In this talk, we discuss a few problems in finite field arithmeticarising from geometry, with a particular emphasis on factoring and cyclotomy. We begin with a discussion of low-dimensional spin groups and K3 surfaces, which allow us to define certain natural geometric polynomials modulo a prime integer whose value set distributions we classify. This also proves variants of Heilbronn's conjecture - including the classical one - via application of E- and G-function techniques (w. Amit Ghosh).

 
We then turn to a problem which is very simple to state: How do we determine a comprehensive (explicit) Galois theory for cubic extensions of a function field, where the constant field is any finite field? In analogy to Kummer and Artin-Schreier theory, we give a standard form, a la Hasse, which can be used when certain roots of unity are missing (w. Sophie Marques).
 
Finally, we conclude with a discussion of geometric cyclotomy in function fields and its role in constructing canonical differentials. Our constructions are p-adic, and are described completely in terms of Galois actions on rank 1 Drinfel'd modules. A classical proof allows for description of this action and such bases in Drinfel'd modules of higher rank (w. Aristides Kontogeorgis).
 
Some open questions on periods and circle packing in function fields will also be mentioned.