Arithmetic & Algebraic Geometry Seminar: Factors of A-polynomials are Rare (3rd Year Candidacy Requirement)

Abstract: Let M be a compact 3-manifold whose boundary is a torus. Eigenvalues of representations pi_1(T²) --> SL_2(C) which extend to pi_1(M) trace out a collection of algebraic curves in (C*)² whose defining equations are called the A-factors of M. There is a certain 1-form on the normalization of each curve which has a primitive given by volumes of representations, implying that A-factors are exact polynomials. I will use Hodge-theoretic methods to show that there are finitely many exact polynomials in an appropriate moduli space of elliptic curves. I will also present partial results for higher genus.