Deformation-enhanced diffusive loss of titanium in quartz during greenschist-facies crystal-plastic deformation, Wasatch Fault Zone, UT

Bryant, DS; Karns, HN; Stearns, MS

Let

K / Q_{p}

be a finite unramified extension,

\overset{¯<#comment/>}{ρ<#comment/>} : G a l ({\overset{¯<#comment/>}{Q}}_{p} / K) \to<#comment/> {G L}_{n} ({\overset{¯<#comment/>}{F}}_{p})

a continuous representation, and

τ<#comment/>

a tame inertial type of dimension

n

. We explicitly determine, under mild regularity conditions on

τ<#comment/>

, the potentially crystalline deformation ring

R_{\overset{¯<#comment/>}{ρ<#comment/>}}^{η<#comment/>, τ<#comment/>}

in parallel Hodge–Tate weights

η<#comment/> = (n -<#comment/> 1, \dots<#comment/>, 1, 0)

and inertial type

τ<#comment/>

when theshapeof

\overset{¯<#comment/>}{ρ<#comment/>}

with respect to

τ<#comment/>

has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre’s conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [Mém. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150].

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