$p$-adic estimates for exponential sums on curves
A central problem in number theory is that of finding rational or integer solutions to systems of polynomials in several variables. This leads one naturally to the slightly easier problems of finding solutions modulo a prime $p$. Using a discrete analogue of the Fourier transformation, this modulo $p$ problem can be reformulated in terms of exponential sums. We will discuss $p$-adic properties of such exponential sums in the case of higher genus curves as well as connections to complex differential equations.