On the torsion of the Jacobian varieties of X(p^n)
Chris Rasmussen

In this talk, we study the representation of the absolute Galois group, into the outer automorphisms of the (pro-p) fundamental group of the projective line minus three points. Although well studied, many properties of this representation are still unknown, such as the size of the field \Omega_p fixed by its kernel.

We will present new work, following the techniques of Anderson and Ihara, demonstrating fields of p-power torsion of the Jacobian varieties of modular curves of level p^N are rational over this field, in the case p = 3. The result rests on both the arithmetic and geometry of X(p^N), when viewed as a cover of the projective line minus three points. This work is joint with Matt Papanikolas.