Joint IAS/PU Number Theory Seminar

The circle method has been a versatile tool in the study of rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to information about moduli spaces of rational curves on hypersurfaces. I will report on joint work with Margaret Bilu on implementing a circle method with an even more geometric flavour, where the computations take place in a suitable Grothendieck ring of varieties. We establish analogues for the key steps of the method, enabling us to approximate the classes of the above moduli spaces directly without recourse to point counting.