Moduli space of bundles and Kloosterman sums

Abstract: The relation between analytic properties of modular forms and arithmetic results has led to many famous results and conjectures. In the geometric analogue of this conjectural relation - called geometric Langlands correspondence quotients of the upper half plane are replaced by moduli spaces of bundles on the curve. We will try to motivate this analogy. 
Since the geometry of these spaces is complicated in general, very few explicit examples of such modular forms are known. In joint work with B.C. Ngô and Z. Yun – which was motivated by work of Gross and Frenkel - we found an explicit series of such forms which turn out to be closely related to classical Kloosterman sums. This gives an example of the (wild) geometric Langlands correspondence.