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⟶GmxrightarrowιGqxrightarrowπA ⟶0 be the extension represented by q∈ A∨(K), and let Rβ(q)∈ Gq(K) be the normalized Ribet point associated with a homomorphism β:A∨→ A. We set δ:=β−β̂ and assume that δ is an isogeny and that Z(δq) is Zariski dense in A. For a torsion point t∈Gm(K)tors, identify t with ι(t) and set P=Rβ(q)+t. Then P has Zariski-dense cyclic orbit in the geometrically nonsplit extension Gq. There exists an explicit integer Nδ,t such that if Nδ,t>1, then Nδ,t| ord(P̅v) at all but finitely many places v. Consequently, there is a squarefree integer QP>1 such that (n,QP)=1 Longrightarrow d𝒩(nP)=d𝒩(P), where d𝒩 denotes the full denominator ideal on the Néron lft-model 𝒩. In particular, we construct explicitly a geometrically nonsplit semiabelian surface G/Q and a semiabelian threefold over Q satisfying the Silverman conjecture. It follows that we can construct instances satisfying the Silverman conjecture for every dimension at least two.
Related
BibTeX
@misc{nguyendang2026ribet,
title = {{Ribet Points, geometric divisibility sequence and order of reductions on semiabelian varieties}},
author = {Khai-Hoan Nguyen-Dang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.19742},
eprint = {2608.19742},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19742},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ribet Points, geometric divisibility sequence and order of reductions on semiabelian varieties
AU - Khai-Hoan Nguyen-Dang
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.19742
UR - https://arxiv.org/abs/2608.19742
AB - 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⟶G_mxrightarrowιG_qxrightarrowπA ⟶0 be the extension represented by q∈ A^∨(K), and let R_β(q)∈ G_q(K) be the normalized Ribet point associated with a homomorphism β:A^∨→ A. We set δ:=β−β̂ and assume that δ is an isogeny and that Z(δq) is Zariski dense in A. For a torsion point t∈G_m(K)_tors, identify t with ι(t) and set P=R_β(q)+t. Then P has Zariski-dense cyclic orbit in the geometrically nonsplit extension G_q. There exists an explicit integer N_δ,t such that if N_δ,t>1, then N_δ,t| ord(P̅_v) at all but finitely many places v. Consequently, there is a squarefree integer Q_P>1 such that (n,Q_P)=1 Longrightarrow d_𝒩(nP)=d_𝒩(P), where d_𝒩 denotes the full denominator ideal on the Néron lft-model 𝒩. In particular, we construct explicitly a geometrically nonsplit semiabelian surface G/Q and a semiabelian threefold over Q satisfying the Silverman conjecture. It follows that we can construct instances satisfying the Silverman conjecture for every dimension at least two.
ER -