Algebra/Number Theory Seminar

p-adic estimates of exponential sums and exponential differential equations

Joe Kramer-Miller (UC Irvine)

Thursday, February 25th, 2021
2:00 PM - 3:00 PM

The discrete analogue of the Gaussian integral, namely the integral over the real line of the exponential of -x2, is the quadratic Gauss sum over the finite field with p elements. It is miraculous and bizarre how similar these quantities are: the Gaussian integral equals the square root of pi, while the quadratic Gauss sum equals the square root of p.

An important generalization of quadratic Gauss sums are exponential sums, which are fundamental to modern arithmetic geometry. In general, closed form expressions for exponential sums are not possible. However, in the 1970s and 1980s, there was a flurry of work establishing formulas and estimates for a certain class of exponential sums over finite feilds. This raises a natural question: are there integrals, generalizing the Gaussian integral, corresponding to these more general exponential sums? This question motivated Pierre Deligne to define the “irregular Hodge filtration”, which can be associated to a certain type of differential equation. Deligne conjectured that the irregular Hodge filtration should force estimates on a wide class of exponential sums. We prove the entirety of Deligne’s conjecture. If time permits, we will explain how to adapt these methods to other types of exponential sums.

Event contact: c dot d dot haessig at rochester dot edu