Families of wild 1-motives

TL;DR

This paper introduces families of wild 1-motives over rings of S-integers that share the same torsion reductions everywhere outside S, using a local-global principle related to a discrete logarithm problem.

Contribution

It constructs explicit families of wild 1-motives with identical torsion reductions, advancing understanding of their classification over rings of S-integers.

Findings

Families of wild 1-motives with identical reductions are constructed.

A local-global principle for a multiple base discrete logarithm problem is established.

The approach links 1-motive theory with discrete logarithm problems.

Abstract

In this paper we present families of wild 1-motives, i.e., families of pairwise non-isomorphic Deligne 1-motives, over rings of $S$-integers $\mathcal{O}_{F,S}$, which have the same reductions to torsion 1-motives for all $v\notin S$. Our proof is based on a technical result concerning a local to global principle for multiple base discrete logarithm problem for arbitrary big bases.