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Borcherds products and arithmetic intersection theory on Hilbert modular surfaces. (English) Zbl 1208.11077

It is the aim of the article under review to study the arithmetic intersection theory on Hilbert modular surfaces associated to a real quadratic field \(K\). There are three main results in this article: the authors proved an arithmetic version of a theorem of Hirzebruch and Zagier which states that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight \(2\), moreover they determined the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface and the Faltings heights of arithmetic Hirzebruch-Zagier divisors which are disjoint to the boundary.
To be more precise, assuming that the discriminant \(D\) of \(K\) is a prime, the authors considered \(\Gamma\) a subgroup of finite index of the Hilbert modular group \(\Gamma_K=\text{SL}_2(\mathcal{O}_K)\) where \(\mathcal{O}_K\) is the ring of integers in \(K\) and a desingularization \(\widetilde{X}(\Gamma)\) of the Baily-Borel compactification \(X(\Gamma)\) of \(\Gamma\backslash\mathbb{H}^2\). Let \(\mathcal{M}_k(\Gamma)\) be the line bundle of modular forms of weight \(k\) with its Petersson metric \(\|\cdot\|_{\text{Pet}}\) which is singular along the normal crossing divisor formed by the exceptional curves of the desingularization. And for any positive integer \(m\), let \(T(m)\) be the divisor in the projective variety \(\widetilde{X}(\Gamma)\) which is the pull-back of the algebraic Hirzebruch-Zagier divisor in \(X(\Gamma)\) of discriminant \(m\). Next, let \(\widetilde{\mathcal{H}}(N)\) be the moduli scheme associated with the Hilbert modular variety for the principal congruence subgroup \(\Gamma_K(N)\) of arbitrary level \(N\geq3\). The authors defined \(\mathcal{T}_N(m)\subseteq \widetilde{\mathcal{H}}(N)\) to be the Zariski closure of the Hirzebruch-Zagier divisor \(T(m)\) on the generic fibre. Since there exists no arithmetic intersection theory for the stack \(\widetilde{\mathcal{H}}(1)\), the authors worked with the tower of schemes \(\{\widetilde{\mathcal{H}}(N)\}_{N\geq3}\) as a substitute for \(\widetilde{\mathcal{H}}(1)\) by following the suggestion of Kudla. They defined the arithmetic Hirzebruch-Zagier divisor \(\widehat{\mathcal{T}}(m)\) as the class in the arithmetic Chow group \[ \widehat{\text{CH}}^1(\widetilde{\mathcal{H}},\mathcal{D}_{\text{pre}}):=\lim_\leftarrow\widehat{\text{CH}}^1(\widetilde{\mathcal{H}}(N),\mathcal{D}_{\text{pre}}) \] which is determined by the sequence \((\widehat{\mathcal{T}}_N(m))_{N\geq3}\). Here \(\widehat{\mathcal{T}}_N(m)\) is the pair \((\mathcal{T}_N(m),g_N(m))\), where \(g_N(m)\) is the pull-back of certain automorphic Green object. Similarly, the authors defined the first arithmetic Chern class of \(\overline{\mathcal{M}}_k\) as \(\widehat{c_1}(\overline{\mathcal{M}}_k):=\big(\widehat{c_1}(\overline{\mathcal{M}}_k(\Gamma_K(N)))\big)_{N\geq3}\).
Let \(M_2^+(D,\chi_D)\) be the space of holomorphic modular forms of weight \(2\) for the congruence subgroup \(\Gamma_0(D)\subseteq\text{SL}_2(\mathbb{Z})\) with character \(\chi_D=(\frac{D}{\cdot})\) satisfying certain condition described by Hirzebruch and Zagier. Then the first main result in the article under review states that the arithmetic generating series \[ \widehat{A}(\tau)=\widehat{c_1}(\overline{\mathcal{M}}_{1/2}^\vee)+\sum_{m>0}\widehat{\mathcal{T}}(m)q^m \] is a holomorphic modular form in \(M_2^+(D,\chi_D)\) with values in \(\widehat{\text{CH}}^1(\widetilde{\mathcal{H}},\mathcal{D}_{\text{pre}})_\mathbb{Q}\).
The second main result in the article under review showed that the arithmetic self-intersection number of the line bundle of modular forms is essentially given by the logarithmic derivative at \(s=-1\) of the Dedekind zeta function \(\zeta_K(s)\) of \(K\). The precise statement is the following identities: \[ \widehat{A}(\tau)\cdot\widehat{c_1}(\overline{\mathcal{M}}_k)^2=\frac{k^2}{2}\zeta_K(-1)\big(\frac{\zeta'_K(-1)}{\zeta_K(-1)} +\frac{\zeta'(-1)}{\zeta(-1)}+\frac{3}{2}+\frac{1}{2}\log(D)\big)\cdot E(\tau) \] where \(E(\tau)\) is certain Eisenstein series described in the text. In particular, the authors obtained the arithmetic self-intersection number of \(\overline{\mathcal{M}}_k\): \[ \overline{\mathcal{M}}_k^3=-k^3\zeta_K(-1)\big(\frac{\zeta'_K(-1)}{\zeta_K(-1)} +\frac{\zeta'(-1)}{\zeta(-1)}+\frac{3}{2}+\frac{1}{2}\log(D)\big). \]
At last, suppose that \(T(m)\) is disjoint to the boundary, the third main result in the article under review gave the Faltings height of \(\mathcal{T}(m)\) with respect to \(\overline{\mathcal{M}}_k\): \[ \text{ht}_{\overline{\mathcal{M}}_k}(\mathcal{T}(m))=-(2k)^2\text{vol}(T(m))\big(\frac{\zeta'(-1)}{\zeta(-1)}+\frac{1}{2}+\frac{1}{2}\frac{\sigma_m'(-1)}{\sigma_m(-1)}\big). \] This result provides further evidence for the conjecture of Kramer that the arithmetic volume is essentially the derivative of the zeta value for the volume of the fundamental domain, the conjecture of Kudla on the constant term of the derivative of certain Eisenstein series, and the conjecture of Maillot and Roessler on special values of logarithmic derivatives of Artin \(L\)-functions.
Reviewer: Shun Tang (Orsay)

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
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