Local Langlands correspondences in $\mathrm{\ell }$-adic coefficients

Let $\mathrm{\ell }$ be a prime number different from the residue characteristic of a non-archimedean local field $F$. We give formulations of $\mathrm{\ell }$-adic local Langlands correspondences for connected reductive algebraic groups over $F$, which we conjecture to be independent of a choice of an isomorphism between the $\mathrm{\ell }$-adic coefficient field and the complex number field.