Towards the Sato-Tate Conjecture for pairs of elliptic curves

6 June 2008
Prof. Michael Harris
Let E be an elliptic curve defined by a cubic equation with rational coefficients. <br />The Sato-Tate Conjecture is a statistical assertion about the variation of the number of points of E over finite fields. I review some of the main steps in my proof of this conjecture with Clozel, Shepherd-Barron, and Taylor, in the case when E has non-integral j-invariant. Emphasis will be placed on the steps involving moduli spaces of certain Calabi-Yau hypersurfaces with level structure.<br /><br />If one admits a version of the stable trace formula that should soon be available, the same techniques imply that, when E and E' are two elliptic curves that are not isogenous, then the numbers of their points over finite fields are statistically independent. For reasons that have everything to do with the current limits to our understanding of the Langlands program, the analogous conjectures for three or more non-isogenous elliptic curves are entirely out of reach.<br /><br />