Correspondence of Hermitian modular forms to cycles associated to \(\mathrm{SU}(p,2)\).

*(English)*Zbl 0559.10027This paper is one of a series of papers by the authors concerning a fascinating relation between certain families of cycles, defined by totally geodesic submanifolds, in one of a certain class of locally symmetric spaces, and certain automorphic forms, defind by theta-series, on a related symmetric space. Previous papers of the authors on the same subject are [Math. Ann. 258, 289–318 (1982; Zbl 0466.58004)] and [Invent. Math. 71, 467–499 (1983; Zbl 0506.10024)]. The original example of this phenomenon was exposed in [F. Hirzebruch and D. Zagier, Invent. Math. 36, 57–113 (1976; Zbl 0332.14009)], and besides the work of the authors, it has been most notably studied by S. S. Kudla [Invent. Math. 47, 189–208 (1978; Zbl 0399.10030)], and by S. S. Kudla and J. J. Millson [Compos. Math. 45, 207–271 (1982; Zbl 0495.10016)].

The phenomenon is as follows. Consider the indefinite unitary group \(U(p,q)=G\). Its symmetric space \(X=U(p,q)/(U(p)\times U(q))\) can be identified with the set of positive \(p\)-dimensional subspaces of \(\mathbb C^{p+q}\) equipped with the standard Hermitian form of signature \((p,q)\). Let \(U\subseteq\mathbb C^{p+q}\) be a positive \(r\)-dimensional space, and let \(X_ U\subseteq X\) be the subset of \(X\) consisting of \(p\)-planes which contain \(U\). Then \(X_ U\) is a totally geodesic submanifold of \(X\) of codimension \(rq\). If \(G_ U\subseteq G\) is the stabilizer of \(V\), then \(G_ U\simeq U(r)\times U(p-r,q)\), and \(X_ U\) is identifiable to the symmetric space of \(G_ U\).

Suppose \(\Gamma\subseteq G\) is a torsion-free discrete group. Assume for simplicity that \(U(p,q)/\Gamma\) is compact. If \(G_ U/(\Gamma \cup G_ U)\) is also compact then \(X_ U\) projects to a smooth submanifold \(C_ U\) in the quotient space \(X/\Gamma\), and so defines a codimension \(rq\) cycle in \(X/\Gamma\). These cycles \(C_ U\) are the cycles involved in the story.

Now let \(k_ 1\) be a totally real number field and let \(k\) be an imaginary quadratic extension of \(k_ 1\). Let \(Y_ k\) be a vector space over \(k\) equipped with a Hermitian form \((\;,\;)\). Assume the form \((\;,\;)\) is anisotropic over \(k\). Let \(v\) be a real place of \(k_ 1\), and \((k_ 1)_ v\), \(k_ v\) the completions of \(k_ 1\) and \(k\) at \(v\) so that \((k_ 1)_ v\simeq\mathbb R\) and \(k_ v\simeq\mathbb C\). Let \(Y_ v\simeq Y_ k\otimes_ k k_ v\) be the completion of \(Y_ k\) at \(v\). The form \((\;,\;)\) extends to a \(k_ v\)-Hermitian form, denoted \((\;,\;)_ v\) on \(Y_ v\). Let \(G_ k\) be the isometry group of \((\;,\;)\) on \(Y_ k\), and let \(G_ v\) be the isometry group of \((\;,\;)_ v\) on \(Y_ v\). Set \(Y_{\mathbb R}=Y=\prod_{v}Y_ v\) and \(G_{\mathbb R}=G=\prod_{v}G_ v\), the product being taken over all real places. Each \(G_ v\) is a unitary group, and so \(G\) is a product of unitary groups \(U(p_ v,q_ v)\). The symmetric space \(X\) of \(G\) will be a product of the symmetric groups \(X_ v\) of the factors: \(X=\prod_{v}X_ v\). If all but one of the forms \((\;,\;)_ v\) are positive definite, then their symmetric spaces reduce to points, so X is essentially the symmetric space of a single \(U(p,q)\); but if more than one of the \((\;,\;)_ v\) are indefinite, the opening remarks above concerning the construction of cycles are easily modified to cover a product of symmetric spaces.

Let \(L\subseteq Y_ k\) be a lattice such that the pairing \((\;,\;)\) restricted to \(L\) takes on integer values. Let \(\Gamma \subseteq G_ k\) be the stabilizer of \(L\). Then \(\Gamma\) may be regarded as a subgroup of \(G\), and G/\(\Gamma\) is compact. Let \(U_ k\subseteq Y_ k\) be a subspace which is totally positive, in the sense the \(U_ v\) is a positive subspace of \(Y_ v\) for all \(v\). Then the hypotheses necessary to form the cycle \(C_ U\) are satisfied. The homology class represented by \(C_ U\) gives rise to a cohomology class by Poincaré duality, and since \(X\) carries a canonical metric, there is by Hodge theory a well defined harmonic differential form, of degree equal to the codimension of \(C_ U\), representing the Poincaré dual class of \(C_ U\) on X/\(\Gamma\). Denote this form \(C^*_ U\).

There is a very different way to construct differential forms on X/\(\Gamma\). Consider the space \({\mathcal S}(Y^ r)\) of Schwartz functions on \(Y^ r\), and further, the space \({\mathcal S}(Y^ r)\otimes \Omega^.(X)\) of smooth differential forms on X with values in \({\mathcal S}(Y^ r)\). The subspace (\({\mathcal S}(Y^ r)\otimes \Omega^.(X))^ G\) of G-invariant forms may be identified with (\({\mathcal S}(Y^ r)\otimes \Lambda^.({\mathfrak p}^*))^ K\) where \({\mathfrak g}=k\oplus {\mathfrak p}\) is the Cartan decomposition of the Lie algebra \({\mathfrak g}\) of G, and \(\Lambda^.({\mathfrak p}^*)\) is the exterior algebra of the dual \({\mathfrak p}^*\) of \({\mathfrak p}\).

Let \(B=(\ell_ 1,...,\ell_ r)\in Y^ r\) be an r-tuple of points from the lattice \(L\). Let \({\mathcal O}\subseteq L^ r\), the \(r\)-fold copy of \(L\) be the \(\Gamma\)-orbit of \[ B: {\mathcal O}=\{\gamma B=(\gamma \ell_ 1,...,\gamma \ell_ r): \gamma \in \Gamma \}. \] Then if \(\phi\) belongs to \((\mathcal S(Y^ r)\otimes \Lambda^.({\mathfrak p}^*))^ K\), the sum \[ \theta_{\phi,{\mathcal O}}(g)=\sum_{\tilde B\in {\mathcal O}}\phi (g\tilde B),\quad g\in G, \] defines a differential form on \(X/\Gamma\). Further, if \(\phi\) is closed as an \(\mathcal S(Y^ r)\)-valued differential form on \(X\), the form \(\theta_{\phi,{\mathcal O}}\) will be closed and so will define a cohomology class on \(X/\Gamma\).

It turns out that there is a very close relationship between the forms \(C^*_ U\) dual to cycles and the form \(\theta_{\phi,{\mathcal O}}\) constructed as above by averaging. More precisely, appropriate finite linear combinations of the cohomology classes defined by the \(C^*_ U\) are equal to appropriate finite linear combinations of the classes defined by the \(\theta_{\phi,{\mathcal O}}\), for a fixed, properly chosen \(\phi\). This is the central phenomenon studied by this paper and the other papers cited above, as well as others uncited. Unfortunately a precise description of the linear combinations involved would greatly lengthen this review. The construction of \(\phi\) is also a considerably involved matter.

The differential forms \(C^*_ U\) and \(\theta_{\phi,{\mathcal O}}\) are of course examples of automorphic forms on \(G/\Gamma\). A further important aspect of the situation under consideration is a relation between these automorphic forms on \(G\) and certain automorphic forms on a group \(G'\) which is the real points of the isometry group \(G'_ k\) of the split Hermitian form of dimension \(2r\) over \(k\).

Indeed following A. Weil [Acta Math. 111, 143–211 (1964; Zbl 0203.03305)] we know there is defined on the space \({\mathcal S}(Y^ r)\) a representation of a two-fold \(^{\sim}\) of the group \(\mathrm{Sp}=\mathrm{Sp}_{2n}(\mathbb R)\) where \(n=\dim Y^ r\). This representation is such that the obvious diagonal action of \(G\) on \(Y^ r\) (as used above to define the orbit \(\mathcal O)\) embeds \(G\) as a subgroup of \(^{\sim}\). Furthermore \(G'\), or rather a 2-fold cover \(\tilde G'\) of it, forms the centralizer of \(G\) in \(^{\sim}\), and the pair \((G,G')\) of groups forms a dual pair [the reviewer, Proc. Symp. Pure Math. 33, part 1, 275–285 (1979; Zbl 0423.22016)] in \(\mathrm{Sp}\).

The \(r\)-fold copy \(L^ r\) of \(L\) defines a lattice in \(Y^ r\). Given \(\phi\in ({\mathcal S}(Y^ r)\otimes \Lambda^.({\mathfrak p}^*))^ K\), we form the sum \(\theta_{\phi,L^ r}(g,g')=\sum_{B\in L^ r}\omega (gg')(\phi)(B),\quad g\in G,\quad g'\in \tilde G',\) where \(\omega\) denotes the action of \(^{\sim}\) on \({\mathcal S}(Y^ r)\). Then evidently, for fixed \(g'\), as a function of \(G\), \(\theta_{\phi,L^ r}\) is a(n infinite) sum of the forms \(\theta_{\phi,{\mathcal O}}\) constructed above. From the general theory of the representation \(\omega\), we further know that \(\theta_{\phi,L^ r}\) will be an automorphic form on \(\tilde G'\), with respect to an appropriate arithmetic subgroup \(\Gamma'\subseteq G'_ k\subseteq G'\).

The group \(G'_ k\simeq U_{r,r}(k)\) is split. Let \(P'_ k\subseteq G'_ k\) denote the maximal parabolic subgroup stabilizing some isotropic subspace of dimension r, and let \(N'_ k\) be the unipotent radical of \(P'_ k\). Then \(N'_ k\) is abelian, isomorphic to the space of \(r\times r\) Hermitian matrices over \(k\). We denote by \(N'\) the closure of \(N'_ k\) in \(G'\). A standard method for studying automorphic forms on \(G'/\Gamma'\) is to expand them in a Fourier series with respect to \(N'\). Thus we may consider the Fourier coefficients \(\theta_{\phi,\chi}\) of \(\theta_{\phi,L^ r}\) for \(\chi\in (N'/N'\cap \Gamma '){\hat{\;}}\). This group is a lattice in \(\hat N,\) and may be identified with a certain class of \(r\times r\) Hermitian matrices over \(k\). At least when the Hermitian matrix associated to \(\chi\) is nondegenerate, the Fourier coefficient \(\theta_{\phi,\chi}\), as a function of \(g\in G\), will by standard finiteness results of adélic geometry be a finite sum of the \(\theta_{\phi,{\mathcal O}}\), and will define a class which is a finite linear combination of the \(C^*_ U\).

This fact has the following geometric interpretation. For each \(\chi\), let \(C_{\chi}\) be the cycle dual to \(\theta_{\phi,\chi}\). Let \(E\subseteq X/\Gamma\) be any submanifold of dimension complementary to the \(C_ U\). Then one can consider the intersection numbers \(i(E,C_ U)\). The story told above implies that the sums \(i(E,C_{\chi})\) of the \(i(E,C_ U)\) define the Fourier coefficients of an automorphic form on \(\tilde G'/\Gamma'\). This was how the seminal results of Hirzebruch-Zagier were formulated.

A story similar to that just told for the unitary group \(U=G\) can be told when \(G\) is an orthogonal group, or the isometry group of a Hermitian form over a totally definite quaternion algebra. When \(G\) is an orthogonal group, the group \(G'\) is a symplectic group (of rank equal to the dimension of the subspaces used to define the cycles). When \(G\) is quaternionic, \(G'\) is the isometry group of a skew-Hermitian form for the same quaternion algebra.

This, then, is the general context of the paper under review. This paper treats the special case when the Hermitian forms \((\;,\;)_ v\) are all definite except for one place \(v\) of \(k\), at which \((\;,\;)_ v\) has signature \((p,2)\). Furthermore only cycles corresponding to positive definite lines \((r=1)\) are considered. The proof relies throughout on explicit computations. In particular the construction of the \(\phi\) used to form \(\theta_{\phi,L^ r}\) is quite ad hoc and would be difficult to generalize to higher codimension. In more recent, as yet unpublished work, the authors have developed a more conceptual approach to this problem. An alternative approach has also been developed by Kudla and Millson. The account given in this review is based on the formulation of Kudla and Millson.

The phenomenon is as follows. Consider the indefinite unitary group \(U(p,q)=G\). Its symmetric space \(X=U(p,q)/(U(p)\times U(q))\) can be identified with the set of positive \(p\)-dimensional subspaces of \(\mathbb C^{p+q}\) equipped with the standard Hermitian form of signature \((p,q)\). Let \(U\subseteq\mathbb C^{p+q}\) be a positive \(r\)-dimensional space, and let \(X_ U\subseteq X\) be the subset of \(X\) consisting of \(p\)-planes which contain \(U\). Then \(X_ U\) is a totally geodesic submanifold of \(X\) of codimension \(rq\). If \(G_ U\subseteq G\) is the stabilizer of \(V\), then \(G_ U\simeq U(r)\times U(p-r,q)\), and \(X_ U\) is identifiable to the symmetric space of \(G_ U\).

Suppose \(\Gamma\subseteq G\) is a torsion-free discrete group. Assume for simplicity that \(U(p,q)/\Gamma\) is compact. If \(G_ U/(\Gamma \cup G_ U)\) is also compact then \(X_ U\) projects to a smooth submanifold \(C_ U\) in the quotient space \(X/\Gamma\), and so defines a codimension \(rq\) cycle in \(X/\Gamma\). These cycles \(C_ U\) are the cycles involved in the story.

Now let \(k_ 1\) be a totally real number field and let \(k\) be an imaginary quadratic extension of \(k_ 1\). Let \(Y_ k\) be a vector space over \(k\) equipped with a Hermitian form \((\;,\;)\). Assume the form \((\;,\;)\) is anisotropic over \(k\). Let \(v\) be a real place of \(k_ 1\), and \((k_ 1)_ v\), \(k_ v\) the completions of \(k_ 1\) and \(k\) at \(v\) so that \((k_ 1)_ v\simeq\mathbb R\) and \(k_ v\simeq\mathbb C\). Let \(Y_ v\simeq Y_ k\otimes_ k k_ v\) be the completion of \(Y_ k\) at \(v\). The form \((\;,\;)\) extends to a \(k_ v\)-Hermitian form, denoted \((\;,\;)_ v\) on \(Y_ v\). Let \(G_ k\) be the isometry group of \((\;,\;)\) on \(Y_ k\), and let \(G_ v\) be the isometry group of \((\;,\;)_ v\) on \(Y_ v\). Set \(Y_{\mathbb R}=Y=\prod_{v}Y_ v\) and \(G_{\mathbb R}=G=\prod_{v}G_ v\), the product being taken over all real places. Each \(G_ v\) is a unitary group, and so \(G\) is a product of unitary groups \(U(p_ v,q_ v)\). The symmetric space \(X\) of \(G\) will be a product of the symmetric groups \(X_ v\) of the factors: \(X=\prod_{v}X_ v\). If all but one of the forms \((\;,\;)_ v\) are positive definite, then their symmetric spaces reduce to points, so X is essentially the symmetric space of a single \(U(p,q)\); but if more than one of the \((\;,\;)_ v\) are indefinite, the opening remarks above concerning the construction of cycles are easily modified to cover a product of symmetric spaces.

Let \(L\subseteq Y_ k\) be a lattice such that the pairing \((\;,\;)\) restricted to \(L\) takes on integer values. Let \(\Gamma \subseteq G_ k\) be the stabilizer of \(L\). Then \(\Gamma\) may be regarded as a subgroup of \(G\), and G/\(\Gamma\) is compact. Let \(U_ k\subseteq Y_ k\) be a subspace which is totally positive, in the sense the \(U_ v\) is a positive subspace of \(Y_ v\) for all \(v\). Then the hypotheses necessary to form the cycle \(C_ U\) are satisfied. The homology class represented by \(C_ U\) gives rise to a cohomology class by Poincaré duality, and since \(X\) carries a canonical metric, there is by Hodge theory a well defined harmonic differential form, of degree equal to the codimension of \(C_ U\), representing the Poincaré dual class of \(C_ U\) on X/\(\Gamma\). Denote this form \(C^*_ U\).

There is a very different way to construct differential forms on X/\(\Gamma\). Consider the space \({\mathcal S}(Y^ r)\) of Schwartz functions on \(Y^ r\), and further, the space \({\mathcal S}(Y^ r)\otimes \Omega^.(X)\) of smooth differential forms on X with values in \({\mathcal S}(Y^ r)\). The subspace (\({\mathcal S}(Y^ r)\otimes \Omega^.(X))^ G\) of G-invariant forms may be identified with (\({\mathcal S}(Y^ r)\otimes \Lambda^.({\mathfrak p}^*))^ K\) where \({\mathfrak g}=k\oplus {\mathfrak p}\) is the Cartan decomposition of the Lie algebra \({\mathfrak g}\) of G, and \(\Lambda^.({\mathfrak p}^*)\) is the exterior algebra of the dual \({\mathfrak p}^*\) of \({\mathfrak p}\).

Let \(B=(\ell_ 1,...,\ell_ r)\in Y^ r\) be an r-tuple of points from the lattice \(L\). Let \({\mathcal O}\subseteq L^ r\), the \(r\)-fold copy of \(L\) be the \(\Gamma\)-orbit of \[ B: {\mathcal O}=\{\gamma B=(\gamma \ell_ 1,...,\gamma \ell_ r): \gamma \in \Gamma \}. \] Then if \(\phi\) belongs to \((\mathcal S(Y^ r)\otimes \Lambda^.({\mathfrak p}^*))^ K\), the sum \[ \theta_{\phi,{\mathcal O}}(g)=\sum_{\tilde B\in {\mathcal O}}\phi (g\tilde B),\quad g\in G, \] defines a differential form on \(X/\Gamma\). Further, if \(\phi\) is closed as an \(\mathcal S(Y^ r)\)-valued differential form on \(X\), the form \(\theta_{\phi,{\mathcal O}}\) will be closed and so will define a cohomology class on \(X/\Gamma\).

It turns out that there is a very close relationship between the forms \(C^*_ U\) dual to cycles and the form \(\theta_{\phi,{\mathcal O}}\) constructed as above by averaging. More precisely, appropriate finite linear combinations of the cohomology classes defined by the \(C^*_ U\) are equal to appropriate finite linear combinations of the classes defined by the \(\theta_{\phi,{\mathcal O}}\), for a fixed, properly chosen \(\phi\). This is the central phenomenon studied by this paper and the other papers cited above, as well as others uncited. Unfortunately a precise description of the linear combinations involved would greatly lengthen this review. The construction of \(\phi\) is also a considerably involved matter.

The differential forms \(C^*_ U\) and \(\theta_{\phi,{\mathcal O}}\) are of course examples of automorphic forms on \(G/\Gamma\). A further important aspect of the situation under consideration is a relation between these automorphic forms on \(G\) and certain automorphic forms on a group \(G'\) which is the real points of the isometry group \(G'_ k\) of the split Hermitian form of dimension \(2r\) over \(k\).

Indeed following A. Weil [Acta Math. 111, 143–211 (1964; Zbl 0203.03305)] we know there is defined on the space \({\mathcal S}(Y^ r)\) a representation of a two-fold \(^{\sim}\) of the group \(\mathrm{Sp}=\mathrm{Sp}_{2n}(\mathbb R)\) where \(n=\dim Y^ r\). This representation is such that the obvious diagonal action of \(G\) on \(Y^ r\) (as used above to define the orbit \(\mathcal O)\) embeds \(G\) as a subgroup of \(^{\sim}\). Furthermore \(G'\), or rather a 2-fold cover \(\tilde G'\) of it, forms the centralizer of \(G\) in \(^{\sim}\), and the pair \((G,G')\) of groups forms a dual pair [the reviewer, Proc. Symp. Pure Math. 33, part 1, 275–285 (1979; Zbl 0423.22016)] in \(\mathrm{Sp}\).

The \(r\)-fold copy \(L^ r\) of \(L\) defines a lattice in \(Y^ r\). Given \(\phi\in ({\mathcal S}(Y^ r)\otimes \Lambda^.({\mathfrak p}^*))^ K\), we form the sum \(\theta_{\phi,L^ r}(g,g')=\sum_{B\in L^ r}\omega (gg')(\phi)(B),\quad g\in G,\quad g'\in \tilde G',\) where \(\omega\) denotes the action of \(^{\sim}\) on \({\mathcal S}(Y^ r)\). Then evidently, for fixed \(g'\), as a function of \(G\), \(\theta_{\phi,L^ r}\) is a(n infinite) sum of the forms \(\theta_{\phi,{\mathcal O}}\) constructed above. From the general theory of the representation \(\omega\), we further know that \(\theta_{\phi,L^ r}\) will be an automorphic form on \(\tilde G'\), with respect to an appropriate arithmetic subgroup \(\Gamma'\subseteq G'_ k\subseteq G'\).

The group \(G'_ k\simeq U_{r,r}(k)\) is split. Let \(P'_ k\subseteq G'_ k\) denote the maximal parabolic subgroup stabilizing some isotropic subspace of dimension r, and let \(N'_ k\) be the unipotent radical of \(P'_ k\). Then \(N'_ k\) is abelian, isomorphic to the space of \(r\times r\) Hermitian matrices over \(k\). We denote by \(N'\) the closure of \(N'_ k\) in \(G'\). A standard method for studying automorphic forms on \(G'/\Gamma'\) is to expand them in a Fourier series with respect to \(N'\). Thus we may consider the Fourier coefficients \(\theta_{\phi,\chi}\) of \(\theta_{\phi,L^ r}\) for \(\chi\in (N'/N'\cap \Gamma '){\hat{\;}}\). This group is a lattice in \(\hat N,\) and may be identified with a certain class of \(r\times r\) Hermitian matrices over \(k\). At least when the Hermitian matrix associated to \(\chi\) is nondegenerate, the Fourier coefficient \(\theta_{\phi,\chi}\), as a function of \(g\in G\), will by standard finiteness results of adélic geometry be a finite sum of the \(\theta_{\phi,{\mathcal O}}\), and will define a class which is a finite linear combination of the \(C^*_ U\).

This fact has the following geometric interpretation. For each \(\chi\), let \(C_{\chi}\) be the cycle dual to \(\theta_{\phi,\chi}\). Let \(E\subseteq X/\Gamma\) be any submanifold of dimension complementary to the \(C_ U\). Then one can consider the intersection numbers \(i(E,C_ U)\). The story told above implies that the sums \(i(E,C_{\chi})\) of the \(i(E,C_ U)\) define the Fourier coefficients of an automorphic form on \(\tilde G'/\Gamma'\). This was how the seminal results of Hirzebruch-Zagier were formulated.

A story similar to that just told for the unitary group \(U=G\) can be told when \(G\) is an orthogonal group, or the isometry group of a Hermitian form over a totally definite quaternion algebra. When \(G\) is an orthogonal group, the group \(G'\) is a symplectic group (of rank equal to the dimension of the subspaces used to define the cycles). When \(G\) is quaternionic, \(G'\) is the isometry group of a skew-Hermitian form for the same quaternion algebra.

This, then, is the general context of the paper under review. This paper treats the special case when the Hermitian forms \((\;,\;)_ v\) are all definite except for one place \(v\) of \(k\), at which \((\;,\;)_ v\) has signature \((p,2)\). Furthermore only cycles corresponding to positive definite lines \((r=1)\) are considered. The proof relies throughout on explicit computations. In particular the construction of the \(\phi\) used to form \(\theta_{\phi,L^ r}\) is quite ad hoc and would be difficult to generalize to higher codimension. In more recent, as yet unpublished work, the authors have developed a more conceptual approach to this problem. An alternative approach has also been developed by Kudla and Millson. The account given in this review is based on the formulation of Kudla and Millson.

Reviewer: Roger Howe (New Haven)

##### MSC:

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

58A14 | Hodge theory in global analysis |

53C35 | Differential geometry of symmetric spaces |

32Q99 | Complex manifolds |

32C37 | Duality theorems for analytic spaces |

22E50 | Representations of Lie and linear algebraic groups over local fields |

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |