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 [11121314] 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 X̃ of the Hilbert modular surface X with the (real) Borel-Serre compactification 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() × O(V ) on 𝒮(V ), the Schwartz functions on V . Let G = SO 0(V ) and let K be a maximal compact subgroup. We let 𝔤 and 𝔨 be their respective Lie algebras and let 𝔤 = 𝔭 𝔨 be the associated Cartan decomposition. Suppose

φ (𝒮(V ) r𝔭)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 φ̃ on the symmetric space D = GK of dimension pq with values in 𝒮(V ). For a coset of a lattice in V , we define the theta distribution Θ = Θ by Θ = δ, where δ is the delta measure concentrated at . It is obvious that Θ is invariant under Γ = Stab() G. There is also a congruence subgroup Γ of SL(2, )) such that Θ is also invariant under Γ. Hence we can apply the theta distribution to φ̃ to obtain a closed r-form 𝜃φ on X = ΓD given by

𝜃φ() = Θ,φ̃.

Assume now in addition that φ has weight k under the maximal compact subgroup SO(2) SL 2(). Then 𝜃φ also gives rise to a (in general) non-holomorphic function on the upper half place which is modular of weight k for Γ. We may then use 𝜃φ as the kernel of a pairing of modular forms f with (closed) differential (pq 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 [2122] is that they found (in greater generality) a family of cocycles φqV in (𝒮(V ) q𝔭)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 q) in the cocompact (standard arithmetic) case that the geometric theta correspondence specialized to φqV induces on the adelic level an isomorphism from the appropriate space of classical modular forms to Hq(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 φqV 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 -rank 1. Then D × , and X is a Hilbert modular surface. We let 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 X¯ a 4-manifold with boundary. For simplicity, we assume that X has only one cusp so that X¯ = e(P), and we write k : X¯X¯ for the inclusion. The special cycles Cn2 in question are now embedded modular and Shimura curves, and are parameterized by n . They define relative homology classes in H2(X,X, ).

The geometric theta correspondence of Kudla-Millson [24] for the cocycle φ2V in this situation takes the following shape. For a compact cycle C in X, we have that

𝜃φ2V ,C =C𝜃φ2V = n0(Cn C)qn (1.1)

is a holomorphic modular form of weight 2 and is equal to the generating series of the intersection numbers with Cn. Here q = e2πiτ with τ . (There is a similar statement for the pairing of 𝜃φ2V with a closed compactly supported differential 2-form on X representing a class in Hc2(X), see Theorem 7.1). Our first result is

Theorem 1.1. (Theorem 7.3) The differential form 𝜃φ2V on X extends to a form on X¯, and the restriction k of 𝜃φ2V to X¯ gives an exact differential form on X¯. Moreover, there exists a theta series 𝜃ϕ1W for a space W of signature (1, 1) of weight 2 with values in the 1-forms on X¯ such that 𝜃ϕ1W is a primitive for k𝜃 φ2V :

d(𝜃ϕ1W) = k𝜃 φ2V .

Considering the mapping cone for the inclusion k : X¯X¯ (see Section 3.3) we then view the pair [𝜃φ2V ,𝜃ϕ1W] as an element of the compactly supported cohomology Hc2(X). Explictly, let C be a relative cycle in X¯ representing a class in H2(X,X, ). Then the Kronecker pairing between [𝜃φ2V ,𝜃ϕ1W] and C is given by

[𝜃φ2V ,𝜃ϕ1W],C =C𝜃φ2V C𝜃ϕ1W. (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 Cn at the boundary X¯ (Section 4). The intersection of Cn with 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 An in X¯ whose boundary is the boundary of Cn in X¯. “Capping” off Cn by An, we obtain a closed cycle Cnc in X¯ defining a class in H2(X, ). Our main result is the extension of (1.1):

Theorem 1.2. (Theorem 7.7) Let C be a relative cycle in X¯. Then

[𝜃φ2V ,𝜃ϕ1W],C = n0(Cnc C)qn

is a holomorphic modular form of weight 2 and is equal to the generating series of the intersection numbers with the capped cycles Cnc. (Similarly for the pairing with an arbitrary closed 2-form on X¯ representing a class in H2(X)).

Note that in view of (1.2) the lift of classes of H2(X,X) or H2(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 𝜃φqV (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 𝜃ϕ1W 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 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 X¯ which is disjoint from the torus fibers containing components of Cn. Then the holomorphic part of the weight 2 non-holomorphic modular form c𝜃ϕ1W is given by the generating series of the linking numbers n>0 Lk(Cn,c)qn.

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 n>0 Lk(Cn,c)qn is a “mixed Mock modular form” of weight 2; it is the product of a Mock modular form of weight 32 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 : XX̃ to modular forms. They introduced the Hirzebruch-Zagier curves Tn in X, which are given by the closure of the cycles Cn in X̃. They then defined “truncated” cycles Tnc as the projections of Tn orthogonal to the subspace of H2(X̃, ) spanned by the compactifying divisors of X̃. The principal result of [16] was that n0[Tnc]qn defines a holomorphic modular form of weight 2 with values in H2(X̃, ). We show jCnc = T nc (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) = n=0(T nc T m)qn

for the intersection numbers in X̃ of Tnc with a fixed Tm 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 Tnc T m as the sum of two terms, Tnc T m = (Tn Tm)X + (Tn Tm), where (Tn Tm)X is the geometric intersection number of Tn and Tm in the interior of X and (Tn Tm) which they called the “contribution from infinity”. They then proved both generating functions n=0(T n Tm)Xqn and n=0(T n Tm)qn are the holomorphic parts of two non-holomorphic forms FX and F 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 = Cm. 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 (Tn Tm)X encoded in FX. So via Theorem 1.2 the second term on the right hand side of (1.2) must match the boundary contribution F in [16], that is, we obtain

Theorem 1.4.

(Tn Tm) = Lk(Cn,Cm).

Hence we give an interpretation for the boundary contribution in [16] in terms of linking numbers in X¯. In fact, the construction of 𝜃ϕ1W owes a great deal to Section 2.3 in [16], where a scalar-valued version of 𝜃ϕ1W 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 φ2V is that the n-th Fourier coefficients of 𝜃φ2V represents the Poincaré dual class for the cycle Cn. Kudla-Millson establish this by showing that φ2V gives rise to a Thom form for the normal bundle of each of the components of Cn. 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 Cn constructed by Kudla [1819]. 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 Tnc in X̃. 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 G̲ = SO(V ), viewed as an algebraic group over . We let G = G̲0() SO 0(2, 2) be the connected component of the identity of the real points of G̲. It is most convenient to identify the associated symmetric space D = DV with the space of negative 2-planes in V () on which the bilinear form (,) is negative definite:

D = {z V ; dim z = 2 and (,)|z < 0}.

We pick an orthogonal basis {e1,e2,e3,e4} of V with (e1,e1) = (e2,e2) = 1 and (e3,e3) = (e4,e4) = 1. We denote the coordinates of a vector x with respect to this basis by xi. We pick as base point of D the plane z0 = [e3,e4] spanned by e3 and e4, and we let K SO(2) × SO(2) be the maximal compact subgroup of G stabilizing z0. Thus D GK. Of course, D × , the product of two upper half planes.

We let P̲ be a rational parabolic subgroup stabilizing a rational isotropic line and define P = P̲0() as before. We let N̲ be its unipotent subgroup and N = N̲(). We let u = (e1 + e4)2 and u = (e 1 e4)2 be two isotropic vectors so that (u,u) = 1. We assume that u,u are defined over and obtain a rational Witt decomposition

V = W

with = u, = u, and a subspace W = such that W = span (e2,e3). The choice of u gives a Levi splitting of P̲, and we write


for the Langlands decomposition. Here, with respect to the basis u,e2,e3,u, we have

N = n(w) = 1 (,w) (w,w)2 1W w 1 ; w W , A = a(t) = t 1W t1 ; t + , M = m(s) = 1 cosh(s) sinh(s) sinh(s) cosh(s) 1 ; s .

Note N W. We obtain coordinates for D by z = z(t,s,w) where z is the negative two-plane in V with z = [n(w)a(t)m(s)e3,n(w)a(t)m(s)e4].

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

X = ΓD.

Example 2.1.

An important example is the following. Let d > 0 be the discriminant of the real quadratic field K = (d) over , 𝒪K its ring of integers. We denote by xx the Galois involution on K. We let V M2(K) be the space of skew-hermitian matrices in M2(K), i.e., which satisfy tx = x. Then the determinant on M2(K) gives V the structure of a non-degenerate rational quadratic space of signature (2, 2) and -rank 1. We define the integral skew-hermitian matrices by

L = x = ad λ λ bd : a,b ,λ 𝒪K .

Then L is a lattice of level d. We embed SL 2(K) into SL 2 × SL 2() by g(g,g) so that SL 2(𝒪K) acts on L by γ.x = γxtγ as isometries. Hirzebruch and Zagier actually considered this case for d 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 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 ΓP = Γ P and ΓN = ΓP N. Then the quotient ΓP ΓN is a non-trivial arithmetic subgroup of P̲N̲ and lies inside the connected component of the identity of the real points of P̲N̲. Furthermore, ΓP ΓN acts as isometries of spinor norm 1 on the anisotropic quadratic space of signature (1, 1). Hence ΓP ΓN is infinite cyclic. Therefore the exact sequence


splits. We fix g ΓP such that its image ḡ generates ΓP ΓN. Then g defines a Levi subgroup M. In fact, the element g generates ΓM := ΓP M. Hence


We will say a Levi decomposition P = NAM is admissible if ΓP = (M ΓP ) ΓN. 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 D¯ be the (rational) Borel-Serre enlargement of D, see [3] or [2], III.9. For any parabolic P̲ as before with admissible Levi decomposition P = NAM, we define the boundary component

e(P) = MN DW × W.

Here DW M is the symmetric space associated to the orthogonal group of W. Then D¯ is given by

D¯ = D P̲e(P),

where P̲ varies over all rational parabolics. The action of Γ on D extends to D¯ in a natural way, and we let

X¯ := ΓD¯

be the Borel-Serre compactification of X = ΓD. This makes X¯ a manifold with boundary such that

X¯ = [P̲]e(P),

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

e(P) = Γ P e(P).

Here [P̲] runs over all Γ-conjugacy classes. The space XW := ΓMDW is a circle. Hence e(P) is a torus bundle over the circle, where the torus T2 is given by ΓNN. That is, e(P) = X W × T2, and we have the natural map κ : e(P) X W . We have a natural product neighborhood of e(P) in D¯ and hence for e(P) in X¯ given by [(T,] × e(P)] for T sufficiently large given by z(t,s,w) with t > T. We let i : XX¯ and iP : e(P)X¯ be the natural inclusions.

It is one of the fundamental properties of the Borel-Serre compactification 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 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 X̃ be Hirzebruch’s smooth resolution of the cusp singularities and π : X̃ X be the natural map collapsing the compactifying divisors for each cusp. We let j : XX̃ be the natural embedding. Note that the Borel-Serre boundary separates X̃ into two pieces, the (connected) inside Xin, which is isomorphic to X and the (disconnected) outside Xout, which for each cusp is a neighborhood of the compactifying divisors. Note that we can view e(P) as lying in both Xin and Xout since the intersection Xin Xout is equal to P̲e(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 ΓN = π1(T2) is a rational multiple of a commutator in ΓP and accordingly the image of H1(T2, ) in H1(e(P), ) is trivial. Let aP H1(e(P), ) be the class of the identity section of κ : e(P) X W and bP H2(e(P), ) be the class of the torus fiber of κ. It is clear that the intersection number of aP and bP is 1 (up to sign) whence aP and bP are nontrivial primitive classes. Furthermore, aP generates H1(e(P), ) and H2(e(P), ), generated by bP . So

Lemma 3.1.

  1. The first rational homology group of e(P) is generated by aP .
  2. The second homology group of e(P) is generated by bP .

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

Let ΩP be the unique P-invariant 2-form on e(P) such that

bPΩP = 1. (3.1)

Since bP is the image of the fundamental class of T2 inside H2(e(P), ), we see that that the restriction of ΩP to T2 lifts to the area form on W N normalized such that T2 = Γ NN has area 1.

3.2. Homology and cohomology of X and X̃.

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

0 P H2(e(P)) H 2(X) (P SP ) H2(X̃) 0.

Here SP denotes the span of the classes defined by compactifying divisors at the cusp associated to P. The zero on the left comes from H3(X̃) = 0 and the zero on the right comes from the fact that for each P the class aP injects into H1(Xout), see [28], II.3. Since the generator bP has trivial intersection with each of the compactifying divisors, bP bounds on the outside so a fortiori it bounds in X̃. Thus the above short exact sequence is the sum of the two short exact sequences P H2(e(P)) H 2(X) jH2(X) and 0 P SP P SP . By adding the third terms of the two sequences and equating them to H2(X̃) we obtain the orthogonal splittings (for the intersection pairing) - see also [28], p.123,

H2(X̃) = jH2(X) [P]SP ,H2(X̃) = j #Hc2(X) [P]SP .

Here j# is the push-forward map. Furthermore, the pairings on each summand are non-degenerate. Considering P H2(e(P)) H 2(X) jH2(X) we also obtain

Proposition 3.3. H2(X¯) is the kernel of j so that

jH2(X) H2(X) [P]H2(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 Ac(X) be the complex of compactly supported differential forms on X which gives rise to Hc(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 : XX¯. However, we will change the sign of the differential on C and shift the grading down by one. Thus we have

Ci = {(a,b),a Ai(X¯),b Ai1(X¯)}

with d(a,b) = (da,ia db). 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 Ac(X) C given by c(c, 0) is a quasi-isomorphism.

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

a d(fπb) = α + dμ.

We define the cohomology class [a,b] in the compactly supported cohomology Hci(X) to be the class of α, and the assignment [[a,b]][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 X¯ and C a relative cycle in X¯ of appropriate degree. Then

[a,b], [η] =X¯a η X¯b iη,and[a,b],C =Ca Cb.

4. Capped special cycles and linking numbers in Sol

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

Dx = {z D; z x}.

Then Dx is an embedded upper half plane in D. We let Γx Γ be the stabilizer of x and define the special or Hirzebruch-Zagier cycle by

Cx = ΓxDx,

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

Cn = xΓnCx.

Since the divisors define in general relative cycles, we take the sum in H2(X,X, ).

4.1. The closure of special cycles in the Borel-Serre boundary and the capped cycle Cxc.

We now study the closure of Cx in X¯, which is the same as the intersection of C¯x or Cx with the union of the hypersurfaces e(P). A straightforward calculation gives

Proposition 4.1. If (x,u)0 then there exists a neighborhood U of e(P) such that

Dx U = .

If (x,u) = 0, then D¯x e(P) is contained in the fiber of p over s(x), where s(x) is the unique element of satifying

(x,m(s(x))e3) = 0.

At s(x) the intersection D¯x e(P) is the affine line in W given by

{w W : (x,w) = (u,x)}.

We define cx Cx to be the closed geodesic in the fiber over s(x) which is the image of D¯x e(P) under the covering e(P) e(P). We have

Proposition 4.2.

  1. The 1-cycle Cx is a finite union of circles.
  2. At a cusp associated to P, each circle is contained in a fiber of the map κ : e(P) X W and hence is a rational boundary (by Lemma 3.1).
  3. Two boundary circles cx and cy are parallel if they are contained in the same fiber. In particular, cx cycx = cy.

We now describe the intersection of C¯n or Cn with e(P). For V = = L + h we can write

W = WP = ( u)( u) k LW,k + hW,k

for some lattices LW,k W and vectors hW,k LW,k#.

Via the isomorphism W N, we can identify ΓN = N Γ with a lattice ΛW in W. Since u is primitive in L and n(w)x = x + (w,x)u for a vector x u we see that W is contained in the dual lattice of ΛW .

Lemma 4.3. The intersection Cn e(P) is given by

xΓMW (x,x)=2n 0k<min λΛ W|(λ,x)|cx+ku.

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


We will first prove Cn,P := Cn e(P) is a disjoint union

Cn,P = yΓPn,ucy, (4.1)

where n,u = {x u; (x,x) = 2n}. Indeed, first note that by Proposition 4.1 only vectors in n,u can contribute to Cn,P . The action of Γ on V induces an equivalence relation Γ on the set Γpn,u V which is consequently a union of equivalence classes [xi] = [xi]P , 1 i k. We may accordingly organize the union R on the right-hand side of (4.1) as R = i=1k y[xi]cy. But it is clear that (Cxi)P = y[xi]cy and hence we have the equality of 1-cycles in e(P) and X

(Cxi)P = y[xi]PcyandCxi = [P] y[xi]Pcy, (4.2)

since an element y [xi] gives rise to the lift Dy of Cxi to D that intersects e(P) and this intersection projects to cy. Thus we may rewrite the right-hand side of(4.1) as R = Γn,uCxi. But it is clear that this latter union is Cn,P and (4.1) follows. Finally, we easily see that xΓMW (x,x)=2n 0k<min λΛ W|(λ,x)|x+kuis a complete set of representatives of ΓP -equivalence classes in n,u. These give the circles cx+ku above.

Proposition 4.4. Let x n,u with n > 0. Then there exists a rational 2-chain ax in e(P) such that

  1. ax = cx
  2. axΩP = 0, here ΩP is the area form for the fibers (see (3.1))


Except for the rationality of the cap this follows immediately from Proposition 4.2. The problem is to find a cap ax such that axΩP . We will prove this in Section 4.3 below.

We will define (Ax)P by (Ax)P = y[x]ax. Then sum over the components e(P) to obtain Ax a rational 2-chain in X. Then we have (noting that (Cx)P = y[x]cy)

Ax = Cx.

Definition 4.5. We define the rational absolute 2-cycle in X¯ by

Cxc = C x (Ax)

with the 2-chain Ax in X¯ as in Proposition 4.4. In particular, Cxc defines a class in H2(X¯) = H2(X). In the same way we obtain Cnc.

4.2. The closure of the special cycles in X̃ and the cycle Tnc.

Following Hirzebruch-Zagier we let Tn be the cycle in X̃ given by the closure of the cycle Cn in X̃. Hence Tn defines a class in H2(X̃).

Definition 4.6. Consider the decomposition H2(X̃) = jH2(X) [P]Sp, which is orthogonal with respect to the intersection pairing on X̃. We let Tnc be the image of Tn under orthogonal projection onto the summand jH2(X).

Proposition 4.7. We have

jCnc = T nc.

Proof. For simplicity, we assume that X has only one cusp. The 3-manifold e(P) separates Tn and we can write Tn = Tn Xin + T n Xout as (appropriately oriented) 2-chains in X̃. It is obvious that we have jC¯n = Tn Xin as 2-chains. We write Bn = Tn Xout. We have Cn = Bn. Hence we can write Tn = jCnc + B nc, the sum of two 2-cycles in X̃. Here Bnc is obtained by ‘capping’ Bn in e(P) with the negative of the cap An of Cnc. Since jCnc is clearly orthogonal to SP (since it lies in Xin) and Bnc S P (since it lies in Xout) the decomposition Tn = jCnc + B nc is just the decomposition of Tn relative to the splitting H2(X̃) = jH2(X) SP . Hence Tnc = j Cnc, 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 Cx, a 2-chain A with that component as boundary so that A is a sum P + T + (γ0) 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” (γ0) 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 C1(T2) on T2. 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 Ck(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 SL(2, ) be a hyperbolic element. We will then consider the 3-manifold M obtained from × T2 (with the 2-torus T2 = W2) given by the relation

(s,w) (s + 1,f(w)). (4.3)

We let π : × T2 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 T2, a subscript c will indicate that the geodesic starts at the point c on T2. 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 T2. If x and y are points on W we will use xy¯ to denote the oriented line segment joining x to y and xy to denote the corresponding (free) vector i.e. the equivalence class of xy¯ 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 α0 be the parallel translate of α to the origin. Then we can find a cylinder P, image of an oriented parallelogram P̃ under the universal cover W T2 with rational vertices, such that in Z1(T2, ), the group of rational 1-cycles, we have

P = α α0. (4.4)

Since P̃ has rational vertices we find P Ω =P̃Ω .

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 γ0 T2 starting at 0 we define the monodromy 2-chain (γ0) to be the image of the cylinder γ0 × [0, 1] T2 × in M. The reader will verify using (4.3) that in Z1(T2, ) we have

(γ0) = f1(γ 0) γ0. (4.5)

Since f preserves the origin, the geodesic f1(γ 0) is also a closed geodesic starting at the origin. Since f1 is hyperbolic we have | tr(f1)| > 2 and hence det(f1 I) = det(I f) = tr(f) 20. Put N = det(f1 I) and define [γ0] H1(T2, ) by

f1([γ 0]) [γ0] = N[α0]. (4.6)

Note that [γ0] = N{(f1 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 Z1(T2, ). Since any homology class contains a unique closed geodesic starting at the origin we obtain a closed geodesic γ0 [γ0] and a corresponding monodromy 2-chain (γ0) whence (4.5) holds in Z1(T2, ). We now solve

Problem 4.8. Find an equation in Z1(T2, ) which descends to (4.6).

Let h1 resp. h2 denote the covering transformation of π corresponding to the element α0 resp γ0 of the fundamental group of T2. Define c1 and c2 in W by c1 = Nh1(0) and c2 = h2(0). Define d W by d = f1(c 2) in W. Let T̃ be the oriented triangle with vertices 0,c2,d. Then

(i)π(0c1¯) = Nα0(ii)π(0c2¯) = γ0(iii)π(0d¯) = π(f1(0c 2¯)) = f1(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

h2(0c1¯) = c2d¯. (4.7)

With this we can solve the problem. We see that if we consider T̃ as an oriented 2-simplex we have the following equality of one chains

T̃ = 0c2¯ + c2d¯ 0d¯.

Let T be the image of T̃ under π. Take the direct image of the previous equation under π and use equation (4.7) which implies that the second edge c2d¯ is equivalent under h2 in the covering group to the directed line segment 0c1¯ which maps to Nα0. Hence c2d¯ also maps to Nα0. We obtain the following equation in Z1(T2, )

T = γ0 + Nα0 f1(γ 0), (4.8)

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

((γ0) + T) = f1(γ 0) γ0 + γ0 + α0 f1(γ 0) = Nα0.

Combining this with (4.4) and setting where A0 = (γ0) + T we obtain

(NP + A0) = Nα, (4.9)

in Z1(M, ). Hence if we define A to be the rational chain A = 1 N(NP + A0) = P + 1 NT + 1 N(γ0) in M we have the following equation in Z1(M, ):

A = α.

Finally, the integral of Ω over A is rational. Indeed, the integral over P is rational. Since all vertices of T̃ are integral the area of T̃ is integral, the integral of Ω over T is integral. Thus it suffices to observe that the restriction of Ω to (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 H1(T2, ), and in this section we are allowed to confuse a and b with their representatives in the lattice 2 and the unique closed geodesic in T2 passing through the origin that represents them. We will write for the image of a and b in × T2 and M a = a(0) = 0 × a and b = b(𝜀) = 𝜀 × b. Our goal is to compute the linking number Lk(a,b(𝜖)). By the explicit construction of the cap A in Section 4.3 we obtain

Lemma 4.9.

Lk(a,b(𝜖)) = M(c) b(𝜖) = c(𝜖) b(𝜖) = c b.

Here c is the rational one cycle obtained by solving (f1 I)(c) = a and M(c) is the (rational) monodromy 2-chain associated to c (see above) with boundary M(c) = (f1 I)(c) = a. Here the first is the intersection of chains in M, the next is the intersection number of 1-cycles in the fiber 𝜖 × T2 and the last is the intersection number of 1-cycles in 0 × T2.

Noting that this last intersection number coincides with the intersection number of the underlying homology classes which in term coincides with the symplectic form , on H1(T2, ) we have found our desired formula for the linking number.

Theorem 4.10. Lk(a,b(𝜖)) = (f1 I)1(a),b.

It is a remarkable fact that there is a simple formula involving only the action of the glueing homeomorphism f SL(2, ) on H1(T2, ) for linking numbers for 1-cycles contained in fiber tori T2 of in Sol (unlike the case of linking numbers in 3).

This immediately leads to an explicit formula for the numbers Lk(Cn,Cm). Using Lemma 4.3 we obtain

Theorem 4.11. Let g = (f1 I)1. Then

Lk((Cn)P , (Cm)P ) = xΓMW (x,x)=2n xΓMW (x,x)=2m (min λΛW |(λ,x)|)(min μΛW |(μ,x)|)g(Jx),Jx.

Here Jx is properly oriented primitive vector in ΛW such that (Jx,x) = 0.

Example 4.12. We consider the integral skew Hermitian matrices in Example 2.1. Let u = p 0 0 0 , so that W = { 0 λ λ 0 ; λ K} K. The symplectic form on K is given by λ,μ = 1 p(λμ λμ). The action of the unipotent radical N = n(λ) = 1 λ 0 1 on a vector μ K is now slightly different, namely, n(λ)μ = μ + λ,μu. Hence in these coordinates, Cμ is given by the image of the line μ = {λ K; λ,μ = 0}, and (min λ𝒪K|λ,μ|)μ is a primitive generator in 𝒪K for that line. We let 𝜀 be a generator of U+, the totally positive units in 𝒪K, and we assume that the glueing map f is realized by multiplication with 𝜀. For d 1(mod 4) a prime and m = 1, C1 has only component arising from x = 1 K and C1 SL 2(). Then Theorem 4.11 becomes (the min -term is now wrt ,)

Lk((Cn)P , (C1)P ) = 2 μU+𝒪K μμ=n,μ0 μ 𝜀1, 1 = 2 μU+𝒪K μμ=n,μ0 = 2 p μ + μ𝜀 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) with maximal compact subgroup K and write D = GK for the associated symmetric space. We let 𝒮(U) be the space of Schwartz functions on U on which SL 2() acts via the Weil representation ω.

5.1. Extending certain Schwartz functions to functions of τ and z D.

Let φ 𝒮(U) be an eigenfunction under the maximal compact SO(2) of SL 2() of weight r. Define gτ SL 2() by gτ = 1 u 0 1 v12 0 0 v12 . Then we have ω(gτ)φ(x) = vm4φ(vx)eπi(x,x)u. Accordingly we define

φ(x,τ) = vr2ω(g τ)φ(x) = vr2+m4φ0(vx)eπi(x,x)τ. (5.1)

Here we have also defined φ0(x) = φ(x)eπ(x,x). Let E be a G-module and let gz G be any element that carries the basepoint z0 in D to z D. Then define for φ [𝒮(U) E]K, the E-valued K-invariant Schwartz functions on U, the functions φ(x,z) and φ(x,τ,z) for x U,z D,τ by

φ(x,z) = gzφ(gz1x)andφ(x,τ,z) = g zφ(gz1x,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 𝔭 be the Cartan decomposition of 𝔤 associated to K. We identify 𝔤 2V as usual via (v1 v2)(v) = (v1,v)v2 (v2,v)v1. We write Xij = ei ej 𝔤 and note that 𝔭 is spanned by Xij with 1 i 2 and 3 j 4. We write ωij for their dual. We orient D such that ω13 ω14 ω23 ω24 gives rise to the G-invariant volume element on D.

5.2.1. Special forms for V . The Kudla-Millson form φ2 is an element in

[𝒮(V ) 𝒜2(D)]G [𝒮(V ) 2 𝔭]K,

where the isomorphism is given by evaluation at the base point. Here 𝒜2(D) denotes the differential 2-forms on D. Note that G acts diagonally in the natural fashion. At the base point φ2 is given by

φ2 = 1 2 μ=34 α=12 x α 1 2π xα φ0 ωαμ.

Here φ0(x) := eπ(x,x)0, where (x,x)0 = i=14x i2 is the minimal majorant associated to the base point in D. Note that φ2 has weight 2, see [21]. There is another Schwartz form ψ1 of weight 0 which lies in [𝒮(V ) 𝒜1(D)]G [𝒮(V ) 𝔭]K and is given by

ψ1 = x1x3φ0(x) ω14 + x1x4φ0(x) ω13 x2x3φ0(x) ω24 + x2x4φ0(x) ω23. (5.2)

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

Theorem 5.1.

ω(L)φ2 = dψ1.

Here ω(L) is the Weil representation action of the SL 2-lowering operator L = 1 2 1 i i 1 𝔰𝔩2() on 𝒮(V ), while d denotes the exterior differentiation on D.

On the upper half plane , 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 , we have

Lf = 2iv2 τ̄f.

When made explicit using (5.1) Theorem 5.1 translates to

v vφ20(vx) = d ψ 10(vx) . (5.3)

5.2.2. The singular form ψ̃1. We define the singular form ψ̃1 by

ψ̃1(x) = 1ψ 10(rx)dr r eπ(x,x) = 1 2π(x32 + x42)ψ1(x). (5.4) 

for x0, and as before ψ̃2,00(x) = ψ̃ 1(x)eπ(x,x) and ψ̃1(x,z). We see that ψ̃1 is defined for x span[e3,e4]. Formulated differently, ψ̃1(x,z) for fixed x is defined for zDx. Furthermore, as if ψ̃1 was a Schwartz function of weight 2, we define

ψ̃1(x,τ,z) = ψ̃10(vx,z)eπi(x,x)τ = vψ 10(rx,z)dr r eπi(x,x)τ. (5.5) 

Proposition 5.2.

ψ̃1(x,z) is a differential 1-form with singularities along Dx. Outside Dx, we have

dψ̃1(x,z) = φ2(x,z).

Here d denotes the exterior differentiation on D. In particular, for (x,x) 0, we see that φ2(x) is exact. Furthermore,

Lψ̃1(x,τ) = ψ1(x,τ).


Using (5.4) and (5.3), we see

dψ̃10(x,z) = 1d ψ 10(rx,z) dr r = 1 r φ20(rx,z) dr r = φ20(x,z),

as claimed. The formula Lψ̃1(x,τ) = ψ1(x,τ) follows easily from (5.5).

Remark 5.3. The construction of the singular form ψ̃ 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

dψ = Lφ.

Then the analogous construction of ψ̃ then immediately yields dψ̃ = φ outside a singular set. The main example for this are the general Kudla-Millson forms φq and ψq1, see [24]. For these forms, this construction is already implicit in [5]. In particular, the proof of Theorem 7.2 in [5] shows that ψ̃ gives rise to a differential character for the analogous cycle Cx, see also Section 8 of this paper. The unitary case will be considered in [10].

5.3. Schwartz forms for W.

Let W 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 for the Lie algebra of M = SO 0(W). Then X23 = e2 e3 is its natural generator with dual ω23. We identify the associated symmetric space DW to M with the space of lines in W on which the bilinear form (,) is negative definite:

DW = {s W; dim s = 1 and (,)|s < 0}.

We pick as base point of DW the line s0 spanned by e3. We set x(s) := m(s)e3 = sinh(s)e2 + cosh(s)e3. This realizes the isomorphism DW . Namely, s = span x(s). Accordingly, we frequently write s for s and vice versa. A vector x W of positive length defines a point DW,x in D via DW,x = {s D; s x}. So s = DW,x if and only if (x,x(s)) = 0. We also write s(x) = DW,x.

5.3.1. Special forms for W. We carry over the conventions from section 5.1. We first consider the Schwartz form φ1,1 on W constructed in [12] (in much greater generality) with values in 𝒜1(D W ) W. More precisely,

φ1,1 [𝒮(W) 𝒜1(D W ) W]M [𝒮(W ) 𝔪 W ],

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

φ1,1(x) = 1 232 4x22 1 πeπ(x22+x32) ω 23 e2.

Note that φ1,1 has weight 2, see [12], Theorem 6.2. We define φ1,1(x,s) and φ1,10 as before. There is another Schwartz function ψ0,1 of weight 0 given by

ψ0,1(x) = 1 2x2x3eπ(x22+x32) 1 e 2 + 1 42πeπ(x22+x32) 1 e 3 [𝒮(W) 0 𝔪 W ],

and also ψ0,1(x,s) and ψ0,10. Note that the notation differs from [12], section 6.5. The function ψ0,1 defined here is the term ψ1,1 1 2Λ1,1 given in Theorem 6.11 in [12]. The key relation between φ1,1 and ψ0,1 (correcting a sign mistake in [12]) is given by

Theorem 5.4. ([12], Theorem 6.2)

ω(L)φ1,1 = dψ0,1.

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

v32 v v12φ 1,10(vx,s) = d ψ 0,10(vx,s) . (5.6)

5.3.2. The singular Schwartz function ψ̃0,1. In the same way as for V we define

ψ̃0,1(x) = 1ψ 0,10(rx)r32dreπ(x,x) (5.7) 

for all x W, including x = 0. Define ψ̃0,10(x), ψ̃0,10(x,s) as before and also

ψ̃0,1(x,τ,s) = v12ψ̃ 0,10(vx,s)eπi(x,x)τ = vψ 0,10(rx,s)r32dreπi(x,x)τ.

Note that ψ̃0,1(x,s) has a singularity at Dw,x. Define functions A and B by

ψ̃0,1(x) = A(x) 1 e2 + B(x) 1 e3

and note

X23B(x) = A(x). (5.8)

We extend these functions to DW as before. We see by integrating by parts

Lemma 5.5.

A(x) = 1 2πx2 x3 |x3|Γ(1 2, 2πx32)eπ(x,x) B(x) = 1 22πeπ(x22+x32) + 1 2π|x3|Γ(1 2, 2πx32)eπ(x,x).

Here Γ(1 2,a) =aeuu12du is the incomplete Γ-funtion at s = 12.

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

Lemma 5.6.

  1. B(x) (12)|x3|eπ(x,x) is C2 on the Minkowski plane W.
  2. A(x) (12)x2 x3 |x3|eπ(x,x) is C1 on the Minkowski plane W.

Proof. Use Lemma 5.5, expand the incomplete gamma function around x3 = 0, and observe that |x|xn is Cn for n > 0.

The key properties of ψ̃0,1 analogous to Lemma 5.2 are given by

Lemma 5.7. Outside DW,x,

dψ̃0,1(x,s) = φ1,1(x,s)andLψ̃0,1(x,τ) = ψ0,1(x,τ).

5.3.3. The singular function ψ̃0,1. Inspired by [16], section 2.3, we define a functions A(x) and B(x) on W by

B(x) = 1 2 min(|x2 x3|,|x2 + x3|)eπ(x,x)ifx 22 x 32 > 0, 0 otherwise, (5.9)  A(x) = X 23B(x) = sgn(x 2x3)B(x).

Lemma 5.8.

  1. B(x) + 1 2|x3|eπ(x,x) is C2 on the complement of the light-cone in W and C2 on nonzero M-orbits.
  2. A(x) + 1 2x2 x3 |x3|eπ(x,x) is C1 on the complement of the light-cone in W and C1 on nonzero M-orbits.

We define ψ̃0,1 by

ψ̃0,1(x) = A(x) 1 e 2 + B(x) 1 e 3

and ψ̃0,1(x,τ,s) = v12m(s)ψ̃ 0,1(m1(s)vx)eπi(x,x)τ. A little calculation shows that ψ̃0,1(x) is locally constant on DW with a singularity at DW,x and holomorphic in τ:

Lemma 5.9. Outside DW,x we have

dψ̃0,1(x) = 0andLψ̃ 0,1(x,τ) = 0.

Remark 5.10. The functions ψ̃0,1(x) and ψ̃0,1(x) define currents on DW . One can show, similarly to Section 6.5, that for (x,x) > 0 we have

d[ψ̃0,1(x)] = δDW,xx + [φ1,1(x)],d[ψ̃0,1(x)] = δ DW,xx,

where DW,x x is the 0-cycle DW,x ‘with coefficient x W’ defined in [12].

5.3.4. The form ϕ0,1 on W. We now combine ψ̃0,1 and ψ̃0,1 to obtain an integrable and also square-integrable W-valued function

ϕ0,1 [L2(W ) 0 𝔪 W ]


ϕ0,1(x) = ψ̃0,1(x) + ψ̃0,1(x)

and then also ϕ0,1(x,s). Combining Lemmas 5.6, 5.8 and (5.8), (5.9) we obtain

Proposition 5.11.

So for given x, the function ϕ0,1(x,s) is a C1-function on DW with values in W.

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 φ1,1 on DW is exact. Namely,

dϕ0,1 = φ1,1.


Lϕ0,1 = dψ0,1.

Proposition 5.13. The function ϕ0,1 is an eigenfunction of K = SO(2) of weight 2 under the Weil representation. More precisely,

ω(k)ϕ 0,1 = χ2(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 ϕ0,1, 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 L1, we see that ω(k)(B + B) is continuous. Hence it suffices to establish the corresponding current equality [ω(k)(B + B)] = χ2(k)[B + B], since continuous functions coincide when they induce the same current. The infinitesimal generator of K acts by H := i 4π 2 x22 2 x32 + πi(x22 x 32), and a straightforward calculation immediately shows

HB = 2iBandHB = 2iB,

outside the singularity x22 x 32 = 0. Now we consider the currents H[B] and H[B]. An easy calculation using that B and B are C2 up to |x3|eπ(x22x32) shows that for a test function f on W we have

H[B](f) = [HB](f) +0eπx22 f(x2, 0)dx2, H[B](f) = [HB](f) 0eπx22 f(x2, 0)dx2.

Thus H[B + B] = [H(B + B)] = 2i[B + B] as claimed.

5.3.5. The map ιP . We define a map

ιP : 𝒮(W) i 𝔪 W 𝒮(W) i+1 𝔪 𝔫


ιP (φ ω w) = φ ω (w u) .

Here we used the isomorphism 𝔫 W u 2V 𝔤 and identify W with its dual via the bilinear form (,) so that 𝔫 W u. In [13], Section 6.2 we explain that ιP is a map of Lie algebra complexes. Hence we obtain a map of complexes

[𝒮(W) 𝒜i(D W ) W]M [𝒮(W ) 𝒜i+1(e(P))]NM,

which we also denote by ιP . Here N acts trivially on 𝒮(W). Explicitly, the vectors e2 and e3 in W map under ιP to the left-invariant 1-forms

e2 cosh(s)dw2 sinh(s)dw3e3 sinh(s)dw2 cosh(s)dw3

with the coordinate functions w2,w3 on W defined by w = w2e2 + w3e3. We apply ιP to the forms on W of this section, and we obtain φ1,1P , ϕ0,1P , ψ0,1P , and ψ 0,1P .

6. The boundary theta lift and linking numbers in Sol

6.1. Global theta functions for W.

We let W be a ΓP -invariant (coset of a) lattice in W, where ΓN acts trivially on W. For φ1,1, we define its theta function by

𝜃φ1,1(τ,W ) = xWφ1,1(x,τ)

and similarly for ψ0,1, and ϕ0,1. Then the usual theta machinery gives that 𝜃φ1,1(τ,W ) and 𝜃ϕ0,1(W ) both transform like (non)-holomorphic modular forms of weight 2 for some congruence subgroup of SL 2().

Remark 6.1. The claim is not obvious for 𝜃ϕ0,1, since ϕ0,1 is not a Schwartz function. In that case, we use Proposition 5.11. The component B + B of ϕ0,1 is C2 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 X23 to obtain the same for the other component A + A of ϕ0,1.

In fact, if W is isotropic and W intersects non-trivially with the light cone, then 𝜃ϕ0,1 is not quite a modular form. The case, when the -rank of V is 2 is interesting in its own right. We will discuss this elsewhere.

Via the map ιP from Section 5.3.5 we can view all theta functions for W as functions resp. differential forms on e(P). We set 𝜃φ1,1P = 𝜃 φ1,1P, and similarly 𝜃ψ0,1P and 𝜃ϕ0,1P . Since ιP is a map of complexes we immediately see by Theorem 5.4 and Theorem 5.12

Proposition 6.2.

L𝜃φ1,1P = d𝜃 ψ0,1P andL𝜃 ϕ0,1P = 𝜃 ψ0,1P .

We now interpret the (holomorphic) Fourier coefficients of the boundary theta lift associated to 𝜃ϕP (τ, WP). 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 Cn or for c = Cy for Cy one of the components of Cn, we have

c𝜃ϕ0,1P (τ, WP) = n=1 Lk((C n)P ,c)qn + ncψ̃0,1P (n)(τ).

So the Fourier coefficients of the holomorphic part of c𝜃ϕP (τ, WP) are the linking numbers of the cycles c and Cn at the boundary component e(P).

Theorem 6.3 follows from ϕ0,1 = ψ̃0,1 + ψ̃ 0,1 combined with Theorem 6.7 below.

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

C1𝜃ϕ0,1P (τ, WP) = 1 2d λ𝒪K λλ>0 min(|λ|,|λ|)e2πλλτ2 dv λ𝒪Kβ(πv(λλ)2)e2πλλτ,

where β(s) = 1 16π1estt32dt. 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 3, see [8], p.79-81, and [7] for its generalization to S3 and H3. 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

aβ = 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 ηM 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 ηM 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 dβ = ηM. 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 A = a. We may suppose ηM vanishes in a neighborhood V of a which is disjoint from U . Then the restriction ηMV of ηM to M V represents the (relative) Poincaré dual of the absolute cycle c in (M V,(M V )). Using this the reader will show that

AηM =A(MV )ηMV = A c = Lk(a,c).

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

aβ =AηM = Lk(a,c).

Hence we have

Proposition 6.6. β is a linking dual of c.

6.3. The 1-form e2πnψ̃ 0,1(n) is a linking dual of (Cn)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 Fn be the union of the fibers containing components of Cn, and we let Fx be the fiber containing cx. Recall that cx is the image of Dx e(P) in e(P).

Theorem 6.7. Let n > 0. The 1-form e2πnψ̃ 0,1(n) is a linking dual for Cn in e(P) relative to any neighborhood U of Fn. Hence, for c a rational 1-boundary in e(P) which is disjoint from Fn we have

cψ̃ 0,1(n) = Lk(Cn,c)e2πn. (6.1)

Furthermore, (6.1) holds when c = cy contained in one fiber Fx of Cn.

We will first deal with the case in which c is disjoint from Fn (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 = cy (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 Fn. Then

e(P)η ψ̃ 0,1(n) = Anηe2πn. (6.2)

Remark 6.9. Note that (6.2) also holds in case η = ΩP . In this case the right-hand side is zero by the normalization of the cap An and the left-hand side is zero because Ω ψ̃ 0,1(n) = 0 since Ω has bidegree (0, 2) and ψ̃ 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

Anη = xΓMW (x,x)=2n min λΛW |(λ,x)|axη

Proof. We use Lemma 4.3. Write η = dω for some 1-form ω which by the support condition on η is closed in Fn. Since cx+ku and cx are parallel hence homologous circles in Fx, we see ax+kuη =cx+kuω =cxω =axη.


ψ̃ 0,1(n) = xΓMW (x,x)=2n γΓMγψ̃ 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 W

e(P)η γΓMγψ̃ 0,1(x) = (min λΛW |(λ,x)|) axηeπ(x,x).

Proof. By choosing appropriate coordinates we can assume that x = μe2 with μ = ±2n, so that the singularity of γΓMγψ̃ 0,1(x) in e(P) occurs at s = 0. We pick a tubular neighborhood U𝜀 = (𝜀,𝜀) × T2 in e(P) around Fx. Then we have first

e(P)η γΓMγψ̃ 0,1P (x) = lim 𝜖0e(P)U𝜀η γΓMγψ̃ 0,1P (x).

Since η ψ̃ 0,1(x) = d(ω ψ̃ 0,1(x)) outside U𝜀 and (e(P) U 𝜀) = U𝜀 we see by Stokes’ theorem

e(P)U𝜀η γΓMγψ̃ 0,1(x) = U𝜀ω γΓMγψ̃ 0,1(x) (6.3)  = γΓMT2 ω(𝜀,w) ψ 0,1 ̃(γ1x,𝜖,w) ω(𝜀,w) ψ̃ 0,1(γ1x,𝜖,w) .

For γ1 we note that ω(s,w) ψ̃ 0,1(γ1x,s,w) is continuous at s = 0, while for γ = 1, we have

ψ̃ 0,1(μe2,s,w) = 1 2|μ|(sgn(s)dw2 dw3)eπμ2 . (6.4)

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

|μ|eπμ2 T2ω3(0,w)dw2dw3 = |μ|eπμ2 T2ce 2 ce 2ω(0,w2,w3) dw2.

In the expression T2c e2 (and for the rest of this proof) we have abused notation and identified the cycle ce2 with the subgroup 0 × S1 of T2.

Here ω3 is the dw3 component of ω and we used that Dx is the w3-line in W. Note that the inner integral on the right is the period of ω over (homologous) horizontal translates of the cycle ce2. But the restriction of ω to Fx is closed so ce 2ω(0,w2,w3) is independent of w2 and the last integral becomes T2ce 2dw2 ce 2ωeπμ2. But ce 2ω =Ae 2η. The proposition is then a consequence of

|μ|T2ce 2dw2 = |μ| min λΛW |(λ,e 2)| = min λΛW |(λ,x)|,

which follows from the fact that the map W given by w(w,e2) induces an isomorphism T2C e2 (min λΛW|(λ,e 2)|).

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

cψ̃ 0,1(n) = Lk(Cn,c)e2πn = (A n c)e2πn.

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 Fn. Let ηc 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.

Anηc = An c = Lk(Cn,c). (6.5)

We then have

Lemma 6.13.

cψ̃ 0,1(n) = Anηc e2πn.

Proof. To prove the Lemma we compute e(P)ηc ψ̃ 0,1(n) =e(P)ψ̃ 0,1(n) ηc in two different ways. First we apply Proposition 6.8 with η = ηc. We deduce

e(P)ηc ψ̃ 0,1(n) = Anηc e2πn.

Next choose a tubular neighborhood V n of the fibers Fn such that e(P) V n contains N(c). Then ψ̃ 0,1(n) is smooth on e(P) V n supp(ηc). Also, since ηc is the extension of a Thom class by zero, the restriction of ηc to e(P) V n represents the Poincaré dual PD(c) of the absolute cycle c in e(P) V n. The lemma now follows from

e(P)ψ̃ 0,1(n)ηc =e(P)V nψ̃ 0,1(n)ηc =e(P)V nψ̃ 0,1(n)PD(c) =cψ̃ 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 Fn.

It remains to treat case (ii). Thus we now assume that c = cy which is contained in a fiber Fx containing a component of Cn. We first prove

Lemma 6.14.

cψ̃0,1(x) =c(𝜖)ψ̃0,1(x).

Proof. We can take x = μe2 and hence c is contained in the fiber over the image of e3 W. Hence, by Proposition 4.1, c is the circle in the torus fiber at s(x) = 0 in the e3-direction, i.e., parallel to the image of (0, e3) in e(P). We note that by (6.4) even though ψ̃0,1(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 γΓMγψ̃ 0,1(x) is closed on the cylinder [0,𝜖] × c, and its integrals over the circles s × c all coincide. But 𝜀 × c = c(𝜖). The lemma follows.

Summing over x and using case (i) we obtain

cψ̃0,1(n) =c(𝜖)ψ̃0,1(n) = Lk(C n,c(𝜖)),

since c(𝜖) is disjoint from all the components of Fn. Thus it suffices to prove

Lk(Cn,c) = Lk(Cn,c(𝜖)). (6.6)

To this end suppose that c Fx Fn and c1,,ck are the components of Cn contained in Fx. Hence c and ci, 1 i k, are all parallel. Since the fibers containing all other components of Cn are disjoint from c, (6.6) will follow from

Lk(ci,c) = Lk(ci,c(𝜖)), 1 i k.

If ci = c then the previous equation is the definition of Lk(c,c). Thus we may assume ci is parallel to and disjoint from c. In this case their linking number is already topologically defined. But since c is disjoint from ci the circles c and c(𝜀) are homologous in the complement of ci (by the product homology c × [0,𝜀]) and since the linking number with ci is a homological invariant of the complement of ci in e(P) we have Lk(ci,c(𝜀)) = Lk(ci,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 Cnc gives rise to a modular form, extending Theorem 7.1 to a lift of the full cohomology H2(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 φ2.

We define the theta series

𝜃φ2(τ,) = xφ2(x,τ,z).

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

φ2(n) = nn,x0φ2(x).

Clearly, 𝜃φ2(τ,) and φ2(n) descend to closed differential 2-forms on X. Furthermore, 𝜃φ2(τ,) is a non-holomorphic modular form in τ of weight 2 for the principal congruence subgroup Γ(N). In fact, for = L as in Example 2.1, 𝜃φ2(τ,) transforms like a form for Γ0(d) of nebentypus.

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

[𝜃φ2(τ)] = 1 2πδh0[ω] + n>0 PD[Cn]qn H2(X, ) M 2(Γ(N)).

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

Λ(η,τ) :=Xη 𝜃φ2(τ,) = 1 2πδh0Xη ω + n>0 Cnηqn.

Here δh0 is Kronecker delta, and ω is the Kähler form on D normalized such that its restriction to the base point is given by ω13 ω14 + ω23 ω24. We obtain a map

Λ : Hc2(X, ) 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) SL 2(). Alternatively, for C an absolute 2-cycle in X defining a class in H2(X, ), the lift Λ(C,τ) is given by (1.1) with C0 the class given by 1 2πδh0[ω].

The key fact for the proof of the Fourier expansion is that for n > 0, the form φ2(n) is a Poincaré dual form of Cn, while φ2(n) is exact for n 0, see also Section 8.

7.2. The restrictions of the global theta functions.

Theorem 7.2.

The differential forms 𝜃φ2(V ) and 𝜃ψ1(V ) on X extend to the Borel-Serre compactification X¯. More precisely, for the restriction iP to the boundary face e(P) of X¯, we have

iP 𝜃 φ2(V ) = 𝜃φ1,1P ( WP)andiP 𝜃 ψ1(V ) = 𝜃ψ0,1P ( WP).

Proof. The restriction of 𝜃φ2(V ) is the theme (in much greater generality) of [13]. For 𝜃ψ1(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 𝜃φ2(V ) to the boundary of X¯ is exact and

iP 𝜃 φ2(V ) = d 𝜃ϕ0,1P ( WP) .

We also have a crucial restriction result for the singular form ψ̃0,1. However, one needs to be careful in forming the naive theta series associated to ψ̃0,1 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 ψ̃2,0(n) in the same way as for φ2(n) by summing over all non-zero x V of length n . This gives a 1-form on X which for n > 0 has singularities along the locally finite cycle Cn. Similarly, we define

ψ̃0,1P (n) = xW P,(x,x)=2n ψ̃0,1P (x),

which descends to a 1-form on e(P) with singularities. We also define ψ̃ 0,1(n) and ϕ0,1P (n) in the same way. We have

Proposition 7.4. The restriction of the 1-form ψ̃1(n) to e(P) is given by

iP ψ̃ 1(n) = ψ̃0,1P (n).

Proof. We assume that P is the stabilizer of the isotropic line = u. For x = au + xW + bu, we have for the majorant at z = (w,t,s) the formula

(x,x)z = 1 t2(a (xW ,w) bq(w))2 + (x w + bw,xw + bw)s + b2t2.

Here (,)s is the majorant associated to W. Hence by (5.4) and (5.2) we see that the sum of all x V with b0 in ψ̃1(n) is uniformly rapidly decreasing as t . Now fix an element xW W . Then xW + (a + h)u V for all a for some h ; in fact all elements in V u are of this form. We consider aψ̃1(xW + (a + h)u,z) as t . 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 and obtain

aψ̃1(xW + au,z) = k 1P(x,t,r)e2πx32r+t2k2rdr r e2πikheπ(xW,xW),


P(x,t,r) = x2x3r 2 dw2 + 1 22 1 2π t2k2 r dw3 ix3k 2 dt + ix2kt 2 ds.

Now the sum over all k0 is rapidly decreasing while for k = 0 we obtain ψ̃0,1(xW ). If xW = 0, i.e., for n = 0 one needs to argue slightly differently. Then we have

a0ψ̃1(au,z) = 1 22π a0eπa2t2 dw3 t = 1 22π keπt2k2 dw3 1 22π dw3 t ,

which goes to 1 22πdw3 = ψ̃0,1(0). This proves the proposition.

7.3. Main result.

In the previous sections, we constructed a closed 2-form 𝜃φ2 on X¯ such that the restriction of 𝜃φ2 to the boundary X¯ was exact with primitive [P̲]𝜃ϕ0,1P . From now on we usually write φ for φ2 and ϕ for ϕ0,1 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 (𝜃φ2(V ), [P]𝜃ϕ0,1P ( WP)) is a 2-cocycle in C.

We write for short (𝜃φ,𝜃ϕ). We obtain a class [[𝜃φ,𝜃ϕ]] in H2(C) and hence a class [𝜃φ,𝜃ϕ] in Hc2(X). The pairing with [𝜃φ,𝜃ϕ] then defines a lift Λc on differential 2-forms on X¯, which factors through H2(X¯) = H2(X). By Lemma 3.5 it is given by

Λc(η,τ) =X¯η 𝜃φ2 [P]e(P)iη 𝜃 ϕ0,1P .

Theorem 7.6. The class [[𝜃φ,𝜃ϕ]] is holomorphic, that is,

L 𝜃φ,𝜃ϕ = d(𝜃ψ1, 0).

Hence [𝜃φ,𝜃ϕ] is a holomorphic modular form with values in the compactly supported cohomology of X, so that the lift Λc takes values in the holomorphic modular forms.

Proof. By Theorem 7.2 and Theorem 6.2 we calculate

d(𝜃ψ1, 0) = (d𝜃ψ1,i𝜃 ψ1) = L𝜃φ2, [P]𝜃ψ0,1P = L 𝜃 φ2, [P]𝜃ϕ0,1P .

It remains to compute the Fourier expansion in τ of [𝜃φ,𝜃ϕ](τ). We will carry this out in Section 8.

Theorem 7.7. We have

[𝜃φ,𝜃ϕ](τ) = 1 2πδh0[ω] + n>0 PD[Cnc]qn H c2(X, ) M 2(Γ(N)).

That is, for any closed 2-form η on X¯

Λc(η,τ) = 1 2πδh0Xη ω + n>0 Cncηqn,

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

Λc : H2(X) 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) SL 2(). Alternatively, for C any relative 2-cycle in X defining a class in H2(X¯,X¯, ), we have

Λc(C,τ) = 1 2πδh0 vol(C) + n>0(Cnc C)qn 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 [𝜃φ,𝜃ϕ] as a class in H2(X̃) via the map j# : Hc2(X) H2(X̃). We recover the Hirzebruch-Zagier-Theorem.

Theorem 7.9. We have

j#[𝜃φ,𝜃ϕ](τ) = 1 2πδh0[ω] + n>0[Tnc]qn H2(X̃, ) M 2(Γ(N)).

In particular,

1 2πδh0 vol(Tm) + n>0(Tnc T m)X̃qn 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 Tm Tnc.


This follows from Theorem 7.7 since jCnc = T nc (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(jC). To see this we have only to replace ω by a cohomologous ‘Thom representative’ of PD(C), namely a closed form ω̃ supported in a tubular neighborhood N(C) of C in X such that the integral of ω̃ 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 ω̃ represents the Poincaré dual of C in any manifold M containing N(C), in particular for M = X̃.

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 H2(X) (Theorem 7.1) factors through the quotient of H2(X) by H2(X) since the restriction of 𝜃φ2 is exact (Theorem 7.3). But by Proposition 3.3 we have jH2(X) H2(X)H2(X), and the Hirzebruch-Zagier result exactly stipulates the modularity of the lift of classes in jH2(X). However, in that way one misses the remarkable extra structure coming from 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 Cy. By Theorem 7.7 and Lemma 3.5 we see

Λc(C y,τ,V ) = 1 2πδh0 vol(Cy) + n>0(Cnc C y)qn (7.3)  =Cy𝜃φ2(τ,V ) [P](Cy)P𝜃ϕ0,1P (τ, WP).

The two terms on the right, the integrals over Cy and Cy, are both non-holomorphic modular forms (see below) whose difference is holomorphic (by Theorem 7.6). So the generating series series of (Cnc 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

(Cn Cy)X = (Cn Cy)tr + vol(C n Cy),

the sum of the transversal intersections and the volume of the 1-dimensional (complex) intersection of Cn and Cy which occur if one of the components of Cn is equal to Cy.

Theorem 7.11. We have

Cy𝜃φ2(τ,V ) = 1 2πδh0 vol(Cy)+ n=1(C nCy)Xqn+ n [P](Cy)Pψ̃0,1P (n)(τ).

So the Fourier coefficients of the holomorphic part of the non-holomorphic modular form Cy𝜃φ2 are the interior intersection numbers of the cycles Cy and Cn.

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 φ2 and ψ̃1 (see Proposition 5.2 and Section 8) and the restriction formula for ψ̃1(n) (Theorem 7.4).

By slight abuse of notation we write Lk(Cn,Cy) = [P] Lk((Cn)P , (Cy)P ) for the total linking number of Cn and Cy. Then by Theorem 6.3 we obtain

Theorem 7.12.

[P](Cy)P𝜃ϕ0,1P (τ, WP) = n>0 Lk(Cn,Cy)qn+ n [P](Cy)Pψ̃0,1P (n)(τ).

So the Fourier coefficients of the holomorphic part of (Cy)P𝜃ϕP (τ, WP) are the linking numbers of the cycles Cy and Cn at the boundary component e(P).

Remark 7.13. There is also another “global” proof for Theorem 7.12. The cycle Cy intersects e(P) transversally (when pushed inside) and hence also the cap An. From this it is not hard to see that we can split the intersection number Cnc C y as

Cnc C y = (Cn Cy)X Lk(Cn,Cy).

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 Tnc T m. They split the intersection number into the interior part (Tn Tm)X and a ‘boundary contribution’ (Tn Tm) given by

(Tn Tm) = (Tn Tm)X̃X (Tm Tmc) (T n Tnc).

Now by Theorem 7.9 and its proof we have

Tnc T m = Cnc C m.

We have (per definition) (Tn Tm)X = (Cn Cm)X, so Theorem 7.11 gives the generating series for (Tn Tm)X. Note that Theorem 5.4 in [9] also compares the explicit formulas in [16] for (Tn Tm)X with the ones obtained via Cy𝜃φ2(τ,V ). All this implies

(Tn Tm) = Lk(Cn Cm).

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 Λc. 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 Cnc.

The key step for the entire Kudla-Millson theory is that for n > 0 the form φ2(n) is a Poincaré dual form for the cycle Cn, i.e.,

Theorem 8.1 ([2223]). Let η be a closed rapidly decreasing 2-form. Then

Xη φ2(n) = Cnηe2πn.

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 X¯, their approach is not applicable in our case. Instead, we use a differential character argument for φ2 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 ψ̃1(n) is a differential character in the sense of Cheeger-Simons for the cycle Cn. More precisely, ψ̃1(n) is a locally integrable 1-form on X, and for any compactly supported 2-form η we have

Xη φ2(n) = Cnηe2πn Xdη ψ̃1(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 ψ̃1. The form ψ̃ there is indeed the form ψ̃1 of this paper.

Remark 8.3. The form ψ̃1 is closely related to Kudla’s Green function ξ [1819] (more generally for O(p, 2)) which is given by

ξ(x) = 1φ 00(rx)dr r eπ(x,x).

Then Ξ(n) = xnξ(x) gives rise to a Green’s function for the divisor Cn and moreover ddcξ = φ 2. Here dc = 1 4πi( ¯). This suggests dcξ = ψ̃ 1, which indeed follows from dcφ 0 = ψ1, see [5], Remark 4.5.

For n we define

φ2c(n) := φ 2(n) [P]d(fπϕ 0,1P (n))

and follow the current approach to show that for n > 0 the form φ2c(n) is a Poincaré dual form for the cycle Cnc. Here we follow the notation of subsection 3.3. That is, πϕ 0,1P (n) is the pullback to a product neighborhood V of X¯, and f is a smooth function on V of the geodesic flow coordinate t which is 1 near t = and zero else. Note that φ2c(n) is exactly the n-th Fourier coefficient of the mapping cone element [𝜃φ,𝜃ϕ], when realized as a rapidly decreasing form on X. We also define

ψ̃1c(n) = ψ̃ 1(n) fπϕ 0,1P (n).

We call a differential form η on X¯ special if in a neighborhood of each boundary component e(P) it is the pullback of a form ηP on e(P) under the geodesic retraction and if the pullback of the form ηP 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 X¯. Note that the proof of Theorem 7.2 shows that 𝜃φ2 is ‘almost’ special; it only differs from a special form by a rapidly decreasing form.

Theorem 8.4. Let n > 0. The form ψ̃1c(n) is a differential character for the cycle Cnc. More precisely, ψ̃1c(n) is a locally integrable 1-form on X and satisfies the following current equation on special 2 forms on X¯:

d[ψ̃1c(n)] + δ Cne2πn = [φ 2c(n)].

That is, for any special 2-form η on X¯ we have

Xη φ2c(n) = Cncηe2πn Xdη ψ̃2c(n).

This implies Theorem 7.7 for the positive Fourier coefficients. For n 0, the form φ2c(n) is exact with primitive ψ̃2c(n) which by Theorem 7.4 is decaying. So Theorem 8.4 holds also for n 0 with Cnc = . 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 Tnc in X̃. Extensive discussions with Kühn suggest that (if X has only one cusp)

Ξ(n)t xW (x,x)=2n fπ(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 ρT be a family of smooth functions on a standard fundamental domain of Γ in D only depending on t which is 1 for t T and 0 for T + 1. We then have

Xη φ2c(n) = lim TXρT η φ2(n) d(fπϕ 0,1(n)) .

We apply Theorem 8.2 for the compactly supported form ρT η and obtain

Xη φ2c(n) = lim T[ CnρT ηe2πn Xd(ρT η) ψ̃1(n) (8.1)  Xd ρT η (fπϕ 0,1(n)) d(ρT η) fπϕ 0,1(n)]

The first term on the right hand side of (8.1) goes to Cnηe2πn as T , while the third vanishes for any T by Stokes’ theorem. For the two remaining terms of (8.1) we first note d(ρT η) = ρT (t)dt η + ρ T dη and ρT (t) = 0 outside [T,T + 1]. We obtain for these two terms

X(dη) ψ̃1(n) fπϕ 0,1(n) lim TT T+1e(P)ρT (t)dt η ψ̃ 1(n) fπϕ 0,1(n) .(8.2)

It remains to compute the second term in the previous equation. For T sufficiently large we have f 1. Furthermore by Theorem 7.4 and its proof we have ψ̃1(n) = πψ̃ 0,1(n) + O(eCt). As ϕ0,1(n) = ψ̃0,1(n) + ψ̃0,1(n), we can replace ψ̃1(n) fπϕ 0,1(n) by πψ̃ 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 TT T+1ρ T (t)dte(P)ηπψ̃ 0,1(n) = e(P)ηψ̃ 0,1(n) = Anηe2πn.

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 Cnc = C n (An) 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

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