Intersections of Hirzebruch–Zagier divisors and CM cycles.

*(English)*Zbl 1238.11069
Lecture Notes in Mathematics 2041. Berlin: Springer (ISBN 978-3-642-23978-6/pbk; 978-3-642-23979-3/ebook). viii, 140 p. (2012).

The celebrated theorem of Gross-Zagier [B. Gross and D. Zagier, “Heegner points and derivatives of \(L\)-series”, Invent. Math. 84, 225–320 (1986; Zbl 0608.14019)] gives a formula relating certain linear combinations of Heegner points on integral models of the modular curve \(X_0(N)\) to Fourier coefficents of Eisenstein series that appear in the integral representation of an associated Rankin-Selberg \(L\)-function at its central value. A series of papers of S. S. Kudla, e.g. [“Central derivatives of Eisenstein series and height pairings”, Ann. Math. (2) 146, No. 3, 545–646 (1997; Zbl 0990.11032)] and [“Integrals of Borcherds forms”, Compos. Math. 137, No. 3, 293–349 (2003; Zbl 1046.11027)], propose to extend such a formula to the general setting of Shimura varieties and Eisenstein series on higher rank groups. The authors give some compelling evidence for a special case of this conjecture, namely in the setting of Hilbert modular surfaces, where a moduli interpretation is available. In particular, by developing some techniques in deformation theory (e.g. the theory of displays due to T. Zink [“The display of a formal \(p\)-divisible group”, Astérisque 278, 127–248 (2002; Zbl 1008.14008)]), the authors are able to perform the calculations required to prove some special cases of Kudla’s proposed conjecture, building on previous works of J. H. Bruinier and second author ([“CM values of Hilbert modular functions”, Invent. Math. 163, No. 2, 229–288 (2006; Zbl 1093.11041)], [T. Yang, “An arithmetic intersection formula on Hilbert modular surfaces”, Am. J. Math. 132, No. 5, 1275–1309 (2010; Zbl 1206.14049)], and [T. Yang, “Arithmetic intersection on a Hilbert modular surface and the Faltings height” (preprint)]).

Roughly, the authors prove the following type of result. Let \(F\) be a real quadratic field, with \(\sigma \in \text{Gal}(F/{\mathbb{Q}})\) the nontrivial automorphism of its Galois group over \({\mathbb{Q}}\). Associated to \(F\) is a Hilbert modular surface \(M\), which can be defined as the moduli space of all \(\mathcal{O}_F\)-polarized RM abelian surfaces \(\mathcal{A}\). Given a positive integer \(m\), Hirzebruch and Zagier constructed a divisor on \(M({\mathbb{C}})\), which was later given a moduli theoretic interpretation by Kudla-Rapoport [S. S. Kudla and M. Rapoport, “Arithmetic Hirzebruch-Zagier cycles”, J. Reine Angew. Math. 515, 155–244 (1999; Zbl 1048.11048)]. To describe this latter interpretation, let \(T(m)\) denote the moduli space of pairs \((\mathcal{A}, j)\), where \(\mathcal{A}\) is an \(\mathcal{O}_F\)-polarized RM abelian surface, and \(j\) is a Rosati fixed endomorphism satisfying \(j \circ j = m\) and \(j \circ x = x^{\sigma} \circ j\) for all \(x \in \mathcal{O}_F\). The “forget-\(j\)” morphism \(T(m) \rightarrow M\) has image given by a codimension one cycle whose complex fibre is equal to the divisor constructed by Hirzebruch and Zagier. The moduli space \(T(m)\) can then justifiably be referred to as a Hirzebrich-Zagier divisor. The main aim of the work is to relate the intersection multiplicities of three such divisors to the Fourier coefficients of the central derivative of a certain Siegel Eisenstein series of genus three.

To be somewhat more precise, the authors establish the following main result (Theorem E). Let \(E\) be a quartic CM field containing \(F\) as a real quadratic subfield. Let us suppose that \(E\) is either Galois over \({\mathbb{Q}}\), with Galois group \(\text{Gal}(E/{\mathbb{Q}}) \cong {\mathbb{Z}}/4{\mathbb{Z}}\), or else that \(E\) is not Galois over \({\mathbb{Q}}\). Given a CM type \(\Sigma\) of \(E\), let us write \(E_{\Sigma}\) to denote its reflex field, with \(\mathcal{O}_{\Sigma}\) the ring of integers of \(E_{\Sigma}\). One can associate to \(E\) a reflex algebra \(E^{\sharp}\) of \(E\), which is the \({\mathbb{Q}}\)-algebra defined up to isomorphism by the existence of an \(\text{Aut}({\mathbb{C}}/{\mathbb{Q}})\)-equivariant bijection \[ \{ \text{CM types of \(E\)}\} \longrightarrow \text{Hom}(E^{\sharp}, {\mathbb{C}}), ~~~ \Sigma \longmapsto \phi_{\Sigma} \] for which \(\phi_{\Sigma}(E^{\sharp}) = E_{\Sigma}\). Let \(F^{\sharp}\) denote the maximal totally real subalgebra of \(E^{\sharp}\). In the setting described above, \(E^{\sharp}\) is a certain quartic CM field, with \(F^{\sharp}\) its maximal totally real subfield. Anyhow, one can associate to the pair \((E, \Sigma)\) an algebraic stack \(CM_{\Sigma}\), defined as the moduli space of principally polarized abelian varieties over \(\mathcal{O}_{\Sigma}\)-schemes with CM by \(\mathcal{O}_{\Sigma}\), satisfying a certain technical condition (the so-called Kottwitz condition of \(\S 3.2\)). The forgetful morphism \(CM_{\Sigma} \longrightarrow M_{\mathcal{O}_{\Sigma}}\) contains a codimension two cycle on \(M_{\mathcal{O}_{\Sigma}}\). To a given pair \(T(m)\) and \(CM_{\Sigma}\), one can associate a finite intersection pairing \(\langle T(m): CM_{\Sigma} \rangle_{\text{fin}}\), which is essentially the sum of lengths of all local rings in the intersection \(T(m) \bigcap CM_{\Sigma}\) (\(\S 5.4\)). Following the constructions of Kudla [loc. cit.] and Bruinier [J. H. Bruinier, “Borcherds products and Chern classes of Hirzebruch-Zagier divisors”, Invent. Math. 138, No. 1, 51–83 (1999; Zbl 1011.11027)], one can also define for any integer \(m\) an associated Green function \(G(m, v, \cdot)\) on \(M({\mathbb{C}})\), where \(v \in {\mathbb{R}}\) is some fixed parameter. Taking the (finite) sum over all complex points of \(CM_{\Sigma}\) then gives rise to a function \(G(m, v, CM_{\Sigma})\) on \(M({\mathbb{C}})\). One can then define the intersection of the formal pair \(\widehat{T}(m, v) := (T(m), G(m, v, \cdot))\) and \(CM_{\Sigma}\) to be the sum \[ \langle \widehat{T}(m, v): CM_{\Sigma} \rangle_{\text{fin}} = \langle T(m): CM_{\Sigma} \rangle_{\text{fin}} + \frac{1}{2}G(m, v, CM_{\Sigma}). \] Now, turning to the automorphic side, the authors define an \(\mathcal{O}_F\)-polarized CM module \({\mathbb{T}}\) to be a projective \(\mathcal{O}_E\)-module of rank one equipped with a perfect \({\mathbb{Z}}\)-valued symplectic form (satisfying some compatibility condition with respect to the \(\mathcal{O}_E\)-action). To such a module \({\mathbb{T}}\), there is an associated CM-type \(\Sigma\). The set \(X_{\Sigma}\) of isomorphism classes of such modules \({\mathbb{T}}\) with associated CM type \(\Sigma\) is finite. To each \({\mathbb{T}} \in X_{\Sigma}\), one can attach a quadratic space \(\mathcal{C}({\mathbb{T}})\) of rank two over the adele ring \({\mathbb{A}}_{F^{}\sharp}\), which is incoherent in the sense that it does not arise from any quadratic space over \(F^{\sharp}\). Using the Weil representation, one can associate to this quadratic space \(\mathcal{C}({\mathbb{T}})\) a Hilbert modular Eisenstein series \(E(\tau, s, {\mathbb{T}})\) of parallel weight one on \(\text{GL}_2({\mathbb{A}}_{F^{\sharp}})\). Here, \(\tau = u+iv\) denotes an element in the \(F^{\sharp}\) upper half space \(\mathfrak{H}_{F^{\sharp}}\), which can be identified with two copies on the usual upper-half space \(\mathfrak{H} = \{ z \in {\mathbb{C}}: \operatorname{Im}(z) > 0 \}\). One can then define \[ E(s, \tau, \Sigma) = \sum_{ {\mathbb{T}} \in X_{\Sigma}} E(\tau, s, {\mathbb{T}}) \] to denote the finite sum over modules \({\mathbb{T}} \in X_{\Sigma}\) of such Eisenstein series, which in the setting described above turns out not to depend on the choice of \(\Sigma\). The incoherence of the underlying quadratic space \(\mathcal{C}({\mathbb{T}})\) implies that \(E(\tau, s, \Sigma)\) vanishes at \(s=0\), whence the derivative at \(s=0\) is given by the sum \[ E'(\tau, 0, \Sigma) = \sum_{\alpha \in F^{\sharp}} c_{\Sigma}(\alpha, v) q^{\alpha} \] for some parameter \(v \in {\mathbb{R}}\). Here, \(q^{\alpha} = e(\text{Tr}_{F^{\sharp}/{\mathbb{Q}}} (\alpha \tau))\), where \(e(x) = \exp(2\pi i x )\). Writing \(\iota: \mathfrak{H} \longrightarrow \mathfrak{H}_{F^{\sharp}}\) to denote the diagonal embedding, the pullback of \(E'(\tau, 0, \Sigma)\) to the complex upper-half space \(\mathfrak{H}\) is a non-holomorphic modular form of weight two, with Fourier series expansion \[ E'(\iota(\tau), 0, \Sigma) = \sum_{m \in {\mathbb{Z}}} b_{\Sigma}(m, v) q^m, \] where the coefficients \(b_{\Sigma}(m, v)\) are given by the formula \[ b_{\Sigma}(m,v) = \sum _{\substack{ \alpha \in F^{\sharp} \\ \text{Tr}_{ F^\sharp/{\mathbb{Q}}^{(\alpha)}} =m }} c_{\Sigma}(\alpha, v). \] The authors prove, under some mild technical hypothesis on the ramification of \(E\) (Hypothesis B), that for any parameter \(v \in {\mathbb{R}}\) and any nonzero integer \(m\), \[ \langle \widehat{T}(m, v): CM_{\Sigma} \rangle = -\frac{1}{W_E} \cdot b_{\sigma}(m, v), \] where \(W_E\) denotes the number of roots of unity in \(E^{\times}\).

The first chapter of the monograph explains the main results. The second chapter introduces the notion of \(\mathfrak{c}\)-polarized RM and CM modules for \(\mathfrak{c} \supset \mathcal{O}_F\) a fractional \(\mathcal{O}_F\)-ideal, then describes how to construct quadratic forms from some associated spaces of endomorphisms. The third chapter defines \(\mathfrak{c}\)-polarized RM and CM abelian surfaces. The fourth chapter gives a construction of the Eisenstein series \(E(\tau, s, {\mathbb{T}})\). The fifth chapter describes the main technical results in more detail. The sixth and final chapter makes up the technical core of the work, developing techniques from the theory of Dieudonné modules, the Grothendieck-Messing deformation theory of \(p\)-divisible groups and Zink’s theory of displays to perform various local calculations required for the proofs.

Though formidably technical, the reviewer recommends this beautiful monograph to anyone interested in the circle of conjecture proposed by Kudla et al., particularly from the point of view of arithmetic geometry. The work contains many useful references and intricate proofs that do not appear elsewhere, and is likely to be extremely useful to future progress in the area.

Roughly, the authors prove the following type of result. Let \(F\) be a real quadratic field, with \(\sigma \in \text{Gal}(F/{\mathbb{Q}})\) the nontrivial automorphism of its Galois group over \({\mathbb{Q}}\). Associated to \(F\) is a Hilbert modular surface \(M\), which can be defined as the moduli space of all \(\mathcal{O}_F\)-polarized RM abelian surfaces \(\mathcal{A}\). Given a positive integer \(m\), Hirzebruch and Zagier constructed a divisor on \(M({\mathbb{C}})\), which was later given a moduli theoretic interpretation by Kudla-Rapoport [S. S. Kudla and M. Rapoport, “Arithmetic Hirzebruch-Zagier cycles”, J. Reine Angew. Math. 515, 155–244 (1999; Zbl 1048.11048)]. To describe this latter interpretation, let \(T(m)\) denote the moduli space of pairs \((\mathcal{A}, j)\), where \(\mathcal{A}\) is an \(\mathcal{O}_F\)-polarized RM abelian surface, and \(j\) is a Rosati fixed endomorphism satisfying \(j \circ j = m\) and \(j \circ x = x^{\sigma} \circ j\) for all \(x \in \mathcal{O}_F\). The “forget-\(j\)” morphism \(T(m) \rightarrow M\) has image given by a codimension one cycle whose complex fibre is equal to the divisor constructed by Hirzebruch and Zagier. The moduli space \(T(m)\) can then justifiably be referred to as a Hirzebrich-Zagier divisor. The main aim of the work is to relate the intersection multiplicities of three such divisors to the Fourier coefficients of the central derivative of a certain Siegel Eisenstein series of genus three.

To be somewhat more precise, the authors establish the following main result (Theorem E). Let \(E\) be a quartic CM field containing \(F\) as a real quadratic subfield. Let us suppose that \(E\) is either Galois over \({\mathbb{Q}}\), with Galois group \(\text{Gal}(E/{\mathbb{Q}}) \cong {\mathbb{Z}}/4{\mathbb{Z}}\), or else that \(E\) is not Galois over \({\mathbb{Q}}\). Given a CM type \(\Sigma\) of \(E\), let us write \(E_{\Sigma}\) to denote its reflex field, with \(\mathcal{O}_{\Sigma}\) the ring of integers of \(E_{\Sigma}\). One can associate to \(E\) a reflex algebra \(E^{\sharp}\) of \(E\), which is the \({\mathbb{Q}}\)-algebra defined up to isomorphism by the existence of an \(\text{Aut}({\mathbb{C}}/{\mathbb{Q}})\)-equivariant bijection \[ \{ \text{CM types of \(E\)}\} \longrightarrow \text{Hom}(E^{\sharp}, {\mathbb{C}}), ~~~ \Sigma \longmapsto \phi_{\Sigma} \] for which \(\phi_{\Sigma}(E^{\sharp}) = E_{\Sigma}\). Let \(F^{\sharp}\) denote the maximal totally real subalgebra of \(E^{\sharp}\). In the setting described above, \(E^{\sharp}\) is a certain quartic CM field, with \(F^{\sharp}\) its maximal totally real subfield. Anyhow, one can associate to the pair \((E, \Sigma)\) an algebraic stack \(CM_{\Sigma}\), defined as the moduli space of principally polarized abelian varieties over \(\mathcal{O}_{\Sigma}\)-schemes with CM by \(\mathcal{O}_{\Sigma}\), satisfying a certain technical condition (the so-called Kottwitz condition of \(\S 3.2\)). The forgetful morphism \(CM_{\Sigma} \longrightarrow M_{\mathcal{O}_{\Sigma}}\) contains a codimension two cycle on \(M_{\mathcal{O}_{\Sigma}}\). To a given pair \(T(m)\) and \(CM_{\Sigma}\), one can associate a finite intersection pairing \(\langle T(m): CM_{\Sigma} \rangle_{\text{fin}}\), which is essentially the sum of lengths of all local rings in the intersection \(T(m) \bigcap CM_{\Sigma}\) (\(\S 5.4\)). Following the constructions of Kudla [loc. cit.] and Bruinier [J. H. Bruinier, “Borcherds products and Chern classes of Hirzebruch-Zagier divisors”, Invent. Math. 138, No. 1, 51–83 (1999; Zbl 1011.11027)], one can also define for any integer \(m\) an associated Green function \(G(m, v, \cdot)\) on \(M({\mathbb{C}})\), where \(v \in {\mathbb{R}}\) is some fixed parameter. Taking the (finite) sum over all complex points of \(CM_{\Sigma}\) then gives rise to a function \(G(m, v, CM_{\Sigma})\) on \(M({\mathbb{C}})\). One can then define the intersection of the formal pair \(\widehat{T}(m, v) := (T(m), G(m, v, \cdot))\) and \(CM_{\Sigma}\) to be the sum \[ \langle \widehat{T}(m, v): CM_{\Sigma} \rangle_{\text{fin}} = \langle T(m): CM_{\Sigma} \rangle_{\text{fin}} + \frac{1}{2}G(m, v, CM_{\Sigma}). \] Now, turning to the automorphic side, the authors define an \(\mathcal{O}_F\)-polarized CM module \({\mathbb{T}}\) to be a projective \(\mathcal{O}_E\)-module of rank one equipped with a perfect \({\mathbb{Z}}\)-valued symplectic form (satisfying some compatibility condition with respect to the \(\mathcal{O}_E\)-action). To such a module \({\mathbb{T}}\), there is an associated CM-type \(\Sigma\). The set \(X_{\Sigma}\) of isomorphism classes of such modules \({\mathbb{T}}\) with associated CM type \(\Sigma\) is finite. To each \({\mathbb{T}} \in X_{\Sigma}\), one can attach a quadratic space \(\mathcal{C}({\mathbb{T}})\) of rank two over the adele ring \({\mathbb{A}}_{F^{}\sharp}\), which is incoherent in the sense that it does not arise from any quadratic space over \(F^{\sharp}\). Using the Weil representation, one can associate to this quadratic space \(\mathcal{C}({\mathbb{T}})\) a Hilbert modular Eisenstein series \(E(\tau, s, {\mathbb{T}})\) of parallel weight one on \(\text{GL}_2({\mathbb{A}}_{F^{\sharp}})\). Here, \(\tau = u+iv\) denotes an element in the \(F^{\sharp}\) upper half space \(\mathfrak{H}_{F^{\sharp}}\), which can be identified with two copies on the usual upper-half space \(\mathfrak{H} = \{ z \in {\mathbb{C}}: \operatorname{Im}(z) > 0 \}\). One can then define \[ E(s, \tau, \Sigma) = \sum_{ {\mathbb{T}} \in X_{\Sigma}} E(\tau, s, {\mathbb{T}}) \] to denote the finite sum over modules \({\mathbb{T}} \in X_{\Sigma}\) of such Eisenstein series, which in the setting described above turns out not to depend on the choice of \(\Sigma\). The incoherence of the underlying quadratic space \(\mathcal{C}({\mathbb{T}})\) implies that \(E(\tau, s, \Sigma)\) vanishes at \(s=0\), whence the derivative at \(s=0\) is given by the sum \[ E'(\tau, 0, \Sigma) = \sum_{\alpha \in F^{\sharp}} c_{\Sigma}(\alpha, v) q^{\alpha} \] for some parameter \(v \in {\mathbb{R}}\). Here, \(q^{\alpha} = e(\text{Tr}_{F^{\sharp}/{\mathbb{Q}}} (\alpha \tau))\), where \(e(x) = \exp(2\pi i x )\). Writing \(\iota: \mathfrak{H} \longrightarrow \mathfrak{H}_{F^{\sharp}}\) to denote the diagonal embedding, the pullback of \(E'(\tau, 0, \Sigma)\) to the complex upper-half space \(\mathfrak{H}\) is a non-holomorphic modular form of weight two, with Fourier series expansion \[ E'(\iota(\tau), 0, \Sigma) = \sum_{m \in {\mathbb{Z}}} b_{\Sigma}(m, v) q^m, \] where the coefficients \(b_{\Sigma}(m, v)\) are given by the formula \[ b_{\Sigma}(m,v) = \sum _{\substack{ \alpha \in F^{\sharp} \\ \text{Tr}_{ F^\sharp/{\mathbb{Q}}^{(\alpha)}} =m }} c_{\Sigma}(\alpha, v). \] The authors prove, under some mild technical hypothesis on the ramification of \(E\) (Hypothesis B), that for any parameter \(v \in {\mathbb{R}}\) and any nonzero integer \(m\), \[ \langle \widehat{T}(m, v): CM_{\Sigma} \rangle = -\frac{1}{W_E} \cdot b_{\sigma}(m, v), \] where \(W_E\) denotes the number of roots of unity in \(E^{\times}\).

The first chapter of the monograph explains the main results. The second chapter introduces the notion of \(\mathfrak{c}\)-polarized RM and CM modules for \(\mathfrak{c} \supset \mathcal{O}_F\) a fractional \(\mathcal{O}_F\)-ideal, then describes how to construct quadratic forms from some associated spaces of endomorphisms. The third chapter defines \(\mathfrak{c}\)-polarized RM and CM abelian surfaces. The fourth chapter gives a construction of the Eisenstein series \(E(\tau, s, {\mathbb{T}})\). The fifth chapter describes the main technical results in more detail. The sixth and final chapter makes up the technical core of the work, developing techniques from the theory of Dieudonné modules, the Grothendieck-Messing deformation theory of \(p\)-divisible groups and Zink’s theory of displays to perform various local calculations required for the proofs.

Though formidably technical, the reviewer recommends this beautiful monograph to anyone interested in the circle of conjecture proposed by Kudla et al., particularly from the point of view of arithmetic geometry. The work contains many useful references and intricate proofs that do not appear elsewhere, and is likely to be extremely useful to future progress in the area.

Reviewer: Jeanine Van Order (Lausanne)