Lattices in function fields and applications

Abstract: In recent decades, the use of ideas from Minkowski's Geometry of Numbers has gained recognition as a helpful tool in bounding the number of solutions to modular congruences with variables from short intervals. In 1941, Mahler introduced an analogue to the Geometry of Numbers in function fields over finite fields. Here, we build on Mahler's ideas and develop results useful for bounding the sizes of intersections of lattices and convex bodies in $\mathbb{F}_q((1/T))^d$, which are more precise than what is known over $\mathbb{R}^d$. These results are then applied to various problems regarding bounding the number of solutions to congruences in $\mathbb{F}_q[T]$, such as the number of points on polynomial curves in low dimensional subspaces of finite fields. Our results improve on a number of previous bounds due to Bagshaw, Cilleruelo, Shparlinski and Zumalac\'{a}rregui. We also present previous techniques developed by various authors for estimating certain energy/point counts in a unified manner.