Hartl's dictionary and a transcendence characterization of equicharacteristic local shtukas with complex multiplication

Abstract

We update Hartl's dictionary between $p$-adic Hodge theory and its equicharacteristic analog to include aspects related to the Hodge--Tate filtration in $p$-adic Hodge theory and its equicharacteristic analog, the Hodge--Tate--Taguchi filtration. Using this update, we establish a transcendence characterization of equicharacteristic local shtukas with abelian Tannakian structure group, matching a result of Howe--Klevdal in the $p$-adic setting. A special case of this result says that a formal Drinfeld module over an equicharacteristic local field has complex multiplication if and only if the Hodge--Tate--Taguchi filtration on its $z$-adic Tate module is algebraic.