Galois representations over convergent de Rham period ring

TL;DR

This paper develops a formalism relating Galois representations over a convergent de Rham period ring to regular connections, establishing finiteness and equivalence results under certain conditions.

Contribution

It introduces a Tate--Sen formalism for Galois representations over the convergent de Rham period ring and proves finiteness and categorical equivalences under specific weight conditions.

Findings

Galois cohomology is finite when Sen weights satisfy a p-adic non-Liouville condition.

Categories of representations over the convergent and usual de Rham rings are equivalent for algebraic Sen weights.

A Tate--Sen formalism relates Galois representations to regular connections over convergent functions.

Abstract

Let $\mathbf{B}_{\mathrm{dR}}^{+, \dagger} \subset \mathbf{B}_{\mathrm{dR}}^{+}$ be the ``convergent" de Rham period ring which is the (un-completed) stalk at the de Rham point of the Fargues--Fontaine curve. We develop a Tate--Sen formalism to relate Galois representations over $\mathbf{B}_{\mathrm{dR}}^{+, \dagger}$ to regular connections over convergent functions. As a consequence, when the Sen weights (of the mod $t$ reduction) satisfy a $p$-adic non-Liouville condition, Galois cohomology of a $\mathbf{B}_{\mathrm{dR}}^{+, \dagger}$-representation compares to that of its $\mathbf{B}_{\mathrm{dR}}^{+}$-base change, and hence is finite. In addition, restricted to objects whose Sen weights are algebraic numbers, the categories of $\mathbf{B}_{\mathrm{dR}}^{+, \dagger}$-representations and $\mathbf{B}_{\mathrm{dR}}^{+}$-representations are equivalent.