A geometric version of the circle method

The circle method can be used to estimate the number of rational points on a smooth hypersurface of low degree in projective or affine space, verifying cases of the Manin conjecture. A function field version of the circle method can be used in the same way to estimate the number of rational curves on such a hypersurface defined over a finite field. Rational curves on a hypersurface are parameterized by a moduli space, so this raises the question of whether geometric information about this space can be gleaned from the circle method. Indeed, Browning and Vishe were able to calculate the dimension in this way, and show it is irreducible. In this talk, I will explain how Browning and myself were able to obtain further information, calculating the high-degree cohomology of this space, by developing a geometric analogue of the circle method.