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 firsst time, unconditional examples of this phenomenon on geometrically nonsplit semiabelian varieties over number fields. 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 $\beta:A^\vee\to A$. Assume that $\delta$ is an isogeny and that the cyclic subgroup generated by $\delta q$ is Zariski dense in $A$. Let $t\in\mathbf G_m(K)$ be a torsion point. Then $G_q$ is geometrically nonsplit and $P=R_\beta(q)+\iota(t)$ has Zariski-dense cyclic orbit. Put $\delta=\beta-\widehat{\beta}$. If $e_\delta$ denotes the exponent of $\ker\delta$ and $h=\operatorname{ord}(t)$, define $$ N_{\delta,t} := \prod_{\ell} \ell^{ \max\left\{ 0, \left\lceil \frac{v_\ell(h)-2v_\ell(e_\delta)}{2} \right\rceil \right\}}. $$ If $N_{\delta,t}>1$, then $N_{\delta,t}\mid d_v(P)$ for all but finitely many finite places $v$, where $d_v(P)$ denotes the order of the reduction of $P$. Consequently, there exists a squarefree integer $Q>1$ such that $$ (n,Q)=1 \quad\Longrightarrow\quad \mathfrak d_{\mathcal N}(nP) = \mathfrak d_{\mathcal N}(P), $$ where $\mathfrak d_{\mathcal N}$ denotes the full denominator ideal on the N\'eron lft-model $\mathcal N$.