Title: Abelian Varieties as Algebraic Protori?
by Wayne Lewis (University of Hawaiʻi) as part of Topological Groups

Lecture held in Elysium.

Abstract
An outcome of the structure theory of protori (compact connected abelian groups) is their representability as quotients of $mathbb{A}^n$ for the ring of adeles $mathbb{A}$. $mathbb{A}$ does not contain zeros of rational polynomials, but rather representations of zeros. Investigating the relations between algebraicity of complex tori and algebraicity of protori leads one to the problem of computing the Pontryagin dual of $mathbb{A}/mathbb{Z}$. Applying an approach by Lenstra in the setting of profinite integers to the more general $mathbb{A}$ leads to a definition of the closed maximal $Lenstra$ $ideal$ $E$ of $mathbb{A}$, whence the locally compact field of $adelic$ $numbers$ $mathbb{F}=mathbb{A}/E$, providing a long-sought connection to $mathbb{C}$ enabling one to define a functor from the category of complex tori to the category of protori – is it possible to do so in a way that preserves algebraicity? While $mathbb{F}$ marks tentative progress, much work remains…