On $p$-adic Minkowski's Theorems

Dual lattice is an important concept of Euclidean lattices. In this paper, we first give the right definition of the concept of the dual lattice of a $p$-adic lattice from the duality theory of locally compact abelian groups. The concrete constructions of ``basic characters'' of local fields given in Weil's famous book ``Basic Number Theory'' help us to do so. We then prove some important properties of the dual lattice of a $p$-adic lattice, which can be viewed as $p$-adic analogues of the famous Minkowski's first, second theorems for Euclidean lattices. We do this simultaneously for local fields $\mathbb{Q}_p$ (the field of $p$-adic numbers) and $\mathbb{F}_p((T))$ (the field of formal power-series of one indeterminate with coefficients in the finite field with $p$ elements).