Researchers studied arithmetic versions of certain integrals that arise in mirror symmetry. Instead of integrating oscillatory functions over complex geometric spaces called Coulomb branches, they counted values of analogous functions at points over a finite field F_q. They show that, for the simplest (A1) Coulomb branch, these finite-field “exponential sums” break up into well-known objects in number theory: polynomial Gauss sums and Selberg character sums.

At a high level, the authors replace the continuous ingredients of the usual integrals by finite-field analogues: multiplicative characters stand in for power terms and additive characters stand in for exponential factors. The Coulomb branch admits a natural projection whose base can be identified with monic polynomials of a fixed degree. The fiber over a polynomial f is identified with the unit group of the finite ring F_q[t]/(f). When one restricts the exponential sum to a single fiber, the sum is exactly a polynomial Gauss sum over that finite ring. The paper gives an explicit formula for these fiberwise sums that features the discriminant of f, the quadratic character of F_q, and classical Gauss sums.

The authors also treat a deformed version of the Coulomb branch, where an auxiliary polynomial is allowed. There the fiberwise formula involves the resultant of the two polynomials instead of the discriminant. A particularly clear case is when the deformation polynomial is supported at two points: summing the fiberwise Gauss sums over all monic base polynomials produces the Selberg character sum previously studied by Evans. Using Evans’s evaluation of that Selberg sum, the paper converts the full Coulomb-branch exponential sum into an explicit product of classical Gauss sums.

The arithmetic structure becomes cleaner under field extensions. Applying the Davenport–Hasse lifting formula for Gauss sums, the authors show how the Coulomb-branch sums behave when one replaces F_q by its degree-n extension. They interpret these relations as suggesting a simple cohomological picture: the local system that encodes the characters should behave, at the level of Frobenius traces, like a one-dimensional pure object. The paper also studies ordinary (untwisted) counts of F_q-points on these Coulomb branches and writes an explicit generating function for those counts. In the pure gauge case this generating function simplifies further to a rational form.