The main preoccupations of this project belong to the field of Galois representations attached to modular forms. The project is divided in three main themes, each one of them has its own importance in the field and they represent three of the major directions in contributing to the scientific research:experimenting and formulating conjectures; finding the right (natural, canonical) tools and environments for certain problems; generalizing previous results.
The first direction is theoretical in its essence, but, as in our previous work, it will be accompanied by many explicit computations. This part tries to offer a unified view of some of recent results in the literature using P.I.’s and his collaborator previous work and is proposing a better understanding of the eigencurve and its classical points. The second part is experimental in its essence but is based on strong arguments and deep previous results by many important mathematicians. The experimental computations will create two databases attached to rational elliptic curves: one for the associated Galois representation modulo its conductor and the second one of the coefficients of the associated eigenform in terms of products of Eisenstein series. A study of interlinks between the two databases will be in our opinion of great importance. The third part is theoretical, but many practical experiments are needed. This part is aiming to obtain a generalization of previous results concerning congruences between modular forms

Principalele preocupari din acest proiect se incadreaza in domeniul reprezentarilor Galois atasate formelor modulare. Proiectul este impartit in 3 directii principale, fiecare cu propria sa importanta in domeniu si care reprezinta, de asemenea trei din directiile majore in cercetarea stiintifica: experimentarea si formularea de conjecturi; gasirea metodelor naturale si adecvate pentru anumite probleme; generalizarea rezultatelor anterioare.
Prima directie este teoretica in esenta, dar va fi insotita de multe calcule explicite. Aceasta parte incearca sa ofere o imagine unificata a mai multor rezultate recente din literatura, folosind rezultatele directorului de proiect si a colaboratorilor sai. Ne propunem o mai buna intelegere a spatiului eigencurve si a punctelor sale clasice. A doua parte este experimentala dar este bazata pe argumente solide si rezultate extrem de puternice a mai multor matematicieni foarte importanti. Calculele experimentale vor creea doua baze de date atasate curbelor eliptice rationale: una pentru reprezentarile Galois modulo conductor, iar cealalalta pentru coeficientii formei cuspidale asociate descompusa in termen de produse de serii Eisenstein. O studiere a legaturilor dintre cele doua baze de date este, dupa parerea noastra, foarte importanta. A treia parte este teoretica insa foarte multe experimente practice sunt necesare. Aceasta parte are ca scop obtinerea unor generalizari a unor rezultate importante privind congruentele dintre forme modulare.