RESEARCH TOPIC
The project aims at interconnecting different methods in number theory, namely geometric, algebraic and analytic techniques, in order to attach central problems in diophantine geometry, Unlikely Intersections over algebraic groups, Iwasawa theory, the arithmetic of elliptic curves, motives and motivic Galois groups, p-adic Hodge theory and L-functions.
Our unit will organize joint seminars together with the units of Genova and Padova. We also plan to organize conferences and summer schools within the ALGANT international Master Program. Part of the funds will be employed by the members of the unit in order to strengthen their collaborations, and took part to conferences in order to disseminate the results obtained.
ABSTRACT
Number theory makes use of geometric, algebraic and analytic methods as tools to attack arithmetic problems. This project connects six research teams in Italy, where all these different approaches to number theory are explored. Connection with complex algebraic geometry, as well as abstract algebra, are also considered.
The main research themes are represented by Diophantine geometry, Unlikely Intersections over algebraic groups, Iwasawa theory, the arithmetic of elliptic curves (e.g. Birch and Swinnerton-Dyer conjecture), motives and motivic Galois groups, p-adic Hodge theory, L-functions, in particular the study of the Selberg class and computation of moments on average over certain families; additive problems of Goldbach type. The theory of p-adic L-functions (e.g. in Hida theory, or in the Birch-Swinnerton-Dyer conjecture). Galois representations associated to abelian varieties and rational points on modular curves. Drinfeld modular curves. Relations with algebraic geometry arise e.g. via the theory of deformations and links with complex analytic geometry derive from Nevanlinna theory and the conjectural relation between the distribution of integral points and of entire curves on algebraic varieties.
The project aims at interconnecting different methods in number theory, namely geometric, algebraic and analytic techniques, in order to attach central problems in diophantine geometry, Unlikely Intersections over algebraic groups, Iwasawa theory, the arithmetic of elliptic curves, motives and motivic Galois groups, p-adic Hodge theory and L-functions.
Our unit will organize joint seminars together with the units of Genova and Padova. We also plan to organize conferences and summer schools within the ALGANT international Master Program. Part of the funds will be employed by the members of the unit in order to strengthen their collaborations, and took part to conferences in order to disseminate the results obtained.
ABSTRACT
Number theory makes use of geometric, algebraic and analytic methods as tools to attack arithmetic problems. This project connects six research teams in Italy, where all these different approaches to number theory are explored. Connection with complex algebraic geometry, as well as abstract algebra, are also considered.
The main research themes are represented by Diophantine geometry, Unlikely Intersections over algebraic groups, Iwasawa theory, the arithmetic of elliptic curves (e.g. Birch and Swinnerton-Dyer conjecture), motives and motivic Galois groups, p-adic Hodge theory, L-functions, in particular the study of the Selberg class and computation of moments on average over certain families; additive problems of Goldbach type. The theory of p-adic L-functions (e.g. in Hida theory, or in the Birch-Swinnerton-Dyer conjecture). Galois representations associated to abelian varieties and rational points on modular curves. Drinfeld modular curves. Relations with algebraic geometry arise e.g. via the theory of deformations and links with complex analytic geometry derive from Nevanlinna theory and the conjectural relation between the distribution of integral points and of entire curves on algebraic varieties.