On the rank of the multivariable (, OK^) -modules associated to mod p representations of GL₂ (K)

Let p be a prime number, K a finite unramified extension of Qₚ and F a finite extension of Fₚ. For an admissible smooth representation of GL₂ (K) over F satisfying certain multiplicity-one properties, we compute the rank of the associated \'etale (, OK^) -module DA () defined by Breuil-Herzig-Hu-Morra-Schraen, extending their results.