Duality for the condensed Weil-étale realisation of $1$-motives over $p$-adic fields

Abstract: We extend Tate duality for Galois cohomology of abelian varieties to $1$-motives over a $p$-adic field, improving a result of Harari and Szamuely. To do this, we replace Galois cohomology with the condensed cohomology of the Weil group. This is a topological cohomology theory defined in a previous work, which keeps track of the topology of the $p$-adic field. To see $1$-motives as coefficients of this cohomology theory, we introduce their condensed Weil-\'etale realisation. Our duality takes the form of a Pontryagin duality between locally compact abelian groups.