Arithmetic Siegel-Weil formula on \(X_{0}(N)\) (Q1721959)

In the paper [Compos. Math. 140, No. 4, 887--951 (2004; Zbl 1088.11050)], \textit{S. S. Kudla} et al. have proved an identity between a generating series for the heights of Heegner cycles on the integral model of a Shimura curve, and the second term at some critical point of an Eisenstein series considered previously by D. Zagier. The main new feature of their work is that, in contrast to other previous works, the critical point is not the center of the functional equation, and the leading term at this point does not vanish. The analogous result in the case of the modular curve \(X(1)\) has been worked out by Kudla and Yang. In the present paper, the authors consider the same question for the modular curve \(X_0(N)\) when \(N\) is square free. In particular, they prove the modularity of the generating series of special cycles and an arithmetic Siegel-Weil formula. Some main technical issues arising in the case of \(X_0(N)\) are the study of Kudla's Green function at cusps and the theta lifting method used by \textit{J. H. Bruinier} and \textit{J. Funke} [J. Reine Angew. Math. 594, 1--33 (2006; Zbl 1104.11021)].