# PRIN2010-2011

Coordinator |
B. ChiarellottoTel : +39 049 8271407 Fax : +39 049 8271479 e-mail : chiarbru@math.unipd.it |
Dipartimento di Matematica Università degli Studi di Padova Via Trieste 63 35121 ITALIA |

## Project overview

The interplay between number theory and arithmetic algebraic geometry has been one of the most fruitful interactions in the recent development of mathematics. This interaction has been based not only on shared methods for solving problems, but also, and more deeply,each of the two fields is a source of inspiration for the other,suggesting analogies that lead to innovative insights and unexpected relations.Examples of this can be seen in the ramification theory in the case of non perfect-residue field, in the behavior of L-functions and their special values according to the Beilinson conjectures, in the study of p-adic representations of Galois groups and the associated p-adic Hodge theory, in Langlands philosophy with all its various formulations(classical, geometric..),in the theory of motives and their applications. One might also mention the new proof of theorem of Siegel given by Kim using motivic fundamental group and p-adic Hodge theory. We plan to exploit the various members of the present research teams and their various skills in order to attack some problems in arithmetic algebraic geometry and number theory and, in according with the philosophy outlined above, to obtain further insights on new problems.The problems that we want to address can be roughly grouped in two fields although the two parts interact:

A) Arithmetic algebraic geometry of varieties over local fields and global fields.

B)Diophantine and analytic number theory.

A)To describe the sort of problem we would like to deal with,one may start from the theory of L-functions associated to numbers fields,to modular forms or,more generally,to motives.One expects L-functions to yield information on geometric entities,as in the Birch and Swinnerton-Dyer Conjecture(BSD).A topic,closed related to the BSD conjecture,is the study of integral and rational points on elliptic curves,and more generally, on algebraic varieties (see B).Conversely from arithmetic geometry we would like to get information on the analytic behavior of such L-functions.Along these lines one could approach the problem of the special values of L-functions via Beilinson's conjectures(regulators with values in absolute cohomology). More recently,p-adic counterparts of these objects have also been introduced,such as p-adic L-functions and regulators. As a natural development along these lines one could also consider Iwasawa theory and the associated Main Conjecture, as well as Beilinson's conjecture in the p-adic setting. Another subject that is intimately linked to this line of research is p-adic Hodge theory.In recent years,an impressive amount of work has been done on this field.This work has been inspired by the Langlands' conjecture and by the construction of p-adic L-functions via reciprocity laws.A systematic study of p-adic Galois representations has been made,giving deep insights among expected compatibilities between the cohomologies of different fibers of an arithmetic variety.This research has made use of p-adic differential equations.Recently,the method of perfectoid spaces has given new boost to this area and to the study of the generic fiber of a semistable variety as a differential Berkovich space.With regard to the study of L-functions attached to motives,one aims for a systematic understanding of the properties of the category of motives,in particular of 1-motives.

B) The diophantine problems we would like to deal with are connected with some deep conjectures on the distribution of the integral and rational points on algebraic varieties and the problem of unlikely intersections in algebraic groups.The aim is clear: one would like to attack at least some cases of the Vojta and Andre'-Oort,Zilberg-Pink conjectures.Ideas coming from Diophantine approximation and transcendental number theory have been used recently on problems of unlikely intersections.In this respect,even partial results on Pink's conjecture for abelian schemes of relative dimension two would have a deep meaning. Diophantine approximation can also be studied in the context of irrationality measures for special values of polylogarithms, which on the other hand can be studied in terms of mixed motives(motivic fundamental group). Analytic methods can be used to study L-functions via Selberg classes.From a more analytic point of view,for the study of the properties of L-functions it is important to tackle problems in diophantine approximation,in the distribution of primes and in their additive theory. In particular,the study of the values of linear forms in prime powers is deeply connected with problems in diophantine geometry since it studies special points of some arithmetic varieties. Although the techniques involved here are different,the aims of the study of the Selberg class and the Langlands program are similar, and both could lead to a more effective insight on automorphic L-functions.

## Partnership

Partnership