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.
Categories: math.NT math.AC math.AG