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Families of wild 1-motives

Grzegorz Banaszak, Dorota Blinkiewicz

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.

Families of wild 1-motives

Abstract

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

Paper Structure

This paper contains 7 sections, 9 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. 1-motives and torsion 1-motives
  3. Former constructions of wild 1-motives
  4. New constructions of wild 1-motives
  5. Appendix on multiple base discrete logarithm problem
  6. The local to global multiple base $(\alpha_1,\ldots,\alpha_n)$ discrete logarithm problem
  7. Infinite basis

Key Result

Theorem 3.2

Let $\mathcal{T}=\mathbb{G}_m^2=\mathbb{G}_m\times_{\mathop{\mathrm{spec}}\nolimits \,\mathbb{Z}_S}\mathbb{G}_m$. Consider the following 1-motives over $\rm{spec}\, \mathbb{Z}_S$: $[\Lambda\rightarrow \mathcal{T}]$, $[\Lambda'\rightarrow \mathcal{T}]$, $[\Lambda\cap\Lambda'\rightarrow \mathcal{T}]$

Theorems & Definitions (16)

  • Definition 2.1
  • Definition 2.4
  • Theorem 3.2
  • Theorem 3.3
  • proof
  • Corollary 3.4
  • proof
  • Corollary 3.6
  • proof
  • Theorem 3.8
  • ...and 6 more