Let $F$ be a non-archimedean local field of residual characteristic $p$, $\ell \neq p$ be a prime number, and $\text{W}_{F}$ the Weil group of $F$. We classify equivalence classes of $\text{W}_{F}$-semisimple Deligne $\ell$-modular representations of $\text{W}_{F}$ in terms of irreducible $\ell$-modular representations of $\text{W}_{F}$, and extend constructions of Artin–Deligne local constants to this setting. Finally, we define a variant of the $\ell$-modular local Langlands correspondence which satisfies a preservation of local constants statement for pairs of generic representations.

We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number $p>2$ are compatible with the usual local constants at all primes not dividing $p$ and in two special cases also at primes dividing $p$. We deduce new cases of the $p$-parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

Let $G$ be a finite group acting freely on a smooth projective scheme $X$ over a locally compact field of characteristic 0. We show that the $\varepsilon_0$-constants associated to symplectic representations $V$ of $G$ and the action of $G$ on $X$ may be determined from Pfaffian invariants associated to duality pairings on Hodge cohomology. We also use such Pfaffian invariants, along with equivariant Arakelov Euler characteristics, to determine hermitian Euler characteristics associated to tame actions of finite groups on regular projective schemes over $\mathbb{Z}$.