Local Langlands correspondences in $$\ell $$ -adic coefficients

Let \(\ell \) be a prime number different from the residue characteristic of a non-archimedean local field F. We give formulations of \(\ell \)-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 \(\ell \)-adic coefficient field and the complex number field.