From 5ea79c7ae0d44686f1df05c4a016652afbe58968 Mon Sep 17 00:00:00 2001 From: Sebastiano Tronto Date: Sun, 14 Jun 2026 09:58:21 +0200 Subject: Initial commit --- l-multiples_l-torsion/l-multiples_l-torsion.tex | 148 ++++++++++++++++++++++++ 1 file changed, 148 insertions(+) create mode 100755 l-multiples_l-torsion/l-multiples_l-torsion.tex (limited to 'l-multiples_l-torsion/l-multiples_l-torsion.tex') diff --git a/l-multiples_l-torsion/l-multiples_l-torsion.tex b/l-multiples_l-torsion/l-multiples_l-torsion.tex new file mode 100755 index 0000000..a07078b --- /dev/null +++ b/l-multiples_l-torsion/l-multiples_l-torsion.tex @@ -0,0 +1,148 @@ +\documentclass[10pt,a4paper]{article} +\usepackage[utf8]{inputenc} +\usepackage{amsmath} +\usepackage{amsthm} +\usepackage[all]{xy} +\usepackage{amsfonts} +\usepackage{color} +\usepackage{amssymb} +\usepackage{float} +\usepackage[a4paper, top=3cm, bottom=3cm, left=2.5cm, right=2.5cm]{geometry} + +\DeclareMathOperator{\alg}{alg} +\DeclareMathOperator{\obj}{Obj} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\hol}{Hol} +\DeclareMathOperator{\aut}{Aut} +\DeclareMathOperator{\gal}{Gal} +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\res}{res} +\DeclareMathOperator{\im}{Im} +\DeclareMathOperator{\Id}{Id} +\DeclareMathOperator{\fib}{Fib} +\DeclareMathOperator{\spec}{Spec} +\DeclareMathOperator{\proj}{Proj} +\DeclareMathOperator{\trdeg}{trdeg} +\DeclareMathOperator{\car}{char} +\DeclareMathOperator{\Frac}{Frac} +\DeclareMathOperator{\reduced}{red} +\DeclareMathOperator{\real}{Re} +\DeclareMathOperator{\imag}{Im} +\DeclareMathOperator{\vol}{vol} +\DeclareMathOperator{\den}{den} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\lcm}{lcm} +\DeclareMathOperator{\rad}{rad} +\DeclareMathOperator{\ord}{ord} +\DeclareMathOperator{\Br}{Br} +\DeclareMathOperator{\inv}{inv} +\DeclareMathOperator{\Nm}{Nm} +\DeclareMathOperator{\Tr}{Tr} +\DeclareMathOperator{\an}{an} +\DeclareMathOperator{\op}{op} +\DeclareMathOperator{\sep}{sep} +\DeclareMathOperator{\unr}{unr} +\DeclareMathOperator{\et}{\acute et} +\DeclareMathOperator{\ev}{ev} +\DeclareMathOperator{\gl}{GL} +\DeclareMathOperator{\SL}{SL} +\DeclareMathOperator{\mat}{Mat} +\DeclareMathOperator{\ab}{ab} + +\newcommand{\grp}{\textsc{Grp}} +\newcommand{\set}{\textsc{Set}} +\newcommand{\x}{\mathbf{x}} +\newcommand{\naturalto}{\overset{.}{\to}} +\newcommand{\qbar}{\overline{\mathbb{Q}}} +\newcommand{\zbar}{\overline{\mathbb{Z}}} + +\newcommand{\pro}{\mathbb{P}} +\newcommand{\aff}{\mathbb{A}} +\newcommand{\quat}{\mathbb{H}} +\newcommand{\rea}{\mathbb{R}} +\newcommand{\kiu}{\mathbb{Q}} +\newcommand{\F}{\mathbb{F}} +\newcommand{\zee}{\mathbb{Z}} +\newcommand{\ow}{\mathcal{O}} +\newcommand{\mcx}{\mathcal{X}} +\newcommand{\mcy}{\mathcal{Y}} +\newcommand{\mcs}{\mathcal{S}} +\newcommand{\mca}{\mathcal{A}} +\newcommand{\mcb}{\mathcal{B}} +\newcommand{\mcf}{\mathcal{F}} +\newcommand{\mcg}{\mathcal{G}} +\newcommand{\mct}{\mathcal{T}} +\newcommand{\mcq}{\mathcal{Q}} +\newcommand{\mcr}{\mathcal{R}} +\newcommand{\adl}{\mathbf{A}} +\newcommand{\mbk}{\mathbf{k}} +\newcommand{\m}{\mathfrak{m}} +\newcommand{\p}{\mathfrak{p}} + +\newcommand{\kbar}{\overline{K}} + +\newtheorem{lemma}{Lemma} +\newtheorem{proposition}[lemma]{Proposition} +\newtheorem{corollary}[lemma]{Corollary} +\theoremstyle{definition} +\newtheorem{remark}[lemma]{Remark} + +\title{Rational $\ell$-multiples of points over the $\ell$-torsion field} +\author{Sebastiano Tronto} + +\begin{document} + +\maketitle + +Let $\ell$ be a rational prime and let $A$ be an abelian variety of dimension $d$ over a number field $K$. Let $K_\ell=K(A[\ell])$ and let $\mathcal{T}_1=\gal(K_\ell\,|\,K)$. + +\begin{remark} +For $m\geq 1$ we have $\#\gl_m(\F_\ell)=\prod_{i=0}^{\ell-1}(\ell^m-\ell^i)$. In fact, elements of $\gl_m(\F_\ell)$ are in bijection with bases of $\F^m_\ell$, and counting basis of a vector space over a finite field is a simple combinatorics exercise: first we pick any non-zero vector ($\ell^n-1$ possibilities), then we pick any vector that is not in the $\F_\ell$-span on the first one ($\ell^n-\ell$ possibilities), then a third one that is not in the span of the first two... + +In particular, $v_\ell\left(\#\gl_m(\F_\ell)\right)=\frac{1}{2}m(m-1)$. +\end{remark} + +Compare the next lemma with \cite{jr}, Lemma 3.7. + +\begin{lemma} +There is an exact sequence +\begin{align*} +0\to \ell A(K) \to A(K)\cap \ell A(K_{\ell})\to H^1(\mathcal{T}_1,A[\ell]). +\end{align*} +In particular, if $H^1(\mathcal{T}_1,A[\ell])=0$ we have $A(K)\cap \ell A(K_\ell)=\ell A(K)$. +\begin{proof} +Consider the short exact sequence of $\mathcal{T}_1$-modules +\begin{align*} +0\to A[\ell](K_{\ell})\to A(K_{\ell})\to \ell A(K_{\ell})\to 0 +\end{align*} +and the induced long exact sequence in cohomology (i.e. take $H^*(\mathcal{T}_1,-)$) +\begin{align*} +0\to A[\ell](K)\to A(K)\to A(K)\cap\ell A(K_{\ell})\to H^1(\mathcal{T}_1,A[\ell])\to\cdots +\end{align*} +and the thesis follows by noticing that $A(K)/A[\ell](K)\cong \ell A(K)$. +\end{proof} +\end{lemma} + +This leads us to study the group $H^1(\mathcal{T}_1,A[\ell])$. In particular, we would like to know in which cases it is trivial (and so we are happy). In particular, the case of elliptic curves has been completely solved if $K=\kiu$, and there are rather complete results if $K\cap \kiu(\zeta_\ell)=\kiu$ (see \cite{lawson}). + +It is in fact possible that some point $\alpha\in A(K)$ is not $\ell$-divisible in $A(K)$, but becomes $\ell$-divisible in $A(K_\ell)$. The smallest example I have found is the point +$(23769/400, 3529853/8000)$ on the elliptic curve over $\kiu$ with Cremona Label $17739g1$. A gp script that finds all the 12 examples of elliptic curves with conductor $<10^5$ with a generator of the free part of the group of rational points that becomes $3$-divisible over $K_3$ can be found in \texttt{test3.gp}. +\\ + +The question now becomes: ``How much'' can a point $\alpha\in A(K)$ become $\ell$-divisible in $A(K_{\ell^\infty})$? That is, can we find an (explicit) $N$ such that there is no $\beta\in A(K_{\ell^\infty})$ with $\alpha=\ell^n\beta$ for $n\geq N$? + +%\textbf{We restrict to the case where $A=E$ is an elliptic curve without complex multiplication}. Following the proof of Theorem 5.2 of \cite{jr} (using Lemma 3.6 of the same article), we see that $A(K)\cap \ell A(K_{\ell^n})=A(K)\cap \ell A(K_{\ell^{n-1}})$ for any $n\geq n_0$, where $n_0$ is such that $A$ satisfies maximal growth of the torsion part starting from $n_0$. So we can replace $K_{\ell^{\infty}}$ with the finite extension $K_{\ell^{n_0}}$. + +%Second question: work in progress! (Latest idea: use heights and the fact that that $P$ does not become ``more divisible'' by $\ell$ after a certain point; see descent part in the proof of Mordell-Weil). + +(If we could say $A(K)\cap\ell A(K_{\ell^\infty})=A(K)\cap\ell A(K_{\ell^{n_0}})$ for some $n_0$, then we could use Petsche's results to explicitly bound $N$ in terms of the height of $P$) + +\begin{thebibliography}{[99]} +\bibitem{jr} R. Jones, J. Rouse, \emph{Galois Theory of Iterated Endomorphisms}, preprint(?). +%\bibitem{lom} D. Lombardo, A. Perucca, \emph{Reductions of Points on Algebraic Groups}, preprint. +%\bibitem{milne-ft} J. S. Milne, \emph{Fields and Galois Theory}, Online notes. +\bibitem{lawson} T. Lawson, C. Wuthrich, \emph{Vanishing of some Galois cohomology groups for elliptic curves}, preprint(?). +\end{thebibliography} + +\end{document} \ No newline at end of file -- cgit v1.3