The object of this proposal is the organization of a conference on the topic of modular curves and modular abelian varieties. This topic belongs to Number Theory, one of the oldest areas in Mathematics, and has been an important source of research for the last few decades. Many of the most important results in Number Theory and Arithmetic Geometry of the last years have some relation with modular curves and modular abelian varieties; for instance the determination of the rational torsion of elliptic curves by Mazur or the proof of Fermat's Last Theorem by Wiles (after work by Frey, Ribet, Mazur, Serre, Taylor and others).
The development of computer technology influenced the work of mathematicians and nowadays many computational aspects of modular forms and modular curves are being investigated with the help of computers. Moreover, modular curves and abelian varieties have applications to communication technologies, since they can be used to develop good error correcting codes and digital signature algorithms. The objective of the Conference is to present the latest developments by some of the top researchers on the subject, that will discuss both theoretical and computational results, along with applications. The Conference will be an occasion of contact between research teams working in many European and non-European countries.