Johann Selewa: Modular forms and Galois representations

Abstract: The absolute Galois group G of Q is arguably one of the most important and mysterious groups in existence. Emil Artin showed that complex conjugation is the only element of order 2, up to conjugation. In fact, the only other known element is the identity. A way of obtaining information about G (or any group) is through its representations.

In this talk we will construct a representation r of G from elliptic curves/modular curves/modular forms (the constructions are very similar). We introduce fundamental operators acting on modular forms, which provide further information. We then verify that this construction completely describes r. In addition to describing G this construction provides an intimate link between number theory, geometry, and analysis.