The main goal of this book is to establish and investigate identities between generating series of cycles on the arithmetic surface associated to a Shimura curve, and certain (nonholomorphic) modular forms. It completes previous work of the authors on this instance of Kudla’s general program of relating generating series obtained from arithmetic geometry, and modular forms, [see S. S. Kudla, Ann. Math. (2) 146, No. 3, 545–646 (1997; Zbl 0990.11032)], and the survey [S. S. Kudla, Special cycles and derivatives of Eisenstein series. In: Darmon, Henri (ed.) et al., Heegner points and Rankin \(L\)-series. Cambridge University Press. MSRI Publ. 49, 243–270 (2004; Zbl 1073.11042)].
Let us sketch the main results. Let \(B\) be an indefinite quaternion algebra over \(\mathbb Q\), and let \(O_B\) be a maximal order. The associated Shimura curve \([O_B^\times \backslash (\mathbb C\setminus \mathbb R)]\) admits a modular description as the space of \(2\)-dimensional abelian varieties with \(O_B\)-action (“fake elliptic curves” in the sense of Serre), which serves to define a model \(\mathcal M\) over the integers, an arithmetic surface.
The locus \(\mathcal Z(t)\) where the underlying abelian variety admits an additional “special” endomorphism of trace \(0\) and determinant a fixed natural number \(t\), is called a special cycle in \(\mathcal M\). One can illustrate the above principle by looking at the generating function \[ \phi_1(\tau) = -\mathop{\text vol}(\mathcal M(\mathbb C)) + \sum_{t>0} \deg(\mathcal Z(t)_{\mathbb C}) q^t \in \mathbb C[[q]], \qquad\text{where } q=e^{2\pi i\tau}.\tag{1} \] Namely \(\phi_1(\tau)\) is the \(q\)-expansion of a homolorphic modular form of weight \(3/2\). However, this result [proved in S. S. Kudla, M. Rapoport, T. Yang, Compos. Math. 140, No. 4, 887–951 (2004; Zbl 1088.11050)] is merely a warm-up example for the authors, who are ultimately interested in an arithmetic version. To this end, they equip the divisor \(\mathcal Z(t)\) of \(\mathcal M\) with a Green function and view it as an element \(\hat{\mathcal Z}(z,v)\) of the arithmetic Chow group \(\widehat{ {\text CH} }^1(\mathcal M)\); here \(v\) is a parameter in \(\mathbb R_{>0}\). They prove that for \(\tau = u+iv\), the generating series \[ \hat{\phi}_1(\tau) = \sum_{t\in\mathbb Z} \hat{\mathcal Z}(t,v)q^t \in \widehat{ {\text CH} }^1(\mathcal M)[[ q^{\pm 1} ]] \tag{2} \] is a nonholomorphic modular form of weight \(3/2\) with values in \(\widehat{ {\text CH} }^1(\mathcal M)\).
The second main result of the book gives an analogous result for \(0\)-cycles instead of divisors of \(\mathcal M\). Here, one imposes the condition that the underlying abelian varieties admits a pair of special endomorphisms. Identifying \(\widehat{ {\text CH} }^2(\mathcal M)\) with \(\mathbb R\) by the arithmetic degree map, one obtains a Siegel modular form of genus \(2\) and weight \(3/2\). Furthermore, the relationship between the two results with respect to the height pairing is clarified.
Rather than verifying the functional equations which characterize modular forms, all results of the form (1), (2) are proved by giving a candidate for the modular form to be obtained, and by then explicitly determining, and identifying, both sides. This leads to horrendously complicated formulas, and it must be called a miracle that in the end everything fits together perfectly.
As an application, the authors define an arithmetic theta lift from modular forms of weight \(3/2\) to the Mordell-Weil space of \(\mathcal M\) and prove a nonvanishing criterion, analogous to the one of Waldspurger for the classical theta lift.
For details on the vast history of the subject, we refer to the introduction of the book under review. We just mention that it can be traced back to Hurwitz who computed the degrees of analogous divisors \(Z(t)\) on the modular curve. D. Zagier proved that in this case as well one (almost) gets the \(q\)-expansion of a modular form [see C. R. Acad. Sci., Paris, Sér. A 281, 883–886 (1975; Zbl 0323.10021)].