesic $α$ on $T^{2}$ . If $x$ and $y$ are points on $W$ we will use $\bar{x y}$ to denote the oriented line segment joining $x$ to $y$ and $\vec{x y}$ to denote the corresponding (free) vector i.e. the equivalence class of $\bar{x y}$ modulo parallel translation.

We ﬁrst take care of the fact that $α$ does not necessarily pass through the origin. For convenience we will assume $α$ is in the ﬁber over the base-point $z (x)$ corresponding to $s = 0$ . Let $α_{0}$ be the parallel translate of $α$ to the origin. Then we can ﬁnd a cylinder $P$ , image of an oriented parallelogram $\tilde{P}$ under the universal cover $W \to T^{2}$ with rational vertices, such that in $Z_{1} (T^{2}, ℚ)$ , the group of rational $1$ -cycles, we have

\partial P = α - α_{0} .

(4.4)

Since $\tilde{P}$ has rational vertices we ﬁnd $\int_{P} Ω = \int_{\tilde{P}} Ω \in ℚ$ .

Now we take care of the harder part of ﬁnding $A$ as above. The key is the construction of “monodromy $2$ -chains”. For any closed geodesic $γ_{0} \subset T^{2}$ starting at $0$ we deﬁne the monodromy $2$ -chain $ℳ (γ_{0})$ to be the image of the cylinder $γ_{0} \times [0, 1] \subset T^{2} \times ℝ$ in $M$ . The reader will verify using (4.3) that in $Z_{1} (T^{2}, ℚ)$ we have

\partial ℳ (γ_{0}) = f^{- 1} (γ_{0}) - γ_{0} .

(4.5)

Since $f$ preserves the origin, the geodesic $f^{- 1} (γ_{0})$ is also a closed geodesic starting at the origin. Since $f^{- 1}$ is hyperbolic we have $| tr (f^{- 1}) | > 2$ and hence $det (f^{- 1} - I) = d e t (I - f) = tr (f) - 2 \neq 0$ . Put $N = det (f^{- 1} - I)$ and deﬁne $[γ_{0}] \in H_{1} (T^{2}, ℤ)$ by

f^{- 1} ([γ_{0}]) - [γ_{0}] = N [α_{0}] .

(4.6)

Note that $[γ_{0}] = N {{(f^{- 1} - I)}^{- 1} ([α_{0}])}$ is necessarily an integer homology class. Also note that is an equation in the ﬁrst homology, it is not an equation in the group of $1$ -cycles $Z_{1} (T^{2}, ℚ)$ . Since any homology class contains a unique closed geodesic starting at the origin we obtain a closed geodesic $γ_{0} \in [γ_{0}]$ and a corresponding monodromy $2$ -chain $ℳ (γ_{0})$ whence (4.5) holds in $Z_{1} (T^{2}, ℚ)$ . We now solve

Problem 4.8. Find an equation in $Z_{1} (T^{2}, ℤ)$ which descends to (4.6).

Let $h_{1}$ resp. $h_{2}$ denote the covering transformation of $π$ corresponding to the element $α_{0}$ resp $γ_{0}$ of the fundamental group of $T^{2}$ . Deﬁne $c_{1}$ and $c_{2}$ in $W$ by $c_{1} = N h_{1} (0)$ and $c_{2} = h_{2} (0)$ . Deﬁne $d \in W$ by $d = f^{- 1} (c_{2})$ in $W$ . Let $\tilde{T}$ be the oriented triangle with vertices $0, c_{2}, d$ . Then

(i) π (\bar{0 c_{1}}) = N α_{0} (i i) π (\bar{0 c_{2}}) = γ_{0} (i i i) π (\bar{0 d}) = π (f^{- 1} (\bar{0 c_{2}})) = f^{- 1} (γ_{0}) .

We now leave it to the reader to combine the homology equation (4.6) and the three equations to show the equality of directed line segments

h_{2} (\bar{0 c_{1}}) = \bar{c_{2} d} .

(4.7)

With this we can solve the problem. We see that if we consider $\tilde{T}$ as an oriented $2$ -simplex we have the following equality of one chains

\partial \tilde{T} = \bar{0 c_{2}} + \bar{c_{2} d} - \bar{0 d} .

Let $T$ be the image of $\tilde{T}$ under $π$ . Take the direct image of the previous equation under $π$ and use equation (4.7) which implies that the second edge $\bar{c_{2} d}$ is equivalent under $h_{2}$ in the covering group to the directed line segment $\bar{0 c_{1}}$ which maps to $N α_{0}$ . Hence $\bar{c_{2} d}$ also maps to $N α_{0}$ . We obtain the following equation in $Z_{1} (T^{2}, ℤ)$

\partial T = γ_{0} + N α_{0} - f^{- 1} (γ_{0}),

(4.8)

and we have solved the above problem. Combining (4.5) and (4.8) we have

\partial (ℳ (γ_{0}) + T) = f^{- 1} (γ_{0}) - γ_{0} + γ_{0} + α_{0} - f^{- 1} (γ_{0}) = N α_{0} .

Combining this with (4.4) and setting where $A_{0} = ℳ (γ_{0}) + T$ we obtain

\partial (N P + A_{0}) = N α,

(4.9)

in $Z_{1} (M, ℤ)$ . Hence if we deﬁne $A$ to be the rational chain $A = \frac{1}{N} (N P + A_{0}) = P + \frac{1}{N} T + \frac{1}{N} ℳ (γ_{0})$ in $M$ we have the following equation in $Z_{1} (M, ℚ)$ :

\partial A = α .

Finally, the integral of $Ω$ over $A$ is rational. Indeed, the integral over $P$ is rational. Since all vertices of $\tilde{T}$ are integral the area of $\tilde{T}$ is integral, the integral of $Ω$ over $T$ is integral. Thus it suﬃces to observe that the restriction of $Ω$ to $ℳ (c)$ is zero. With this we have completed the proof of Proposition 4.4.

4.4. Linking numbers in Sol.

In the introduction we deﬁned the linking number of two two disjoint homologically trivial $1$ -cycles $a$ and $b$ in a closed $3$ -manifold $M$ as $Lk (a, b) = ⟨ A, b ⟩$ , where $A$ is any rational $2$ -chain in $M$ with boundary $a$ . Since $b$ deﬁnes a trivial homology class in $M$ , the link is well-deﬁned, ie, does not depend on the choice of $A$ .

We let $M$ be the Sol manifold as before realized as in Section 4.3 via (4.3) and consider the case when $a$ and $b$ are two contained in two torus ﬁbers. Then by the previous section they are homologically trivial. If $a$ and $b$ are contained in the same ﬁber we move $b$ to the right (i.e. in the direction of positive $s$ ) to a nearby ﬁber. We take $a, b \in H_{1} (T^{2}, ℤ)$ , and in this section we are allowed to confuse $a$ and $b$ with their representatives in the lattice $ℤ^{2}$ and the unique closed geodesic in $T^{2}$ passing through the origin that represents them. We will write for the image of $a$ and $b$ in $ℝ \times T^{2}$ and $M$ $a = a (0) = 0 \times a$ and $b = b (𝜀) = 𝜀 \times b$ . Our goal is to compute the linking number $L k (a, b (𝜖))$ . By the explicit construction of the cap $A$ in Section 4.3 we obtain

Lemma 4.9.

L k (a, b (𝜖)) = M (c) \cdot b (𝜖) = c (𝜖) \cdot b (𝜖) = c \cdot b .

Here $c$ is the rational one cycle obtained by solving $(f^{- 1} - I) (c) = a$ and $M (c)$ is the (rational) monodromy $2$ -chain associated to $c$ (see above) with boundary $\partial M (c) = (f^{- 1} - I) (c) = a$ . Here the ﬁrst $\cdot$ is the intersection of chains in $M$ , the next $\cdot$ is the intersection number of $1$ -cycles in the ﬁber $𝜖 \times T^{2}$ and the last $\cdot$ is the intersection number of $1$ -cycles in $0 \times T^{2}$ .

Noting that this last intersection number coincides with the intersection number of the underlying homology classes which in term coincides with the symplectic form $⟨ \cdot, \cdot ⟩$ on $H_{1} (T^{2}, ℚ)$ we have found our desired formula for the linking number.

Theorem 4.10. $L k (a, b (𝜖)) = ⟨ {(f^{- 1} - I)}^{- 1} (a), b ⟩ .$

It is a remarkable fact that there is a simple formula involving only the action of the glueing homeomorphism $f \in SL (2, ℤ)$ on $H_{1} (T^{2}, ℤ)$ for linking numbers for $1$ -cycles contained in ﬁber tori $T^{2}$ of in Sol (unlike the case of linking numbers in $ℝ^{3}$ ).

This immediately leads to an explicit formula for the numbers $L k (\partial C_{n}, \partial C_{m})$ . Using Lemma 4.3 we obtain

Theorem 4.11. Let $g = {(f^{- 1} - I)}^{- 1}$ . Then

L k ({(\partial C_{n})}_{P}, {(\partial C_{m})}_{P}) = \sum_{\begin{array}{c} x \in Γ_{M} ∖ ℒ_{W} \\ (x, x) = 2 n \end{array}} \sum_{\begin{array}{c} x^{'} \in Γ_{M} ∖ ℒ_{W} \\ (x, x) = 2 m \end{array}} (min_{λ \in Λ_{W}}^{'} | (λ, x) |) (min_{μ \in Λ_{W}}^{'} | (μ, x^{'}) |) ⟨ g (J x), J x^{'} ⟩ .

Here $J x$ is properly oriented primitive vector in $Λ_{W}$ such that $(J x, x) = 0$ .

Example 4.12. We consider the integral skew Hermitian matrices in Example 2.1. Let $u = (\begin{matrix} \sqrt{p} & 0 \\ 0 & 0 \end{matrix})$ , so that $W = {(\begin{matrix} 0 & λ \\ - λ^{'} & 0 \end{matrix}); λ \in K} ≃ K$ . The symplectic form on $K$ is given by $⟨ λ, μ ⟩ = \frac{1}{\sqrt{p}} (λ μ^{'} - λ^{'} μ)$ . The action of the unipotent radical $N = \{n (λ) = (\begin{matrix} 1 & λ \\ 0 & 1 \end{matrix})\}$ on a vector $μ \in K$ is now slightly diﬀerent, namely, $n (λ) μ = μ + ⟨ λ, μ ⟩ u$ . Hence in these coordinates, $\partial C_{μ}$ is given by the image of the line $ℝ μ = {λ \in K_{ℝ}; ⟨ λ, μ ⟩ = 0}$ , and $({min}_{λ \in 𝒪_{K}}^{'} | ⟨ λ, μ ⟩ |) μ$ is a primitive generator in $𝒪_{K}$ for that line. We let $𝜀$ be a generator of $U_{+}$ , the totally positive units in $𝒪_{K}$ , and we assume that the glueing map $f$ is realized by multiplication with $𝜀^{'}$ . For $d \equiv 1 (mod 4)$ a prime and $m = 1$ , $C_{1}$ has only component arising from $x = 1 \in K$ and $C_{1} ≃ {SL}_{2} (ℤ) ∖ ℍ$ . Then Theorem 4.11 becomes (the ${min}^{'}$ -term is now wrt $⟨, ⟩$ )

L k ({(\partial C_{n})}_{P}, {(\partial C_{1})}_{P}) = 2 \sum_{\begin{array}{c} μ \in U_{+} ∖ 𝒪_{K} \\ μ μ^{'} = n, μ ≫ 0 \end{array}} ⟨\frac{μ}{𝜀 - 1}, 1⟩ = 2 \sum_{\begin{array}{c} μ \in U_{+} ∖ 𝒪_{K} \\ μ μ^{'} = n, μ ≫ 0 \end{array}} = \frac{2}{\sqrt{p}} \frac{μ + μ^{'} 𝜀}{𝜀 - 1} .

This is (twice) the “boundary contribution” in [16], Section 1.4, see also Section 7.5.

5. Schwartz functions and forms

Let $U$ be a non-degenerate rational quadratic space of signature $(p, q)$ and even dimension $m$ . We will later apply the following to $U = V$ and $U = W$ . Changing notation from before, we let $G = {SO}_{0} (U_{ℝ})$ with maximal compact subgroup $K$ and write $D = G ∕ K$ for the associated symmetric space. We let $𝒮 (U_{ℝ})$ be the space of Schwartz functions on $U_{ℝ}$ on which ${SL}_{2} (ℝ)$ acts via the Weil representation $ω$ .

5.1. Extending certain Schwartz functions to functions of $τ \in ℍ$ and $z \in D$ .

Let $φ \in 𝒮 (U_{ℝ})$ be an eigenfunction under the maximal compact $SO (2)$ of ${SL}_{2} (ℝ)$ of weight $r$ . Deﬁne $g_{τ}^{'} \in {SL}_{2} (ℝ)$ by $g_{τ}^{'} = (\begin{matrix} 1 & u \\ 0 & 1 \end{matrix}) (\begin{matrix} v^{1 ∕ 2} & 0 \\ 0 & v^{- 1 ∕ 2} \end{matrix})$ . Then we have $ω (g_{τ}^{'}) φ (x) = v^{m ∕ 4} φ (\sqrt{v} x) e^{π i (x, x) u}$ . Accordingly we deﬁne

φ (x, τ) = v^{- r ∕ 2} ω (g_{τ}^{'}) φ (x) = v^{- r ∕ 2 + m ∕ 4} φ^{0} (\sqrt{v} x) e^{π i (x, x) τ} .

(5.1)

Here we have also deﬁned $φ^{0} (x) = φ (x) e^{π (x, x)}$ . Let $E$ be a $G$ -module and let $g_{z} \in G$ be any element that carries the basepoint $z_{0}$ in $D$ to $z \in D$ . Then deﬁne for $φ \in {[𝒮 (U_{ℝ}) \otimes E]}^{K}$ , the $E$ -valued $K$ -invariant Schwartz functions on $U_{ℝ}$ , the functions $φ (x, z)$ and $φ (x, τ, z)$ for $x \in U, z \in D, τ \in ℍ$ by

φ (x, z) = g_{z} φ (g_{z}^{- 1} x) and φ (x, τ, z) = g_{z} φ (g_{z}^{- 1} x, τ) .

We will continue to use these notational conventions for other (not necessarily Schwartz) functions that arise in this paper.

5.2. Schwartz forms for $V$ .

Let $𝔤$ be the Lie algebra of $G$ and $𝔤 = 𝔨 \oplus 𝔭$ be the Cartan decomposition of $𝔤$ associated to $K$ . We identify $𝔤 ≃ \land 2 V_{ℝ}$ as usual via $(v_{1} \land v_{2}) (v) = (v_{1}, v) v_{2} - (v_{2}, v) v_{1}$ . We write $X_{i j} = e_{i} \land e_{j} \in 𝔤$ and note that $𝔭$ is spanned by $X_{i j}$ with $1 \leq i \leq 2$ and $3 \leq j \leq 4$ . We write $ω_{i j}$ for their dual. We orient $D$ such that $ω_{13} \land ω_{14} \land ω_{23} \land ω_{24}$ gives rise to the $G$ -invariant volume element on $D$ .

5.2.1. Special forms for $V$ . The Kudla-Millson form $φ_{2}$ is an element in

{[𝒮 (V_{ℝ}) \otimes 𝒜^{2} (D)]}^{G} ≃ {[𝒮 (V_{ℝ}) \otimes \land 2 𝔭^{*}]}^{K},

where the isomorphism is given by evaluation at the base point. Here $𝒜^{2} (D)$ denotes the diﬀerential $2$ -forms on $D$ . Note that $G$ acts diagonally in the natural fashion. At the base point $φ_{2}$ is given by

φ_{2} = \frac{1}{2} \prod_{μ = 3}^{4} \sum_{α = 1}^{2} (x_{α} - \frac{1}{2 π} \frac{\partial}{\partial x_{α}}) φ_{0} \otimes ω_{α μ} .

Here $φ_{0} (x) : = e^{- π {(x, x)}_{0}}$ , where ${(x, x)}_{0} = \sum_{i = 1}^{4} x_{i}^{2}$ is the minimal majorant associated to the base point in $D$ . Note that $φ_{2}$ has weight $2$ , see [21]. There is another Schwartz form $ψ_{1}$ of weight $0$ which lies in ${[𝒮 (V_{ℝ}) \otimes 𝒜^{1} (D)]}^{G} ≃ {[𝒮 (V_{ℝ}) \otimes 𝔭^{*}]}^{K}$ and is given by

ψ_{1} = - x_{1} x_{3} φ_{0} (x) \otimes ω_{14} + x_{1} x_{4} φ_{0} (x) \otimes ω_{13} - x_{2} x_{3} φ_{0} (x) \otimes ω_{24} + x_{2} x_{4} φ_{0} (x) \otimes ω_{23} .

(5.2)

The key relationship is (see [24], §8)

Theorem 5.1.

ω (L) φ_{2} = d ψ_{1} .

Here $ω (L)$ is the Weil representation action of the ${SL}_{2}$ -lowering operator $L = \frac{1}{2} (\begin{matrix} 1 & - i \\ - i & - 1 \end{matrix}) \in {𝔰 𝔩}_{2} (ℝ)$ on $𝒮 (V_{ℝ})$ , while $d$ denotes the exterior diﬀerentiation on $D$ .

On the upper half plane $ℍ$ , the action of $L$ corresponds to the action of the classical Maass lowering operator which we also denote by $L$ . For a function $f$ on $ℍ$ , we have

L f = - 2 i v^{2} \frac{\partial}{\partial \bar{τ}} f .

When made explicit using (5.1) Theorem 5.1 translates to

v \frac{\partial}{\partial v} φ_{2}^{0} (\sqrt{v} x) = d (ψ_{1}^{0} (\sqrt{v} x)) .

(5.3)

5.2.2. The singular form ${\tilde{ψ}}_{1}$ . We deﬁne the singular form ${\tilde{ψ}}_{1}$ by

\begin{align} {\tilde{ψ}}_{1} (x) & = - (\int_{1}^{\infty} ψ_{1}^{0} (\sqrt{r} x) \frac{d r}{r}) e^{- π (x, x)} = - \frac{1}{2 π (x_{3}^{2} + x_{4}^{2})} ψ_{1} (x) . & (5.4) \end{align}

for $x \neq 0$ , and as before ${\tilde{ψ}}_{2, 0}^{0} (x) = {\tilde{ψ}}_{1} (x) e^{π (x, x)}$ and ${\tilde{ψ}}_{1} (x, z)$ . We see that ${\tilde{ψ}}_{1}$ is deﬁned for $x \notin span {[e_{3}, e_{4}]}^{⊥}$ . Formulated diﬀerently, ${\tilde{ψ}}_{1} (x, z)$ for ﬁxed $x$ is deﬁned for $z \notin D_{x}$ . Furthermore, as if ${\tilde{ψ}}_{1}$ was a Schwartz function of weight $2$ , we deﬁne

\begin{align} {\tilde{ψ}}_{1} (x, τ, z) & = {\tilde{ψ}}_{1}^{0} (\sqrt{v} x, z) e^{π i (x, x) τ} = - (\int_{v}^{\infty} ψ_{1}^{0} (\sqrt{r} x, z) \frac{d r}{r}) e^{π i (x, x) τ} . & (5.5) \end{align}

Proposition 5.2.

${\tilde{ψ}}_{1} (x, z)$ is a diﬀerential $1$ -form with singularities along $D_{x}$ . Outside $D_{x}$ , we have

d {\tilde{ψ}}_{1} (x, z) = φ_{2} (x, z) .

Here $d$ denotes the exterior diﬀerentiation on $D$ . In particular, for $(x, x) \leq 0$ , we see that $φ_{2} (x)$ is exact. Furthermore,

L {\tilde{ψ}}_{1} (x, τ) = ψ_{1} (x, τ) .

Proof.

Using (5.4) and (5.3), we see

\begin{array}{l} d {\tilde{ψ}}_{1}^{0} (x, z) & = - \int_{1}^{\infty} d (ψ_{1}^{0} (\sqrt{r} x, z)) \frac{d r}{r} = - \int_{1}^{\infty} \frac{\partial}{\partial r} (φ_{2}^{0} (\sqrt{r} x, z)) \frac{d r}{r} = φ_{2}^{0} (x, z), \end{array}

as claimed. The formula $L {\tilde{ψ}}_{1} (x, τ) = ψ_{1} (x, τ)$ follows easily from (5.5). □

Remark 5.3. The construction of the singular form $\tilde{ψ}$ works in much greater generality for $O (p, q)$ whenever we have two Schwartz forms $ψ$ and $φ$ (of weight $r - 2$ and $r$ resp.) such that

d ψ = L φ .

Then the analogous construction of $\tilde{ψ}$ then immediately yields $d \tilde{ψ} = φ$ outside a singular set. The main example for this are the general Kudla-Millson forms $φ_{q}$ and $ψ_{q - 1}$ , see [24]. For these forms, this construction is already implicit in [5]. In particular, the proof of Theorem 7.2 in [5] shows that $\tilde{ψ}$ gives rise to a diﬀerential character for the analogous cycle $C_{x}$ , see also Section 8 of this paper. The unitary case will be considered in [10].

5.3. Schwartz forms for $W$ .

Let $W \subset V$ be the rational quadratic space of signature $(1, 1)$ obtained from the Witt decomposition of $V$ . We will refer to the nullcone of $W$ as the light-cone. We write $𝔪 ≃ ℝ$ for the Lie algebra of $M = {SO}_{0} (W_{ℝ})$ . Then $X_{23} = e_{2} \land e_{3}$ is its natural generator with dual $ω_{23}$ . We identify the associated symmetric space $D_{W}$ to $M$ with the space of lines in $W_{ℝ}$ on which the bilinear form $(,)$ is negative deﬁnite:

D_{W} = {s \subset W_{ℝ}; dim s = 1 and (,) |_{s} < 0} .

We pick as base point of $D_{W}$ the line $s_{0}$ spanned by $e_{3}$ . We set $x (s) : = m (s) e_{3} = sinh (s) e_{2} + cosh (s) e_{3}$ . This realizes the isomorphism $D_{W} ≃ ℝ$ . Namely, $s = span x (s)$ . Accordingly, we frequently write $s$ for $s$ and vice versa. A vector $x \in W$ of positive length deﬁnes a point $D_{W, x}$ in $D$ via $D_{W, x} = {s \in D; s ⊥ x}$ . So $s = D_{W, x}$ if and only if $(x, x (s)) = 0$ . We also write $s (x) = D_{W, x}$ .

5.3.1. Special forms for $W$ . We carry over the conventions from section 5.1. We ﬁrst consider the Schwartz form $φ_{1, 1}$ on $W_{ℝ}$ constructed in [12] (in much greater generality) with values in $𝒜^{1} (D_{W}) \otimes W_{ℂ}$ . More precisely,

φ_{1, 1} \in {[𝒮 (W_{ℝ}) \otimes 𝒜^{1} (D_{W}) \otimes W_{ℂ}]}^{M} ≃ [𝒮 (W_{ℝ}) \otimes 𝔪^{*} \otimes W_{ℂ}],

Here $M$ acts diagonally on all three factors. Explicitly at the base point, we have

φ_{1, 1} (x) = \frac{1}{2^{3 ∕ 2}} (4 x_{2}^{2} - \frac{1}{π}) e^{- π (x_{2}^{2} + x_{3}^{2})} \otimes ω_{23} \otimes e_{2} .

Note that $φ_{1, 1}$ has weight $2$ , see [12], Theorem 6.2. We deﬁne $φ_{1, 1} (x, s)$ and $φ_{1, 1}^{0}$ as before. There is another Schwartz function $ψ_{0, 1}$ of weight $0$ given by

\begin{matrix} ψ_{0, 1} (x) = - \frac{1}{\sqrt{2}} x_{2} x_{3} e^{- π (x_{2}^{2} + x_{3}^{2})} \otimes 1 \otimes e_{2} + \frac{1}{4 \sqrt{2} π} e^{- π (x_{2}^{2} + x_{3}^{2})} \otimes 1 \otimes e_{3} \\ \in [𝒮 (W_{ℝ}) \otimes \land 0 𝔪^{*} \otimes W_{ℂ}], \end{matrix}

and also $ψ_{0, 1} (x, s)$ and $ψ_{0, 1}^{0}$ . Note that the notation diﬀers from [12], section 6.5. The function $ψ_{0, 1}$ deﬁned here is the term $- ψ_{1, 1} - \frac{1}{2} Λ_{1, 1}$ given in Theorem 6.11 in [12]. The key relation between $φ_{1, 1}$ and $ψ_{0, 1}$ (correcting a sign mistake in [12]) is given by

Theorem 5.4. ([12], Theorem 6.2)

ω (L) φ_{1, 1} = d ψ_{0, 1} .

When made explicit, we have, again using (5.1),

v^{3 ∕ 2} \frac{\partial}{\partial v} (v^{- 1 ∕ 2} φ_{1, 1}^{0} (\sqrt{v} x, s)) = d (ψ_{0, 1}^{0} (\sqrt{v} x, s)) .

(5.6)

5.3.2. The singular Schwartz function ${\tilde{ψ}}_{0, 1}$ . In the same way as for $V$ we deﬁne

\begin{align} {\tilde{ψ}}_{0, 1} (x) & = - (\int_{1}^{\infty} ψ_{0, 1}^{0} (\sqrt{r} x) r^{- 3 ∕ 2} d r) e^{- π (x, x)} & (5.7) \end{align}

for all $x \in W$ , including $x = 0$ . Deﬁne ${\tilde{ψ}}_{0, 1}^{0} (x)$ , ${\tilde{ψ}}_{0, 1}^{0} (x, s)$ as before and also

\begin{align} {\tilde{ψ}}_{0, 1} (x, τ, s) & = v^{- 1 ∕ 2} {\tilde{ψ}}_{0, 1}^{0} (\sqrt{v} x, s) e^{π i (x, x) τ} = - (\int_{v}^{\infty} ψ_{0, 1}^{0} (\sqrt{r} x, s) r^{- 3 ∕ 2} d r) e^{π i (x, x) τ} . \end{align}

Note that ${\tilde{ψ}}_{0, 1} (x, s)$ has a singularity at $D_{w, x}$ . Deﬁne functions $A$ and $B$ by

{\tilde{ψ}}_{0, 1} (x) = A (x) \otimes 1 \otimes e_{2} + B (x) \otimes 1 \otimes e_{3}

and note

- X_{23} B (x) = A (x) .

(5.8)

We extend these functions to $D_{W}$ as before. We see by integrating by parts

Lemma 5.5.

\begin{array}{l} A (x) & = \frac{1}{2 \sqrt{π}} x_{2} \frac{x_{3}}{| x_{3} |} Γ (\frac{1}{2}, 2 π x_{3}^{2}) e^{- π (x, x)} \\ B (x) & = - \frac{1}{2 \sqrt{2} π} e^{- π (x_{2}^{2} + x_{3}^{2})} + \frac{1}{2 \sqrt{π}} | x_{3} | Γ (\frac{1}{2}, 2 π x_{3}^{2}) e^{- π (x, x)} . \end{array}

Here $Γ (\frac{1}{2}, a) = \int_{a}^{\infty} e^{- u} u^{- 1 ∕ 2} d u$ is the incomplete $Γ$ -funtion at $s = 1 ∕ 2$ .

It is now immediate that $B$ is continuous and bounded on $D_{W}$ . Since $A$ is clearly bounded we ﬁnd that $A$ and $B$ are locally integrable on $D_{W}$ and integrable and square-integrable on $W$ . The singularities of $A$ and $B$ are given as follows.

Lemma 5.6.

$B (x) - (1 ∕ 2) | x_{3} | e^{- π (x, x)}$ is $C^{2}$ on the Minkowski plane $W$ .
$A (x) - (1 ∕ 2) x_{2} \frac{x_{3}}{| x_{3} |} e^{- π (x, x)}$ is $C^{1}$ on the Minkowski plane $W$ .

Proof. Use Lemma 5.5, expand the incomplete gamma function around $x_{3} = 0$ , and observe that $| x | x^{n}$ is $C^{n}$ for $n > 0$ . □

The key properties of ${\tilde{ψ}}_{0, 1}$ analogous to Lemma 5.2 are given by

Lemma 5.7. Outside $D_{W, x}$ ,

d {\tilde{ψ}}_{0, 1} (x, s) = φ_{1, 1} (x, s) and L {\tilde{ψ}}_{0, 1} (x, τ) = ψ_{0, 1} (x, τ) .

5.3.3. The singular function ${\tilde{ψ}}_{0, 1}^{'}$ . Inspired by [16], section 2.3, we deﬁne a functions $A^{'} (x)$ and $B^{'} (x)$ on $W$ by

\begin{align} B^{'} (x) & = \{\begin{matrix} \frac{1}{2} min (| x_{2} - x_{3} |, | x_{2} + x_{3} |) e^{- π (x, x)} & if x_{2}^{2} - x_{3}^{2} > 0, \\ 0 & otherwise, \end{matrix} & (5.9) \\ A^{'} (x) & = - X_{23} B^{'} (x) = - sgn (x_{2} x_{3}) B^{'} (x) . \end{align}

Lemma 5.8.

$B^{'} (x) + \frac{1}{2} | x_{3} | e^{- π (x, x)}$ is $C^{2}$ on the complement of the light-cone in $W$ and $C^{2}$ on nonzero $M$ -orbits.
$A^{'} (x) + \frac{1}{2} x_{2} \frac{x_{3}}{| x_{3} |} e^{- π (x, x)}$ is $C^{1}$ on the complement of the light-cone in $W$ and $C^{1}$ on nonzero $M$ -orbits.

We deﬁne ${\tilde{ψ}}_{0, 1}^{'}$ by

{\tilde{ψ}}_{0, 1}^{'} (x) = A^{'} (x) \otimes 1 \otimes e_{2} + B^{'} (x) \otimes 1 \otimes e_{3}

and ${\tilde{ψ}}_{0, 1}^{'} (x, τ, s) = v^{- 1 ∕ 2} m (s) {\tilde{ψ}}_{0, 1}^{'} (m^{- 1} (s) \sqrt{v} x) e^{π i (x, x) τ}$ . A little calculation shows that ${\tilde{ψ}}_{0, 1}^{'} (x)$ is locally constant on $D_{W}$ with a singularity at $D_{W, x}$ and holomorphic in $τ$ :

Lemma 5.9. Outside $D_{W, x}$ we have

d {\tilde{ψ}}_{0, 1}^{'} (x) = 0 and L {\tilde{ψ}}_{0, 1}^{'} (x, τ) = 0 .

Remark 5.10. The functions ${\tilde{ψ}}_{0, 1} (x)$ and ${\tilde{ψ}}_{0, 1}^{'} (x)$ deﬁne currents on $D_{W}$ . One can show, similarly to Section 6.5, that for $(x, x) > 0$ we have

\begin{array}{l} d [{\tilde{ψ}}_{0, 1} (x)] = δ_{D_{W, x} \otimes x} + [φ_{1, 1} (x)], d [{\tilde{ψ}}_{0, 1}^{'} (x)] = - δ_{D_{W, x} \otimes x}, \end{array}

where $D_{W, x} \otimes x$ is the $0$ -cycle $D_{W, x}$ ‘with coeﬃcient $x \in W$ ’ deﬁned in [12].

5.3.4. The form $ϕ_{0, 1}$ on $W$ . We now combine ${\tilde{ψ}}_{0, 1}$ and ${\tilde{ψ}}_{0, 1}^{'}$ to obtain an integrable and also square-integrable $W$ -valued function

ϕ_{0, 1} \in [L^{2} (W_{ℝ}) \otimes \land 0 𝔪^{*} \otimes W_{ℂ}]

ϕ_{0, 1} (x) = {\tilde{ψ}}_{0, 1} (x) + {\tilde{ψ}}_{0, 1}^{'} (x)

and then also $ϕ_{0, 1} (x, s)$ . Combining Lemmas 5.6, 5.8 and (5.8), (5.9) we obtain

Proposition 5.11.

$B (x) + B^{'} (x)$ is $C^{2}$ on the complement of the light-cone in $W$ and $C^{2}$ on nonzero $M$ -orbits.
$A (x) + A^{'} (x)$ is $C^{1}$ on the complement of the light-cone in $W$ and $C^{1}$ on nonzero $M$ orbits.
$X_{23} (B + B^{'}) = - (A + A^{'})$ on all of $W$ .

So for given $x$ , the function $ϕ_{0, 1} (x, s)$ is a $C^{1}$ -function on $D_{W}$ with values in $W_{ℂ}$ .

The following theorem is fundamental for us. It is an immediate consequence of the Lemmas 5.7 and 5.9.

Theorem 5.12. The form $φ_{1, 1}$ on $D_{W}$ is exact. Namely,

d ϕ_{0, 1} = φ_{1, 1} .

Furthermore,

L ϕ_{0, 1} = d ψ_{0, 1} .

Proposition 5.13. The function $ϕ_{0, 1}$ is an eigenfunction of $K^{'} = SO (2)$ of weight $2$ under the Weil representation. More precisely,

ω (k^{'}) ϕ_{0, 1} = χ^{2} (k^{'}) ϕ_{0, 1},

where $χ$ is the standard character of $SO (2) ≃ U (1)$ .

Proof. It suﬃces to show this for one component of $ϕ_{0, 1}$ , that is, the function $B (x) + B^{'} (x)$ . Then the assertion has been already proved in §2.3 by showing that $B (x) + B^{'} (x)$ is an eigenfunction under the Fourier transform. We give here an inﬁnitesimal proof. Since $ω (k^{'})$ acts essentially as Fourier transform and $B + B^{'}$ is $L^{1}$ , we see that $ω (k^{'}) (B + B^{'})$ is continuous. Hence it suﬃces to establish the corresponding current equality $[ω (k^{'}) (B + B^{'})] = χ^{2} (k^{'}) [B + B^{'}]$ , since continuous functions coincide when they induce the same current. The inﬁnitesimal generator of $K^{'}$ acts by $H : = \frac{- i}{4 π} (\frac{\partial^{2}}{\partial x_{2}^{2}} - \frac{\partial^{2}}{\partial x_{3}^{2}}) + π i (x_{2}^{2} - x_{3}^{2})$ , and a straightforward calculation immediately shows

H B^{'} = 2 i B^{'} and H B = 2 i B,

outside the singularity $x_{2}^{2} - x_{3}^{2} = 0$ . Now we consider the currents $H [B]$ and $H [B^{'}]$ . An easy calculation using that $B$ and $B^{'}$ are $C^{2}$ up to $| x_{3} | e^{- π (x_{2}^{2} - x_{3}^{2})}$ shows that for a test function $f$ on $W$ we have

\begin{array}{l} H [B] (f) & = [H B] (f) + \int_{0}^{\infty} e^{- π x_{2}^{2}} f (x_{2}, 0) d x_{2}, \\ H [B^{'}] (f) & = [H B^{'}] (f) - \int_{0}^{\infty} e^{- π x_{2}^{2}} f (x_{2}, 0) d x_{2} . \end{array}

Thus $H [B + B^{'}] = [H (B + B^{'})] = 2 i [B + B^{'}]$ as claimed. □

5.3.5. The map $ι_{P}$ . We deﬁne a map

ι_{P} : 𝒮 (W_{ℝ}) \otimes \land i 𝔪^{*} \otimes W_{ℂ} \to 𝒮 (W_{ℝ}) \otimes \land i + 1 (𝔪^{*} \oplus 𝔫^{*})

ι_{P} (φ \otimes ω \otimes w) = φ \otimes (ω \land (w \land u^{'})) .

Here we used the isomorphism $𝔫 ≃ W \land ℝ u \in \land^{2} V_{ℝ} ≃ 𝔤$ and identify $W$ with its dual via the bilinear form $(,)$ so that $𝔫^{*} ≃ W \land ℝ u^{'}$ . In [13], Section 6.2 we explain that $ι_{P}$ is a map of Lie algebra complexes. Hence we obtain a map of complexes

{[𝒮 (W_{ℝ}) \otimes 𝒜^{i} (D_{W}) \otimes W_{ℂ}]}^{M} \to {[𝒮 (W_{ℝ}) \otimes 𝒜^{i + 1} (e (P))]}^{N M},

which we also denote by $ι_{P}$ . Here $N$ acts trivially on $𝒮 (W_{ℝ})$ . Explicitly, the vectors $e_{2}$ and $e_{3}$ in $W$ map under $ι_{P}$ to the left-invariant $1$ -forms

e_{2} \mapsto cosh (s) d w_{2} - sinh (s) d w_{3} e_{3} \mapsto sinh (s) d w_{2} - cosh (s) d w_{3}

with the coordinate functions $w_{2}, w_{3}$ on $W$ deﬁned by $w = w_{2} e_{2} + w_{3} e_{3}$ . We apply $ι_{P}$ to the forms on $W$ of this section, and we obtain $φ_{1, 1}^{P}$ , $ϕ_{0, 1}^{P}$ , $ψ_{0, 1}^{P}$ , and ${ψ^{'}}_{0, 1}^{P}$ .

6. The boundary theta lift and linking numbers in Sol

6.1. Global theta functions for $W$ .

We let $ℒ_{W}$ be a $Γ_{P}$ -invariant (coset of a) lattice in $W$ , where $Γ_{N}$ acts trivially on $W$ . For $φ_{1, 1}$ , we deﬁne its theta function by

𝜃_{φ_{1, 1}} (τ, ℒ_{W}) = \sum_{x \in ℒ_{W}} φ_{1, 1} (x, τ)

and similarly for $ψ_{0, 1}$ , and $ϕ_{0, 1}$ . Then the usual theta machinery gives that $𝜃_{φ_{1, 1}} (τ, ℒ_{W})$ and $𝜃_{ϕ_{0, 1}} (ℒ_{W})$ both transform like (non)-holomorphic modular forms of weight $2$ for some congruence subgroup of ${SL}_{2} (ℤ)$ .

Remark 6.1. The claim is not obvious for $𝜃_{ϕ_{0, 1}}$ , since $ϕ_{0, 1}$ is not a Schwartz function. In that case, we use Proposition 5.11. The component $B + B^{'}$ of $ϕ_{0, 1}$ is $C^{2}$ outside the light cone. Since $W$ is anisotropic we can then apply Possion summation, and this component transforms like a modular form. Then apply the diﬀerential operator $X_{23}$ to obtain the same for the other component $A + A^{'}$ of $ϕ_{0, 1}$ .

In fact, if $W$ is isotropic and $ℒ_{W}$ intersects non-trivially with the light cone, then $𝜃_{ϕ_{0, 1}}$ is not quite a modular form. The case, when the $ℚ$ -rank of $V$ is $2$ is interesting in its own right. We will discuss this elsewhere.

Via the map $ι_{P}$ from Section 5.3.5 we can view all theta functions for $W$ as functions resp. diﬀerential forms on $e^{'} (P)$ . We set $𝜃_{φ_{1, 1}}^{P} = 𝜃_{φ_{1, 1}^{P}}$ , and similarly $𝜃_{ψ_{0, 1}}^{P}$ and $𝜃_{ϕ_{0, 1}}^{P}$ . Since $ι_{P}$ is a map of complexes we immediately see by Theorem 5.4 and Theorem 5.12

Proposition 6.2.

L 𝜃_{φ_{1, 1}}^{P} = d 𝜃_{ψ_{0, 1}}^{P} and L 𝜃_{ϕ_{0, 1}}^{P} = 𝜃_{ψ_{0, 1}}^{P} .

We now interpret the (holomorphic) Fourier coeﬃcients of the boundary theta lift associated to $𝜃_{ϕ}^{P} (τ, ℒ_{W_{P}})$ . They are given by linking numbers. We have

Theorem 6.3. Let $c$ a homological trivial $1$ -cycle in $e^{'} (P)$ which is disjoint from the torus ﬁbers containing components of $\partial C_{n}$ or for $c = \partial C_{y}$ for $C_{y}$ one of the components of $C_{n}$ , we have

\int_{c} 𝜃_{ϕ_{0, 1}}^{P} (τ, ℒ_{W_{P}}) = \sum_{n = 1}^{\infty} Lk ({(\partial C_{n})}_{P}, c) q^{n} + \sum_{n \in ℚ} \int_{c} {\tilde{ψ}}_{0, 1}^{P} (n) (τ) .

So the Fourier coeﬃcients of the holomorphic part of $\int_{c} 𝜃_{ϕ}^{P} (τ, ℒ_{W_{P}})$ are the linking numbers of the cycles $c$ and $\partial C_{n}$ at the boundary component $e^{'} (P)$ .

Theorem 6.3 follows from $ϕ_{0, 1} = {\tilde{ψ}}_{0, 1} + {\tilde{ψ^{'}}}_{0, 1}$ combined with Theorem 6.7 below.

Example 6.4. In the situation of Examples 2.1 and 4.12, we obtain

\int_{\partial C_{1}} 𝜃_{ϕ_{0, 1}}^{P} (τ, ℒ_{W_{P}}) = \frac{1}{\sqrt{2 d}} \sum_{\begin{array}{c} λ \in 𝒪_{K} \\ λ λ^{'} > 0 \end{array}} min (| λ |, | λ^{'} |) e^{- 2 π λ λ^{'} τ} - \frac{\sqrt{2}}{\sqrt{d v}} \sum_{λ \in 𝒪_{K}} β (π v {(λ - λ^{'})}^{2}) e^{- 2 π λ λ^{'} τ},

where $β (s) = \frac{1}{16 π} \int_{1}^{\infty} e^{- s t} t^{- 3 ∕ 2} d t$ . This is (up to a constant) exactly Zagier’s function $𝒲 (τ)$ in [16], §2.3.

6.2. Linking numbers, de Rham cohomology and linking duals.

We begin with a general discussion of integral formulas for linking numbers. Such formulas go back to the classical Gauss-Ampère formula for $ℝ^{3}$ , see [8], p.79-81, and [7] for its generalization to $S^{3}$ and $H^{3}$ . Suppose now that $c$ is a $1$ -cycle in an oriented compact $3$ -manifold M that is a rational boundary and $U$ is a tubular neighborhood of $c$ .

Deﬁnition 6.5. We will say any closed form $β$ in $M - U$ is a linking dual (relative to $U$ ) of the bounding $1$ -cycle $c$ if for any $1$ -cycle $a$ in $M - U$ which is a rational boundary in $M$ we have

\int_{a} β = Lk (a, c) .

We will prove that given a cycle $c$ that bounds rationally then linking duals for $c$ exist for all tubular neighborhoods $U$ of $c$ . Let $η$ be a Thom form for $c$ compactly supported in $U$ . This means that $η$ is closed and has integral $1$ over any normal disk to $c$ . Let $η_{M}$ be the extension of $η$ to $M$ by zero. It is standard in topology (the extension of the Thom class by zero is the Poincaré dual of the zero section of the normal bundle) that the form $η_{M}$ represents the $2$ -dimensional cohomology class on $M$ which is Poincaré dual to $c$ . Since $c$ is a rational boundary there exists a $1$ -form $β$ on $M$ such that $d β = η_{M}$ . We will now see that $β$ is a linking dual of $c$ . To this end, suppose $a$ is a $1$ -cycle in $M - U$ which is a rational boundary in $M$ , hence there exists a rational chain $A$ with $\partial A = a$ . We may suppose $η_{M}$ vanishes in a neighborhood $V$ of $a$ which is disjoint from $U$ . Then the restriction $η_{M - V}$ of $η_{M}$ to $M - V$ represents the (relative) Poincaré dual of the absolute cycle $c$ in $(M - V, \partial (M - V))$ . Using this the reader will show that

\int_{A} η_{M} = \int_{A \cap (M - V)} η_{M - V} = A \cdot c = Lk (a, c) .

Note that restriction of $β$ to $M - U$ is closed. Then

\int_{a} β = \int_{A} η_{M} = Lk (a, c) .

Hence we have

Proposition 6.6. $β$ is a linking dual of $c$ .

6.3. The $1$ -form $e^{2 π n} {\tilde{ψ^{'}}}_{0, 1} (n)$ is a linking dual of ${(\partial C_{n})}_{P}$ .

We now return to the case in hand. In what follows, we drop subscript and superscript $P$ ’s since we are ﬁxing a boundary component $e (P)$ . We let $F_{n}$ be the union of the ﬁbers containing components of $\partial C_{n}$ , and we let $F_{x}$ be the ﬁber containing $c_{x}$ . Recall that $c_{x}$ is the image of $D_{x} \cap e (P)$ in $e^{'} (P)$ .

Theorem 6.7. Let $n > 0$ . The $1$ -form $e^{2 π n} {\tilde{ψ^{'}}}_{0, 1} (n)$ is a linking dual for $\partial C_{n}$ in $e^{'} (P)$ relative to any neighborhood $U$ of $F_{n}$ . Hence, for $c$ a rational $1$ -boundary in $e^{'} (P)$ which is disjoint from $F_{n}$ we have

\int_{c} {\tilde{ψ^{'}}}_{0, 1} (n) = Lk (\partial C_{n}, c) e^{- 2 π n} .

(6.1)

Furthermore, (6.1) holds when $c = c_{y}$ contained in one ﬁber $F_{x}$ of $\partial C_{n}$ .

We will ﬁrst deal with the case in which $c$ is disjoint from $F_{n}$ (which we will refer to in what follows as case (i)), then at the end of this section we will reduce the case in which $c = c_{y}$ (which we will refer to as case (ii)) to case (i) by a Stokes’ Theorem argument. Thus we will now assume we are in case (i).

The key step is

Proposition 6.8. Let $n > 0$ and let $η$ be an exact $2$ -form in $e^{'} (P)$ which is compactly supported in the complement of $F_{n}$ . Then

\int_{e^{'} (P)} η \land {\tilde{ψ^{'}}}_{0, 1} (n) = (\int_{A_{n}} η) e^{- 2 π n} .

(6.2)

Remark 6.9. Note that (6.2) also holds in case $η = Ω_{P}$ . In this case the right-hand side is zero by the normalization of the cap $A_{n}$ and the left-hand side is zero because $Ω \land {\tilde{ψ^{'}}}_{0, 1} (n) = 0$ since $Ω$ has bidegree $(0, 2)$ and ${\tilde{ψ^{'}}}_{0, 1} (n)$ has bidegree $(0, 1)$ (here we use the obvious base/ﬁber bigrading on the de Rham algebra of $e^{'} (P)$ ).

6.4. Proof of Proposition 6.8.

Lemma 6.10. Under the hypothesis on $η$ in Proposition 6.8 we have

\int_{A_{n}} η = \sum_{\begin{array}{c} x \in Γ_{M} ∖ ℒ_{W} \\ (x, x) = 2 n \end{array}} min_{λ \in Λ_{W}}^{'} | (λ, x) | \int_{a_{x}} η

Proof. We use Lemma 4.3. Write $η = d ω$ for some $1$ -form $ω$ which by the support condition on $η$ is closed in $F_{n}$ . Since $c_{x + k u}$ and $c_{x}$ are parallel hence homologous circles in $F_{x}$ , we see $\int_{a_{x + k u}} η = \int_{c_{x + k u}} ω = \int_{c_{x}} ω = \int_{a_{x}} η$ . □

Since

{\tilde{ψ^{'}}}_{0, 1} (n) = \sum_{\begin{array}{c} x \in Γ_{M} ∖ ℒ_{W} \\ (x, x) = 2 n \end{array}} \sum_{γ \in Γ_{M}} γ^{*} {\tilde{ψ^{'}}}_{0, 1} (x),

Proposition 6.8 will now follow from

Proposition 6.11. Under the hypothesis on $η$ in Proposition 6.8, we have for any positive length vector $x \in ℒ_{W}$

\int_{e^{'} (P)} η \land \sum_{γ \in Γ_{M}} γ^{*} {\tilde{ψ^{'}}}_{0, 1} (x) = (min_{λ \in Λ_{W}}^{'} | (λ, x) |) (\int_{a_{x}} η) e^{- π (x, x)} .

Proof. By choosing appropriate coordinates we can assume that $x = μ e_{2}$ with $μ = \pm \sqrt{2 n}$ , so that the singularity of $\sum_{γ \in Γ_{M}} γ^{*} {\tilde{ψ^{'}}}_{0, 1} (x)$ in $e^{'} (P)$ occurs at $s = 0$ . We pick a tubular neighborhood $U_{𝜀} = (- 𝜀, 𝜀) \times T^{2}$ in $e^{'} (P)$ around $F_{x}$ . Then we have ﬁrst

\int_{e^{'} (P)} η \land \sum_{γ \in Γ_{M}} γ^{*} {\tilde{ψ^{'}}}_{0, 1}^{P} (x) = lim_{𝜖 \to 0} \int_{e^{'} (P) - U_{𝜀}} η \land \sum_{γ \in Γ_{M}} γ^{*} {\tilde{ψ^{'}}}_{0, 1}^{P} (x) .

Since $η \land {\tilde{ψ^{'}}}_{0, 1} (x) = d (ω \land {\tilde{ψ^{'}}}_{0, 1} (x))$ outside $U_{𝜀}$ and $\partial (e^{'} (P) - U_{𝜀}) = - \partial U_{𝜀}$ we see by Stokes’ theorem

\begin{align} \int_{e^{'} (P) - U_{𝜀}} η \land \sum_{γ \in Γ_{M}} γ^{*} {\tilde{ψ^{'}}}_{0, 1} (x) = - \int_{\partial U_{𝜀}} ω \land \sum_{γ \in Γ_{M}} γ^{*} {\tilde{ψ^{'}}}_{0, 1} (x) & (6.3) \\ = \sum_{γ \in Γ_{M}} \int_{T^{2}} [ω (- 𝜀, w) \land \tilde{{ψ^{'}}_{0, 1}} (γ^{- 1} x, - 𝜖, w) - ω (𝜀, w) \land {\tilde{ψ^{'}}}_{0, 1} (γ^{- 1} x, 𝜖, w)] . \end{align}

For $γ \neq 1$ we note that $ω (s, w) \land {\tilde{ψ^{'}}}_{0, 1} (γ^{- 1} x, s, w)$ is continuous at $s = 0$ , while for $γ = 1$ , we have

{\tilde{ψ^{'}}}_{0, 1} (μ e_{2}, s, w) = \frac{1}{2} | μ | (sgn (s) d w_{2} - d w_{3}) e^{- π μ^{2}} .

(6.4)

Hence taking the limit in the last term of (6.3) we obtain

| μ | e^{- π μ^{2}} \int_{T^{2}} ω_{3} (0, w) d w_{2} d w_{3} = | μ | e^{- π μ^{2}} \int_{T^{2} ∕ c_{e_{2}}} (\int_{c_{e_{2}}} ω (0, w_{2}, w_{3})) d w_{2} .

In the expression $T^{2} ∕ c_{e_{2}}$ (and for the rest of this proof) we have abused notation and identiﬁed the cycle $c_{e_{2}}$ with the subgroup $0 \times S^{1}$ of $T^{2}$ .

Here $ω_{3}$ is the $d w_{3}$ component of $ω$ and we used that $\partial D_{x}$ is the $w_{3}$ -line in $W$ . Note that the inner integral on the right is the period of $ω$ over (homologous) horizontal translates of the cycle $c_{e_{2}}$ . But the restriction of $ω$ to $F_{x}$ is closed so $\int_{c_{e_{2}}} ω (0, w_{2}, w_{3})$ is independent of $w_{2}$ and the last integral becomes $(\int_{T^{2} ∕ c_{e_{2}}} d w_{2}) (\int_{c_{e_{2}}} ω) e^{- π μ^{2}}$ . But $\int_{c_{e_{2}}} ω = \int_{A_{e_{2}}} η$ . The proposition is then a consequence of

| μ | \int_{T^{2} ∕ \partial c_{e_{2}}} d w_{2} = | μ | min_{λ \in Λ_{W}}^{'} | (λ, e_{2}) | = min_{λ \in Λ_{W}}^{'} | (λ, x) |,

which follows from the fact that the map $W \to ℝ$ given by $w \mapsto (w, e_{2})$ induces an isomorphism $T^{2} ∕ \partial C_{e_{2}} ≃ ℝ ∕ ({min}_{λ \in Λ_{W}}^{'} | (λ, e_{2}) |) ℤ$ . □

6.5. Proof of Theorem 6.7.

We now prove Theorem 6.7. First we will assume that we are in case(i). We need to show

\int_{c} {\tilde{ψ^{'}}}_{0, 1} (n) = Lk (\partial C_{n}, c) e^{- 2 π n} = (A_{n} \cdot c) e^{- 2 π n} .

The theorem will be a consequence of the following discussion. We may assume that $c$ is an embedded loop in $e^{'} (P)$ (note that since any loop in a manifold of dimension $3$ or more is homotopic to an embedded loop by transversality any homology class of degree $1$ in $e^{'} (P)$ is represented by an embedded loop).

Choose a tubular neighborhood $N (c)$ of $c$ such that $N (c)$ is disjoint from $F_{n}$ . Let $η_{c}$ be a closed $2$ -form which is supported inside $N (c)$ and has integral $1$ on the disk ﬁbers of $N (c)$ (a Thom class for the normal disk bundle $N (c)$ ). Then we have proved in Subsection 6.2

Lemma 6.12.

\int_{A_{n}} η_{c} = A_{n} \cdot c = Lk (\partial C_{n}, c) .

(6.5)

We then have

Lemma 6.13.

\int_{c} {\tilde{ψ^{'}}}_{0, 1} (n) = (\int_{A_{n}} η_{c}) e^{- 2 π n} .

Proof. To prove the Lemma we compute $\int_{e^{'} (P)} η_{c} \land {\tilde{ψ^{'}}}_{0, 1} (n) = \int_{e^{'} (P)} {\tilde{ψ^{'}}}_{0, 1} (n) \land η_{c}$ in two diﬀerent ways. First we apply Proposition 6.8 with $η = η_{c}$ . We deduce

\int_{e^{'} (P)} η_{c} \land {\tilde{ψ^{'}}}_{0, 1} (n) = (\int_{A_{n}} η_{c}) e^{- 2 π n} .

Next choose a tubular neighborhood $V_{n}$ of the ﬁbers $F_{n}$ such that $e^{'} (P) - V_{n}$ contains $N (c)$ . Then ${\tilde{ψ^{'}}}_{0, 1} (n)$ is smooth on $e^{'} (P) - V_{n} \supset supp (η_{c})$ . Also, since $η_{c}$ is the extension of a Thom class by zero, the restriction of $η_{c}$ to $e^{'} (P) - V_{n}$ represents the Poincaré dual $P D (c)$ of the absolute cycle $c$ in $e^{'} (P) - V_{n}$ . The lemma now follows from

\int_{e^{'} (P)} {\tilde{ψ^{'}}}_{0, 1} (n) \land η_{c} = \int_{e^{'} (P) - V_{n}} {\tilde{ψ^{'}}}_{0, 1} (n) \land η_{c} = \int_{e^{'} (P) - V_{n}} {\tilde{ψ^{'}}}_{0, 1} (n) \land P D (c) = \int_{c} {\tilde{ψ^{'}}}_{0, 1} (n) .

□

By Lemma 6.12 this concludes the proof of Theorem 6.7 in the case when $c$ is disjoint from the ﬁbers $F_{n}$ .

It remains to treat case (ii). Thus we now assume that $c = c_{y}$ which is contained in a ﬁber $F_{x}$ containing a component of $\partial C_{n}$ . We ﬁrst prove

Lemma 6.14.

\int_{c} {\tilde{ψ}}_{0, 1}^{'} (x) = \int_{c (𝜖)} {\tilde{ψ}}_{0, 1}^{'} (x) .

Proof. We can take $x = μ e_{2}$ and hence $c$ is contained in the ﬁber over the image of $e_{3} \in W$ . Hence, by Proposition 4.1, $c$ is the circle in the torus ﬁber at $s (x) = 0$ in the $e_{3}$ -direction, i.e., parallel to the image of $(0, ℝ e_{3})$ in $e^{'} (P)$ . We note that by (6.4) even though ${\tilde{ψ}}_{0, 1}^{'} (x)$ is not deﬁned on the whole ﬁber over $s = 0$ its restriction to $c$ is smooth. Hence the left hand side is well-deﬁned since all the other terms in the sum are deﬁned on the whole ﬁber and in fact in a neighborhood of that ﬁber. Hence the locally constant form $\sum_{γ \in Γ_{M}} γ^{*} {\tilde{ψ}}_{0, 1}^{'} (x)$ is closed on the cylinder $[0, 𝜖] \times c$ , and its integrals over the circles $s \times c$ all coincide. But $𝜀 \times c = c (𝜖)$ . The lemma follows. □

Summing over $x$ and using case (i) we obtain

\int_{c} {\tilde{ψ}}_{0, 1}^{'} (n) = \int_{c (𝜖)} {\tilde{ψ}}_{0, 1}^{'} (n) = Lk (\partial C_{n}, c (𝜖)),

since $c (𝜖)$ is disjoint from all the components of $F_{n}$ . Thus it suﬃces to prove

Lk (\partial C_{n}, c) = Lk (\partial C_{n}, c (𝜖)) .

(6.6)

To this end suppose that $c \subset F_{x} \subset F_{n}$ and $c_{1}, \dots, c_{k}$ are the components of $\partial C_{n}$ contained in $F_{x}$ . Hence $c$ and $c_{i}, 1 \leq i \leq k$ , are all parallel. Since the ﬁbers containing all other components of $\partial C_{n}$ are disjoint from $c$ , (6.6) will follow from

Lk (c_{i}, c) = Lk (c_{i}, c (𝜖)), 1 \leq i \leq k .

If $c_{i} = c$ then the previous equation is the deﬁnition of $L k (c, c)$ . Thus we may assume $c_{i}$ is parallel to and disjoint from $c$ . In this case their linking number is already topologically deﬁned. But since $c$ is disjoint from $c_{i}$ the circles $c$ and $c (𝜀)$ are homologous in the complement of $c_{i}$ (by the product homology $c \times [0, 𝜀]$ ) and since the linking number with $c_{i}$ is a homological invariant of the complement of $c_{i}$ in $e^{'} (P)$ we have $Lk (c_{i}, c (𝜀)) = L k (c_{i}, c)$ .

We this Theorem 6.7 is proved.

7. The generating series of the capped cycles

In this section, we show that the generating series of the ‘capped’ cycles $C_{n}^{c}$ gives rise to a modular form, extending Theorem 7.1 to a lift of the full cohomology $H^{2} (X)$ of $X$ . In particular, we give our new proof of the theorem of Hirzebruch and Zagier and show how a remarkable feature of their proof appears from our point of view.

7.1. The theta series associated to $φ_{2}$ .

We deﬁne the theta series

𝜃_{φ_{2}} (τ, ℒ) = \sum_{x \in ℒ} φ_{2} (x, τ, z) .

In the following we will often drop the argument $ℒ = L + h$ . For $n \in ℚ$ , we also set

φ_{2} (n) = \sum_{n \in ℒ_{n}, x \neq 0} φ_{2} (x) .

Clearly, $𝜃_{φ_{2}} (τ, ℒ)$ and $φ_{2} (n)$ descend to closed diﬀerential $2$ -forms on $X$ . Furthermore, $𝜃_{φ_{2}} (τ, ℒ)$ is a non-holomorphic modular form in $τ$ of weight $2$ for the principal congruence subgroup $Γ (N)$ . In fact, for $ℒ = L$ as in Example 2.1, $𝜃_{φ_{2}} (τ, ℒ)$ transforms like a form for $Γ_{0} (d)$ of nebentypus.

Theorem 7.1 (Kudla-Millson [24]). We have

[𝜃_{φ_{2}} (τ)] = - \frac{1}{2 π} δ_{h 0} [ω] + \sum_{n > 0} PD [C_{n}] q^{n} \in H^{2} (X, ℚ) \otimes M_{2} (Γ (N)) .

That is, for any closed $2$ -form $η$ on $X$ with compact support,

Λ (η, τ) : = \int_{X} η \land 𝜃_{φ_{2}} (τ, ℒ) = - \frac{1}{2 π} δ_{h 0} \int_{X} η \land ω + \sum_{n > 0} (\int_{C_{n}} η) q^{n} .

Here $δ_{h 0}$ is Kronecker delta, and $ω$ is the Kähler form on $D$ normalized such that its restriction to the base point is given by $ω_{13} \land ω_{14} + ω_{23} \land ω_{24}$ . We obtain a map

Λ : H_{c}^{2} (X, ℂ) \to M_{2} (Γ (N))

(7.1)

from the cohomology with compact supports to the space of holomorphic modular forms of weight $2$ for the principal congruence subgroup $Γ (N) \subset {SL}_{2} (ℤ)$ . Alternatively, for $C$ an absolute $2$ -cycle in $X$ deﬁning a class in $H_{2} (X, ℤ)$ , the lift $Λ (C, τ)$ is given by (1.1) with $C_{0}$ the class given by $- \frac{1}{2 π} δ_{h 0} [ω]$ .

The key fact for the proof of the Fourier expansion is that for $n > 0$ , the form $φ_{2} (n)$ is a Poincaré dual form of $C_{n}$ , while $φ_{2} (n)$ is exact for $n \leq 0$ , see also Section 8.

7.2. The restrictions of the global theta functions.

Theorem 7.2.

The diﬀerential forms $𝜃_{φ_{2}} (ℒ_{V})$ and $𝜃_{ψ_{1}} (ℒ_{V})$ on $X$ extend to the Borel-Serre compactiﬁcation $\bar{X}$ . More precisely, for the restriction $i_{P}^{*}$ to the boundary face $e^{'} (P)$ of $\bar{X}$ , we have

i_{P}^{*} 𝜃_{φ_{2}} (ℒ_{V}) = 𝜃_{φ_{1, 1}}^{P} (ℒ_{W_{P}}) and i_{P}^{*} 𝜃_{ψ_{1}} (ℒ_{V}) = 𝜃_{ψ_{0, 1}}^{P} (ℒ_{W_{P}}) .

Proof. The restriction of $𝜃_{φ_{2}} (ℒ_{V})$ is the theme (in much greater generality) of [13]. For $𝜃_{ψ_{1}} (ℒ_{V})$ one proceeds in the same way. In short, one detects the boundary behaviour of the theta functions by switching to a mixed model of the Weil representation. For a model calculation see the proof of Theorem 7.4 below. □

We conclude by Proposition 6.2

Theorem 7.3.

The restriction of $𝜃_{φ_{2}} (ℒ_{V})$ to the boundary of $\bar{X}$ is exact and

i_{P}^{*} 𝜃_{φ_{2}} (ℒ_{V}) = d (𝜃_{ϕ_{0, 1}}^{P} (ℒ_{W_{P}})) .

We also have a crucial restriction result for the singular form ${\tilde{ψ}}_{0, 1}$ . However, one needs to be careful in forming the naive theta series associated to ${\tilde{ψ}}_{0, 1}$ by summing over all (non-zero) lattice elements. This would give a form on $X$ with singularities on a dense subset of $X$ . Instead we deﬁne ${\tilde{ψ}}_{2, 0} (n)$ in the same way as for $φ_{2} (n)$ by summing over all non-zero $x \in ℒ_{V}$ of length $n \in ℚ$ . This gives a $1$ -form on $X$ which for $n > 0$ has singularities along the locally ﬁnite cycle $C_{n}$ . Similarly, we deﬁne

{\tilde{ψ}}_{0, 1}^{P} (n) = \sum_{\begin{array}{c} x \in ℒ_{W_{P}}, (x, x) = 2 n \end{array}} {\tilde{ψ}}_{0, 1}^{P} (x),

which descends to a $1$ -form on $e^{'} (P)$ with singularities. We also deﬁne ${\tilde{ψ^{'}}}_{0, 1} (n)$ and $ϕ_{0, 1}^{P} (n)$ in the same way. We have

Proposition 7.4. The restriction of the $1$ -form ${\tilde{ψ}}_{1} (n)$ to $e^{'} (P)$ is given by

i_{P}^{*} {\tilde{ψ}}_{1} (n) = {\tilde{ψ}}_{0, 1}^{P} (n) .

Proof. We assume that $P$ is the stabilizer of the isotropic line $ℓ = ℚ u$ . For $x = a u + x_{W} + b u^{'}$ , we have for the majorant at $z = (w, t, s)$ the formula

{(x, x)}_{z} = \frac{1}{t^{2}} {(a - (x_{W}, w) - b q (w))}^{2} + {(x_{w} + b w, x_{w} + b w)}_{s} + b^{2} t^{2} .

Here ${(,)}_{s}$ is the majorant associated to $W$ . Hence by (5.4) and (5.2) we see that the sum of all $x \in ℒ_{V}$ with $b \neq 0$ in ${\tilde{ψ}}_{1} (n)$ is uniformly rapidly decreasing as $t \to \infty$ . Now ﬁx an element $x_{W} \in ℒ_{W}$ . Then $x_{W} + (a + h) u \in ℒ_{V}$ for all $a \in ℤ$ for some $h \in ℚ ∕ ℤ$ ; in fact all elements in $ℒ_{V} \cap u^{⊥}$ are of this form. We consider $\sum_{a \in ℤ} {\tilde{ψ}}_{1} (x_{W} + (a + h) u, z)$ as $t \to \infty$ . By considerations as in [13], sections 4 and 9, we can assume $w = 0$ and $s = 0$ . We apply Poisson summation for the sum on $a \in ℤ$ and obtain

\begin{array}{l} \sum_{a \in ℤ} {\tilde{ψ}}_{1} (x_{W} + a u, z) = \sum_{k \in ℤ} (\int_{1}^{\infty} P (x, t, r) e^{- 2 π x_{3}^{2} r + t^{2} k^{2} ∕ r} \frac{d r}{r}) e^{- 2 π i k h} e^{- π (x_{W}, x_{W})}, \end{array}

where

P (x, t, r) = \frac{x_{2} x_{3} \sqrt{r}}{\sqrt{2}} d w_{2} + \frac{1}{2 \sqrt{2}} (\frac{1}{2 π} - \frac{t^{2} k^{2}}{r}) d w_{3} - \frac{i x_{3} k}{\sqrt{2}} d t + \frac{i x_{2} k t}{\sqrt{2}} d s .

Now the sum over all $k \neq 0$ is rapidly decreasing while for $k = 0$ we obtain ${\tilde{ψ}}_{0, 1} (x_{W})$ . If $x_{W} = 0$ , i.e., for $n = 0$ one needs to argue slightly diﬀerently. Then we have

\sum_{a \neq 0} {\tilde{ψ}}_{1} (a u, z) = \frac{1}{2 \sqrt{2} π} \sum_{a \neq 0} e^{- π a^{2} ∕ t^{2}} \frac{d w_{3}}{t} = \frac{1}{2 \sqrt{2} π} (\sum_{k \in ℤ} e^{- π t^{2} k^{2}}) d w_{3} - \frac{1}{2 \sqrt{2} π} \frac{d w_{3}}{t},

which goes to $\frac{1}{2 \sqrt{2} π} d w_{3} = {\tilde{ψ}}_{0, 1} (0)$ . This proves the proposition. □

7.3. Main result.

In the previous sections, we constructed a closed $2$ -form $𝜃_{φ_{2}}$ on $\bar{X}$ such that the restriction of $𝜃_{φ_{2}}$ to the boundary $\partial \bar{X}$ was exact with primitive $\sum_{[\underset{̲}{P}]} 𝜃_{ϕ_{0, 1}}^{P}$ . From now on we usually write $φ$ for $φ_{2}$ and $ϕ$ for $ϕ_{0, 1}$ if it does not cause any confusion. By the deﬁnition of the diﬀerential for the mapping cone complex $C^{∙}$ we immediately obtain by Theorem 7.2 and Theorem 7.3

Proposition 7.5. The pair $(𝜃_{φ_{2}} (ℒ_{V}), \sum_{[P]} 𝜃_{ϕ_{0, 1}}^{P} (ℒ_{W_{P}}))$ is a $2$ -cocycle in $C^{∙}$ .

We write for short $(𝜃_{φ}, 𝜃_{ϕ})$ . We obtain a class $[[𝜃_{φ}, 𝜃_{ϕ}]]$ in $H^{2} (C^{∙})$ and hence a class $[𝜃_{φ}, 𝜃_{ϕ}]$ in $H_{c}^{2} (X)$ . The pairing with $[𝜃_{φ}, 𝜃_{ϕ}]$ then deﬁnes a lift $Λ^{c}$ on diﬀerential $2$ -forms on $\bar{X}$ , which factors through $H^{2} (\bar{X}) = H^{2} (X)$ . By Lemma 3.5 it is given by

Λ^{c} (η, τ) = \int_{\bar{X}} η \land 𝜃_{φ_{2}} - \sum_{[P]} \int_{e^{'} (P)} i^{*} η \land 𝜃_{ϕ_{0, 1}}^{P} .

Theorem 7.6. The class $[[𝜃_{φ}, 𝜃_{ϕ}]]$ is holomorphic, that is,

L (𝜃_{φ}, 𝜃_{ϕ}) = d (𝜃_{ψ_{1}}, 0) .

Hence $[𝜃_{φ}, 𝜃_{ϕ}]$ is a holomorphic modular form with values in the compactly supported cohomology of $X$ , so that the lift $Λ^{c}$ takes values in the holomorphic modular forms.

Proof. By Theorem 7.2 and Theorem 6.2 we calculate

d (𝜃_{ψ_{1}}, 0) = (d 𝜃_{ψ_{1}}, i^{*} 𝜃_{ψ_{1}}) = (L 𝜃_{φ_{2}}, \sum_{[P]} 𝜃_{ψ_{0, 1}}^{P}) = L (𝜃_{φ_{2}}, \sum_{[P]} 𝜃_{ϕ_{0, 1}}^{P}) . □

It remains to compute the Fourier expansion in $τ$ of $[𝜃_{φ}, 𝜃_{ϕ}] (τ)$ . We will carry this out in Section 8.

Theorem 7.7. We have

[𝜃_{φ}, 𝜃_{ϕ}] (τ) = - \frac{1}{2 π} δ_{h 0} [ω] + \sum_{n > 0} PD [C_{n}^{c}] q^{n} \in H_{c}^{2} (X, ℚ) \otimes M_{2} (Γ (N)) .

That is, for any closed $2$ -form $η$ on $\bar{X}$

Λ^{c} (η, τ) = - \frac{1}{2 π} δ_{h 0} \int_{X} η \land ω + \sum_{n > 0} (\int_{C_{n}^{c}} η) q^{n},

In particular, the map takes values in the holomorphic modular forms and factors through cohomology. We obtain a map

Λ^{c} : H^{2} (X) \to M_{2} (Γ (N))

(7.2)

from the cohomology with compact supports to the space of holomorphic modular forms of weight $2$ for the principal congruence subgroup $Γ (N) \subseteq {SL}_{2} (ℤ)$ . Alternatively, for $C$ any relative $2$ -cycle in $X$ deﬁning a class in $H_{2} (\bar{X}, \partial \bar{X}, ℤ)$ , we have

Λ^{c} (C, τ) = - \frac{1}{2 π} δ_{h 0} vol (C) + \sum_{n > 0} (C_{n}^{c} \cdot C) q^{n} \in M_{2} (Γ (N)) .

Remark 7.8. In the theorem we now consider the Kähler form $ω$ representing a class in the compactly supported cohomology. In fact, our mapping cone construction gives an explicit coboundary by which $ω$ is modiﬁed to become rapidly decreasing.

7.4. The Hirzebruch-Zagier Theorem.

We now view $[𝜃_{φ}, 𝜃_{ϕ}]$ as a class in $H^{2} (\tilde{X})$ via the map $j_{#} : H_{c}^{2} (X) \to H^{2} (\tilde{X})$ . We recover the Hirzebruch-Zagier-Theorem.

Theorem 7.9. We have

j_{#} [𝜃_{φ}, 𝜃_{ϕ}] (τ) = - \frac{1}{2 π} δ_{h 0} [ω] + \sum_{n > 0} [T_{n}^{c}] q^{n} \in H^{2} (\tilde{X}, ℚ) \otimes M_{2} (Γ (N)) .

In particular,

- \frac{1}{2 π} δ_{h 0} vol (T_{m}) + \sum_{n > 0} {(T_{n}^{c} \cdot T_{m})}_{\tilde{X}} q^{n} \in M_{2} (Γ (N)) .

This is the result Hirzebruch-Zagier proved for certain Hilbert modular surfaces (Example 2.1) by explicitly computing the intersection numbers $T_{m} \cdot T_{n}^{c}$ .

Proof.

This follows from Theorem 7.7 since $j_{*} C_{n}^{c} = T_{n}^{c}$ (Proposition 4.7), combined with the following general principle. Suppose $ω$ is a compactly supported form on $X$ such that the cohomology class of $ω$ is the Poincaré dual of the homology class of a cycle $C$ : $[ω] = PD (C)$ . Then we have $j_{#} [ω] = PD (j_{*} C)$ . To see this we have only to replace $ω$ by a cohomologous ‘Thom representative’ of $PD (C)$ , namely a closed form $\tilde{ω}$ supported in a tubular neighborhood $N (C)$ of $C$ in $X$ such that the integral of $\tilde{ω}$ over any disk of $N (C)$ is one. Then it is a general fact from algebraic topology (extension by zero of a Thom class) that $\tilde{ω}$ represents the Poincaré dual of $C$ in any manifold $M$ containing $N (C)$ , in particular for $M = \tilde{X}$ . □

Remark 7.10. If one is only interested in recovering the statement of this theorem, then there is also a diﬀerent way of deriving this from the Kudla-Millson theory. Namely, the lift $Λ$ on $H_{2} (X)$ (Theorem 7.1) factors through the quotient of $H_{2} (X)$ by $H_{2} (\partial X)$ since the restriction of $𝜃_{φ_{2}}$ is exact (Theorem 7.3). But by Proposition 3.3 we have $j_{*} H_{2} (X) ≃ H_{2} (X) ∕ H_{2} (\partial X)$ , and the Hirzebruch-Zagier result exactly stipulates the modularity of the lift of classes in $j_{*} H_{2} (X)$ . However, in that way one misses the remarkable extra structure coming from $\partial X$ as we will explain in the next subsection.

7.5. The lift of special cycles.

We now consider the lift of a special cycle $C_{y}$ . By Theorem 7.7 and Lemma 3.5 we see

\begin{align} Λ^{c} (C_{y}, τ, ℒ_{V}) & = - \frac{1}{2 π} δ_{h 0} vol (C_{y}) + \sum_{n > 0} (C_{n}^{c} \cdot C_{y}) q^{n} & (7.3) \\ = \int_{C_{y}} 𝜃_{φ_{2}} (τ, ℒ_{V}) - \sum_{[P]} \int_{{(\partial C_{y})}_{P}} 𝜃_{ϕ_{0, 1}}^{P} (τ, ℒ_{W_{P}}) . \end{align}

The two terms on the right, the integrals over $C_{y}$ and $\partial C_{y}$ , are both non-holomorphic modular forms (see below) whose diﬀerence is holomorphic (by Theorem 7.6). So the generating series series of $(C_{n}^{c} \cdot C_{y})$ is the sum of two non-holomorphic modular forms. We now give geometric interpretations for the two individual non-holomorphic forms.

Following [16] we deﬁne the interior intersection number of two special cycles by

{(C_{n} \cdot C_{y})}_{X} = {(C_{n} \cdot C_{y})}^{t r} + vol (C_{n} \cap C_{y}),

the sum of the transversal intersections and the volume of the $1$ -dimensional (complex) intersection of $C_{n}$ and $C_{y}$ which occur if one of the components of $C_{n}$ is equal to $C_{y}$ .

Theorem 7.11. We have

\int_{C_{y}} 𝜃_{φ_{2}} (τ, ℒ_{V}) = - \frac{1}{2 π} δ_{h 0} vol (C_{y}) + \sum_{n = 1}^{\infty} {(C_{n} \cdot C_{y})}_{X} q^{n} + \sum_{n \in ℚ} \sum_{[P]} \int_{{(\partial C_{y})}_{P}} {\tilde{ψ}}_{0, 1}^{P} (n) (τ) .

So the Fourier coeﬃcients of the holomorphic part of the non-holomorphic modular form $\int_{C_{y}} 𝜃_{φ_{2}}$ are the interior intersection numbers of the cycles $C_{y}$ and $C_{n}$ .

Proof. This is essentially [9], section 5, where more generally $O (p, 2)$ is considered. There the interpretation of the holomorphic Fourier coeﬃcients as interior intersection number is given. (For more details of an analogous calculation see [14], section 8). A little calculation using the formulas in [9] gives the non-holomorphic contribution. A more conceptual proof would use the relationship between $φ_{2}$ and ${\tilde{ψ}}_{1}$ (see Proposition 5.2 and Section 8) and the restriction formula for ${\tilde{ψ}}_{1} (n)$ (Theorem 7.4). □

By slight abuse of notation we write $Lk (C_{n}, C_{y}) = \sum_{[P]} Lk ({(\partial C_{n})}_{P}, {(\partial C_{y})}_{P})$ for the total linking number of $\partial C_{n}$ and $\partial C_{y}$ . Then by Theorem 6.3 we obtain

Theorem 7.12.

\sum_{[P]} \int_{{(\partial C_{y})}_{P}} 𝜃_{ϕ_{0, 1}}^{P} (τ, ℒ_{W_{P}}) = \sum_{n > 0} Lk (C_{n}, C_{y}) q^{n} + \sum_{n \in ℚ} \sum_{[P]} \int_{{(\partial C_{y})}_{P}} {\tilde{ψ}}_{0, 1}^{P} (n) (τ) .

So the Fourier coeﬃcients of the holomorphic part of $\int_{{(\partial C_{y})}_{P}} 𝜃_{ϕ}^{P} (τ, ℒ_{W_{P}})$ are the linking numbers of the cycles $\partial C_{y}$ and $\partial C_{n}$ at the boundary component $e^{'} (P)$ .

Remark 7.13. There is also another “global” proof for Theorem 7.12. The cycle $C_{y}$ intersects $e^{'} (P)$ transversally (when pushed inside) and hence also the cap $A_{n}$ . From this it is not hard to see that we can split the intersection number $C_{n}^{c} \cdot C_{y}$ as

C_{n}^{c} \cdot C_{y} = {(C_{n} \cdot C_{y})}_{X} - Lk (C_{n}, C_{y}) .

Hence Theorem 7.12 also follows from combining (7.3) and Theorem 7.11.

Hirzebruch-Zagier also obtain the modularity of the functions given in Theorems 7.11 and 7.12, but by quite diﬀerent methods. In particular, they explicitly calculate the intersection number $T_{n}^{c} \cdot T_{m}$ . They split the intersection number into the interior part ${(T_{n} \cdot T_{m})}_{X}$ and a ‘boundary contribution’ ${(T_{n} \cdot T_{m})}_{\infty}$ given by

{(T_{n} \cdot T_{m})}_{\infty} = {(T_{n} \cdot T_{m})}_{\tilde{X} - X} - (T_{m} - T_{m}^{c}) \cdot (T_{n} - T_{n}^{c}) .

Now by Theorem 7.9 and its proof we have

T_{n}^{c} \cdot T_{m} = C_{n}^{c} \cdot C_{m} .

We have (per deﬁnition) ${(T_{n} \cdot T_{m})}_{X} = {(C_{n} \cdot C_{m})}_{X}$ , so Theorem 7.11 gives the generating series for ${(T_{n} \cdot T_{m})}_{X}$ . Note that Theorem 5.4 in [9] also compares the explicit formulas in [16] for ${(T_{n} \cdot T_{m})}_{X}$ with the ones obtained via $\int_{C_{y}} 𝜃_{φ_{2}} (τ, ℒ_{V})$ . All this implies

{(T_{n} \cdot T_{m})}_{\infty} = Lk (C_{n} \cdot C_{m}) .

Independently, we also obtain this from comparing the explicit formulas for the boundary contribution in [16], Section 1.4 with our formulas for the linking numbers, Theorem 4.11 and Example 4.12.

8. A current approach for the special cycles

In this section we prove Theorem 7.7, the crucial Fourier coeﬃcient formula for our lift $Λ^{c}$ . As a consequence of our approach we will also obtain Theorem 6.7, the linking number interpretation for the lift at the boundary.

8.1. A diﬀerential character for $C_{n}^{c}$ .

The key step for the entire Kudla-Millson theory is that for $n > 0$ the form $φ_{2} (n)$ is a Poincaré dual form for the cycle $C_{n}$ , i.e.,

Theorem 8.1 ([22, 23]). Let $η$ be a closed rapidly decreasing $2$ -form. Then

\int_{X} η \land φ_{2} (n) = (\int_{C_{n}} η) e^{- 2 π n} .

To show this they employ at some point a homotopy argument which requires $η$ to be rapidly decaying. Since we require $η$ to be any closed $2$ -form on the compactiﬁcation $\bar{X}$ , their approach is not applicable in our case. Instead, we use a diﬀerential character argument for $φ_{2}$ which implicitly already occurred in [5], Section 7 for general signature $(p, q)$ . Namely, we have

Theorem 8.2. ([5], Section 7) Let $n > 0$ . The singular form ${\tilde{ψ}}_{1} (n)$ is a diﬀerential character in the sense of Cheeger-Simons for the cycle $C_{n}$ . More precisely, ${\tilde{ψ}}_{1} (n)$ is a locally integrable $1$ -form on $X$ , and for any compactly supported $2$ -form $η$ we have

\int_{X} η \land φ_{2} (n) = (\int_{C_{n}} η) e^{- 2 π n} - \int_{X} d η \land {\tilde{ψ}}_{1} (n) .

Proof. This is the content of the proofs of Theorem 7.1 and Theorem 7.2 in [5]. There the analogous properties for a singular theta lift associated to $ψ$ is established. However, the proofs boil down to establish the claims for ${\tilde{ψ}}_{1}$ . The form $\tilde{ψ}$ there is indeed the form ${\tilde{ψ}}_{1}$ of this paper. □

Remark 8.3. The form ${\tilde{ψ}}_{1}$ is closely related to Kudla’s Green function $ξ$ [18, 19] (more generally for $O (p, 2)$ ) which is given by

ξ (x) = (\int_{1}^{\infty} φ_{0}^{0} (\sqrt{r} x) \frac{d r}{r}) e^{- π (x, x)} .

Then $Ξ (n) = \sum_{x \in ℒ_{n}} ξ (x)$ gives rise to a Green’s function for the divisor $C_{n}$ and moreover $d d^{c} ξ = φ_{2}$ . Here $d^{c} = \frac{1}{4 π i} (\partial - \bar{\partial})$ . This suggests $d^{c} ξ = {\tilde{ψ}}_{1}$ , which indeed follows from $d^{c} φ_{0} = - ψ_{1}$ , see [5], Remark 4.5.

For $n \in ℚ$ we deﬁne

φ_{2}^{c} (n) : = φ_{2} (n) - \sum_{[P]} d (f π^{*} ϕ_{0, 1}^{P} (n))

and follow the current approach to show that for $n > 0$ the form $φ_{2}^{c} (n)$ is a Poincaré dual form for the cycle $C_{n}^{c}$ . Here we follow the notation of subsection 3.3. That is, $π^{*} ϕ_{0, 1}^{P} (n)$ is the pullback to a product neighborhood $V$ of $\partial \bar{X}$ , and $f$ is a smooth function on $V$ of the geodesic ﬂow coordinate $t$ which is $1$ near $t = \infty$ and zero else. Note that $φ_{2}^{c} (n)$ is exactly the $n$ -th Fourier coeﬃcient of the mapping cone element $[𝜃_{φ}, 𝜃_{ϕ}]$ , when realized as a rapidly decreasing form on $X$ . We also deﬁne

{\tilde{ψ}}_{1}^{c} (n) = {\tilde{ψ}}_{1} (n) - f π^{*} ϕ_{0, 1}^{P} (n) .

We call a diﬀerential form $η$ on $\bar{X}$ special if in a neighborhood of each boundary component $e^{'} (P)$ it is the pullback of a form $η_{P}$ on $e^{'} (P)$ under the geodesic retraction and if the pullback of the form $η_{P}$ to the universal cover $e (P)$ is $N$ -left-invariant. The signiﬁcance of the forms lies in the fact that the complex of special forms also computes the cohomology of $\bar{X}$ . Note that the proof of Theorem 7.2 shows that $𝜃_{φ_{2}}$ is ‘almost’ special; it only diﬀers from a special form by a rapidly decreasing form.

Theorem 8.4. Let $n > 0$ . The form ${\tilde{ψ}}_{1}^{c} (n)$ is a diﬀerential character for the cycle $C_{n}^{c}$ . More precisely, ${\tilde{ψ}}_{1}^{c} (n)$ is a locally integrable $1$ -form on $X$ and satisﬁes the following current equation on special $2$ forms on $\bar{X}$ :

d [{\tilde{ψ}}_{1}^{c} (n)] + δ_{C_{n}} e^{- 2 π n} = [φ_{2}^{c} (n)] .

That is, for any special $2$ -form $η$ on $\bar{X}$ we have

\int_{X} η \land φ_{2}^{c} (n) = (\int_{C_{n}^{c}} η) e^{- 2 π n} - \int_{X} d η \land {\tilde{ψ}}_{2}^{c} (n) .

This implies Theorem 7.7 for the positive Fourier coeﬃcients. For $n \leq 0$ , the form $φ_{2}^{c} (n)$ is exact with primitive ${\tilde{ψ}}_{2}^{c} (n)$ which by Theorem 7.4 is decaying. So Theorem 8.4 holds also for $n \leq 0$ with $C_{n}^{c} = \emptyset$ . Hence for the these coeﬃcients only the term $x = 0$ contributes, which gives the integral of $η$ against the Kähler form.

Remark 8.5. In view of Remark 8.3 it is very natural question to ask how one can modify Kudla’s Green’s function $Ξ (n)$ to obtain a Green’s function for the cycle $T_{n}^{c}$ in $\tilde{X}$ . Extensive discussions with Kühn suggest that (if $X$ has only one cusp)

Ξ (n) - t \sum_{\begin{array}{c} x \in ℒ_{W} \\ (x, x) = 2 n \end{array}} f π^{*} (B (x) + B^{'} (x))

is such a Green’s function, but we have not checked all details.

8.2. Proof of Theorem 8.4 .

For simplicity assume that $X$ has only one cusp and continue the drop the superscript $P$ . We let $ρ_{T}$ be a family of smooth functions on a standard fundamental domain $ℱ$ of $Γ$ in $D$ only depending on $t$ which is $1$ for $t \leq T$ and $0$ for $T + 1$ . We then have

\begin{array}{l} \int_{X} η \land φ_{2}^{c} (n) & = lim_{T \to \infty} \int_{X} ρ_{T} η \land (φ_{2} (n) - d (f π^{*} ϕ_{0, 1} (n))) . \end{array}

We apply Theorem 8.2 for the compactly supported form $ρ_{T} η$ and obtain

\begin{align} \int_{X} η \land φ_{2}^{c} (n) & = lim_{T \to \infty} [(\int_{C_{n}} ρ_{T} η) e^{- 2 π n} - \int_{X} d (ρ_{T} η) \land {\tilde{ψ}}_{1} (n) & (8.1) \\ - \int_{X} d (ρ_{T} η \land (f π^{*} ϕ_{0, 1} (n))) - d (ρ_{T} η) \land f π^{*} ϕ_{0, 1} (n)] \end{align}

The ﬁrst term on the right hand side of (8.1) goes to $(\int_{C_{n}} η) e^{- 2 π n}$ as $T \to \infty$ , while the third vanishes for any $T$ by Stokes’ theorem. For the two remaining terms of (8.1) we ﬁrst note $d (ρ_{T} η) = ρ_{T}^{'} (t) d t \land η + ρ_{T} d η$ and $ρ_{T}^{'} (t) = 0$ outside $[T, T + 1]$ . We obtain for these two terms

\begin{matrix} - \int_{X} (d η) \land ({\tilde{ψ}}_{1} (n) - f π^{*} ϕ_{0, 1} (n)) \\ - lim_{T \to \infty} \int_{T}^{T + 1} \int_{e^{'} (P)} ρ_{T}^{'} (t) d t \land η \land ({\tilde{ψ}}_{1} (n) - f π^{*} ϕ_{0, 1} (n)) . & (8.2) \end{matrix}

It remains to compute the second term in the previous equation. For $T$ suﬃciently large we have $f \equiv 1$ . Furthermore by Theorem 7.4 and its proof we have ${\tilde{ψ}}_{1} (n) = π^{*} {\tilde{ψ}}_{0, 1} (n) + O (e^{- C t})$ . As $ϕ_{0, 1} (n) = {\tilde{ψ}}_{0, 1} (n) + {\tilde{ψ}}_{0, 1}^{'} (n)$ , we can replace ${\tilde{ψ}}_{1} (n) - f π^{*} ϕ_{0, 1} (n)$ by $- π^{*} {\tilde{ψ^{'}}}_{0, 1} (n)$ . Since $η$ is special it does not depend on the $t$ -variable near the boundary. For the last term in (2)

lim_{T \to \infty} \int_{T}^{T + 1} ρ_{T}^{'} (t) d t \int_{e^{'} (P)} η \land π^{*} {\tilde{ψ^{'}}}_{0, 1} (n) = - \int_{e^{'} (P)} η \land {\tilde{ψ^{'}}}_{0, 1} (n) = - (\int_{A_{n}} η) e^{- 2 π n} .

Indeed, for $η = Ω$ this is Remark 6.9. Otherwise, $η$ is exact with special primitive $ω$ , and it is not hard to see that the proof of Proposition 6.8 carries over to this situation. Since $C_{n}^{c} = C_{n} ∐ (- A_{n})$ collecting all terms completes the proof of Theorem 8.4.

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Department of Mathematical Sciences, University of Durham, Science Laboratories, South Rd, Durham DH1 3LE, United Kingdom

E-mail address: jens.funke@durham.ac.uk

Department of Mathematics, University of Maryland, College Park, MD 20742, USA

E-mail address: jjm@math.umd.edu