On the arithmetic and geometry of spaces $L_{m+1,n}$
Michel Matignon, Guillaume Pagot, Daniele Turchetti
TL;DR
The paper addresses the problem of classifying spaces $L_{m+1,n}$ of logarithmic differential forms in characteristic $p$ that appear in local lifting problems for elementary abelian $p$-groups. It develops a general framework using Moore determinants and Dickson invariants to characterize when a given $n$-tuple of polynomials yields a space $L_{\lambda p^{n-1},n}$, and proves a sharp obstruction for the $n=2$ case when $p>3\lambda$. Extending to general $n$, it provides constructive criteria (via polynomials $Q_i$) and yields extensive new families of spaces (notably in characteristic 2), together with nonexistence results in odd characteristic. The work gives complete classifications for $(p,n)=(3,2)$ in the cases $\lambda=4,5$, and establishes that $L_{4p,2}$ exists only for $p\in\{2,3,5\}$ with $p=5$ yielding standard spaces; it further clarifies the role of standard spaces and étale pullbacks in generating subspaces. Overall, the paper advances the understanding of the arithmetic and geometric structure of $L_{m+1,n}$, providing powerful algebraic tools for lifting problems and new avenues for constructing and obstructing such spaces.
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.