On the arithmetic and geometry of spaces $L_{m+1,n}$

Abstract

Let $p$ be a prime number. Motivated by the local lifting problem for $(\mathbb{Z}/p\mathbb{Z}{)}^n$ with $n>1$, we prove several new results on certain ${\mathbb{F}}_p$-vector spaces of logarithmic differential forms on the projective line in characteristic $p$, called "spaces $L_{m+1,n}$". Expanding the previous work by the first two authors, we prove positive and negative results for the existence of spaces $L_{m+1,n}$ in many situations. Moreover, we classify all spaces $L_{4p,2}$ for any $p$, and all spaces $L_{15,2}$ for $p=3$. Among the novel tools we use, Moore determinants and computational algebra play a prominent role.

63 pages. Supporting code available at https://github.com/DanieleTurchetti/equidistant. New in v2: rewritten the proof of Lemma 3.12 (former 3.10); modified Sections 4.2 and 4.2.1, which now make use of Dickson's invariants; added Sections 4.2.2 on the non-existence of spaces $L_{p^{n-1},n}$ in odd characteristic and Section 7 on the classification of spaces $L_{4p,2}$