Lifting Galois representations: Variations on a conjecture of Serre

Representations of Galois groups over Q play a fundamental role in number theory and its connections to arithmetic geometry. Serre conjectured that two dimensional mod p Galois representations of a certain type come from modular forms. In general, we would like to know which types of Galois representations come from other objects, such as algebraic varieties or cohomology classes of congruence subgroups.