Title:Multiplicative Relations Among Differences of Singular Moduli
Abstract: An elliptic curve is an algebraic curve which comes endowed with a group structure. Certain elliptic curves have more endomorphisms than expected, these elliptic curves are said to have “complex multiplication”. A singular modulus is a numerical invariant (known as the j-invariant) of an elliptic curve with complex multiplication.
The arithmetic properties of these numbers are of great interest, in particular, there are important results concerning the differences of singular moduli, as well as the (lack of) multiplicative dependencies of singular moduli. In joint work with Vahagn Aslanyan and Guy Fowler we study the multiplicative dependencies that can arise among differences of singular moduli, and give a fairly explicit characterization. This result is a special case of a form of the Zilber-Pink conjecture, and uses methods from algebraic number theory as well as o-minimality.