Ribet Points, geometric divisibility sequence and order of reductions on semiabelian varieties

Abstract

Silverman conjectured that the geometric divisibility sequence attached to a Zariski-dense point on an irreducible commutative algebraic group of dimension at least two, with no unipotent part, returns to its initial value infinitely often. We construct, for the first time, unconditional examples of this phenomenon on geometrically nonsplit semiabelian varieties over number fields using torsion translates of normalized Ribet points. More precisely, let $A/K$ be a positive-dimensional abelian variety over number field $K$, let \[ 1\longrightarrow\mathbf G_m\xrightarrow{\iota}G_q\xrightarrow{\pi}A \longrightarrow0 \] be the extension represented by $q\in A^\vee(K)$, and let $R_\beta(q)\in G_q(K)$ be the normalized Ribet point associated with a homomorphism $\beta:A^\vee\to A$. We set $\delta:=\beta-\widehat\beta$ and assume that $\delta$ is an isogeny and that $\mathbf Z(\delta q)$ is Zariski dense in $A$. For a torsion point $t\in\mathbf G_m(K)_{\mathrm{tors}}$, identify $t$ with $\iota(t)$ and set $P=R_\beta(q)+t$. Then $P$ has Zariski-dense cyclic orbit in the geometrically nonsplit extension $G_q$. There exists an explicit integer $N_{\delta,t}$ such that if $N_{\delta,t}>1$, then $N_{\delta,t}\mid \mathrm{ord}(\overline P_v)$ at all but finitely many places $v$. Consequently, there is a squarefree integer $Q_P>1$ such that \[ (n,Q_P)=1 \quad\Longrightarrow\quad \mathfrak d_{\mathcal N}(nP)=\mathfrak d_{\mathcal N}(P), \] where $\mathfrak d_{\mathcal N}$ denotes the full denominator ideal on the N\'eron lft-model $\mathcal N$. In particular, we construct explicitly a geometrically nonsplit semiabelian surface $G/\mathbf Q$ and a semiabelian threefold over $\mathbf Q$ satisfying the Silverman conjecture. It follows that we can construct instances satisfying the Silverman conjecture for every dimension at least two.