Adelic and 1-adic Galois representations arising from abelian varieties and modular forms

• Applicant: Aaron Greicius, Ph.D.
• Subject Area: Mathematics
• Term: from 2008 to 2012
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 72701598

Associated to any elliptic curve E defined over a number field K is a family of l-adic Galois representations. These representations can be packaged into one adelic representation. Serre proved in 1972 that the image of ρ is open: in particular, it is of finite index. He also showed that if K = Q, the index of the image is at most 2; i.e., the adelic representation is never surjective. The research problems described in my proposal arise more or less directly from this seminal work. They fall into three general categories, which can be summarized as follows.1. In my Ph.D. thesis I proved that there are number fields K and elliptic curves E/K whose corresponding adelic representation is surjective. I would like to strengthen this result by proving that in fact “most” elliptic curves defined over number fields have a surjective adelic Galois representation.2. There are many other arithmo-geometric sources of adelic Galois representations: e.g., modular forms and abelian varieties of higher dimension. I intend to establish open image theorems for these more general adelic representations and determine precisely their image of Galois.3. Serre’s result implies that given an elliptic curve E/K, the l-adic representations are surjective for all l >> 0. Serre has asked whether there is for each number field K a constant S(K) such that for all elliptic curves E/K and all l > S(K) the corresponding l-adic representation is surjective. Much research has been done in the K = Q case which suggests that such a constant does exist, although we are still far from giving a full answer. A long-term research goal of mine is to contribute toward a positive answer to this question in the K = Q case.

• DFG Programme: Research Grants