Grant Details

This project aims to connect two research teams in Italy that share closely related interests in the areas of number theory and arithmetic algebraic geometry. The project centres around the arithmetic of (semi)abelian varieties, both for its intrinsic importance and as a tool to study the absolute Galois groups of fields. Related themes are also explored, such as rational and integral points on modular curves and on more general varieties, modular forms (also over function fields), and pure algebraic number theory, with an emphasis on the recently developed theory of Hopf-Galois extensions. The main motivating problems are questions of Serre, Mumford and Tate concerning the Galois representations attached to abelian varieties and conjectures in the field of ‘Unlikely Intersections’ associated with the names of Bombieri, Masser, Zannier, Zilber and Pink. Furthermore, the conjecture of Birch and Swinnerton-Dyer (and its analogues for function fields) and conjectures by Lang-Vojta and Campana all help inform the research directions of the members of the group. The scientific goals of the project can be roughly summarised as follows. We will classify Galois representations attached to abelian varieties over number fields and deduce consequences for arithmetic questions (isogenies, cases of the Sato-Tate conjecture, number of rational points of curves over finite fields, problems related to the Lang-Trotter conjecture). Concerning Unlikely Intersections, in addition to proving new cases of the Zilber-Pink conjecture, we are going to investigate problems of effectivity, the invariance of the conjectures under field extensions, and applications. We aim to establish further instances of the Lang-Vojta and Campana conjectures, both over number fields and function fields, and of their hyperbolic analogues, and consider applications to various number-theoretic problems. In Hopf-Galois theory, we shall investigate how the properties of a field extension relate to its different Hopf-Galois structures (for field extensions within classes where this is not yet well understood)‚ and in particular, clarify the link between such structures and the ramification properties of the extension. Concerning Drinfeld modular forms, we will investigate the structure of old-, new- and cusp forms and understand the relations between eigenvalues and Fourier coefficients. We will also try to construct v-adic families of modular forms à la Hida-Coleman.