Mordell--Weil groups over large algebraic extensions of fields of characteristic zero
Takuya Asayama, Yuichiro Taguchi
TL;DR
The paper advances understanding of Mordell--Weil groups over two broad families of large algebraic extensions in characteristic zero. It proves that the field $K_{g,\boldsymbol{m}}$, obtained by adjoining $p^{m_p}$-division coordinates across semiabelian varieties, is highly Kummer-faithful and not sub-$p$-adic when infinitely many $m_p\ge1$, while also identifying conditions under which $A(K_{g,\boldsymbol{m}})/A(K_{g,\boldsymbol{m}})_{\mathrm{tor}}$ fails to be free. For the second type of extensions, formed as fixed fields under automorphisms, the work shows that for almost all $\sigma$ with $e\ge2$, $A(L)/A(L)_{\mathrm{tor}}$ is a free $\mathbb{Z}$-module of countable rank over finite extensions $L$ of $\overline{K}[\sigma]$, and obtains Kummer-faithfulness and toral-divisible-vanishing results for $\overline{K}(\sigma)$ and $\overline{K}[\sigma]$. These results uncover a rich landscape where Mordell--Weil groups exhibit strong torsion constraints and free-abelian structures in large, non-sub-$p$-adic ground fields, with implications for anabelian geometry and the arithmetic of semiabelian varieties. The paper also clarifies the limitations of certain claims in the $e=1$ case and outlines connections to prior work on Kummer-faithfulness and Galois representations.
Abstract
We study the structure of the Mordell--Weil groups of semiabelian varieties over large algebraic extensions of a finitely generated field of characteristic zero. We consider two types of algebraic extensions in this paper; one is of extensions obtained by adjoining the coordinates of certain points of various semiabelian varieties; the other is of extensions obtained as the fixed subfield in an algebraically closed field by a finite number of automorphisms. Some of such fields turn out to be new examples of Kummer-faithful fields which are not sub-$p$-adic. Among them, we find both examples of Kummer-faithful fields over which the Mordell--Weil group modulo torsion can be free of infinite rank and not free.