Integral points on coarse Hilbert moduli schemes

Abstract: We continue our study of integral points on moduli schemes by combining the method of Faltings (Arakelov, Paršin, Szpiro) with modularity results and Masser-Wüstholz isogeny estimates. In this work we explicitly bound the height and the number of integral points on coarse Hilbert moduli schemes outside the branch locus.
      In the first part we define and study coarse Hilbert moduli schemes with their heights and branch loci. Building on the foundational works of Rapoport, Katz-Mazur, Deligne-Pappas and using the Faltings height, we first develop the basic definitions. Then we prove some geometric results for the height and the branch locus. In particular we relate the branch locus to automorphism groups of the involved moduli stack which allows us to compute the branch locus in some relevant cases.
      In part two we establish the effective Shafarevich conjecture for abelian varieties \(A\) over a number field \(K\) such that \(A_{\bar{K}}\) has CM or \(A_{\bar{K}}\) is of GL(2)-type and isogenous to all its \(\text{Aut}(\bar{K}/\mathbb{Q})\)-conjugates. If \(A_{\bar{K}}\) has no CM, then we use isogeny estimates and results of Ribet and Wu to reduce via Weil restriction to the known case \(K=\mathbb{Q}\) proven by one of us via Faltings’ method and modularity. We also use isogeny estimates to reduce the CM case to effectively bounding the height \(h(\Phi)\) of simple CM types in terms of the discriminant. To bound \(h(\Phi)\) we follow Tsimerman’s strategy and we combine the averaged Colmez conjecture with explicit analytic estimates for L-functions.
      In the third part we continue our explicit study of the Paršin construction given by the forgetful morphism of Hilbert moduli schemes. We now work out our strategy for arbitrary number fields \(K\) and we explicitly bound the number of polarizations and module structures on abelian varieties over \(K\) with real multiplications.
     In the last part we illustrate our results by applying them to two classical surfaces first studied by Clebsch (1871) and Klein (1873): We explicitly bound the Weil height and the number of their integral points. Hirzebruch proved that both surfaces are models over \(\mathbb{C}\) of a Hilbert modular surface \(Y_\mathbb{C}\). To show that they are coarse Hilbert moduli schemes over \(\mathbb{Z}[\tfrac{1}{30}]\), we go into the construction of Rapoport and Faltings–Chai of the integral minimal compactification of \(Y_\mathbb{C}\) and we make it explicit via Hirzebruch’s work. Here we use geometric results for integral Hirzebruch-Zagier divisors of Bruinier-Burgos-Kühn and Yang, and we compute the Fourier expansion of certain Eisenstein series via analytic results of Klingen, Siegel and Zagier. To explicitly relate the height to the Weil height, we combine Pazuki’s height comparison with formulas for Hilbert theta functions due to Götzky, Gundlach and Lauter-Naehrig-Yang.