arXiv subjects

Marco Artusa

Publications and source records attributed to Marco Artusa.

3 recordsLinked to original sources

arXiv2025

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

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-étale realisation. Our duality takes the form of a Pontryagin duality between locally compact abelian groups.

math.NT↗

arXiv2024

X-matrices

We evidence a family $\mathcal{X}$ of square matrices over a field $\mathbb{K}$, whose elements will be called X-matrices. We show that this family is shape invariant under multiplication as well as transposition. We show that $\mathcal{X}$ is a (in general non-commutative) subring of $GL(n,\mathbb{K})$. Moreover, we analyse the condition for a matrix $A \in \mathcal{X}$ to be invertible in $\mathcal{X}$. We also show that, if one adds a symmetry condition called here bi-symmetry, then the set $\mathcal{X}^b$ of bi-symmetric X-matrices is a commutative subring of $\mathcal{X}$. We propose results for eigenvalue inclusion, showing that for X-matrices eigenvalues lie exactly on the boundary of Cassini ovals. It is shown that any monic polynomial on $ \mathbb{R} $ can be associated with a companion matrix in $ \mathcal{X} $.

math.RA↗

arXiv2024

Duality for condensed cohomology of the Weil group of a p-adic field

We use the theory of Condensed Mathematics to build a condensed cohomology theory for the Weil group of a $p$-adic field. The cohomology groups are proved to be locally compact abelian groups of finite ranks in some special cases. This allows us to enlarge the local Tate Duality to a more general category of non-necessarily discrete coefficients, where it takes the form of a Pontryagin duality between locally compact abelian groups.

math.NT↗