Optimal bases for direct images of $p$-adic differential modules over discs

TL;DR

This paper constructs explicit optimal bases for the direct images of $p$-adic differential modules over discs, using ramification and preimage bases, advancing understanding of $p$-adic differential equations.

Contribution

It provides an explicit method to determine optimal bases for direct images of $p$-adic differential modules under finite morphisms of discs, incorporating ramification data.

Findings

Explicit construction of optimal bases for direct images

Connection between ramification and basis optimality

Enhanced understanding of $p$-adic differential modules

Abstract

Let $(k,|\cdot|)$ be a complete and algebraically closed valued field extension of $(\mathbb{Q}_p,|\cdot|_p)$. Given a finite morphism $\varphi:\mathscr{D}_1\to \mathscr{D}_2$ of unit discs over $k$, a differential module $(M,D)$ on $\mathscr{D}_1$ and a point $b\in \mathscr{D}_2(k)$, we construct explicitly an optimal basis of space of horizontal elements for the direct image $\varphi_*(M,D)$ at $b$ in terms of the suitable chosen optimal bases of $(M,D)$ at preimages of $b$ by $\varphi$ and ramification properties of the morphism.