2023/07/13 by Rafael von Känel, von Kanel, Rafael, Arno Kret +1
Mathematics · Social Sciences · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Vietnamese History and Culture Studies
paper · pdf · doi:10.48550/arxiv.2307.06944
openalex publication_date 2023/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We continue our study of integral points on moduli schemes by combining the method of Faltings (Arakelov, Parsin, 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. In the second part we establish the effective Shafarevich conjecture for abelian varieties A over a number field K such that A_K has CM or A_K is of GL2-type and isogenous to all its G_\mathbb Q-conjugates. In the third part we continue our explicit study of the Parsin 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.