Sums of three cubes over a function field

Abstract

We use a function field version of the circle method to prove that a positive proportion of elements in ${\mathbb{F}}_q[t]$ are representable as a sum of three cubes of minimal degree from ${\mathbb{F}}_q[t]$, assuming a suitable form of the Ratios Conjecture and that the characteristic is greater than 3. The analogue of this conjecture for quadratic Dirichlet $L$-functions is known for large fixed $q$, via recent developments in homological stability.

57 pages