Galois representations over convergent de Rham period ring

Abstract

Let ${\mathbf{B}}_{\mathrm{dR}}^{+,†}\subset {\mathbf{B}}_{\mathrm{dR}}^{+}$ 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 ${\mathbf{B}}_{\mathrm{dR}}^{+,†}$ 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 ${\mathbf{B}}_{\mathrm{dR}}^{+,†}$-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 ${\mathbf{B}}_{\mathrm{dR}}^{+,†}$-representations and ${\mathbf{B}}_{\mathrm{dR}}^{+}$-representations are equivalent.

arXiv commentv2: very minor revisions/simplifications. 40 pages. Comments welcome!