Exact realization of nonsquarefree eventual reduction divisors

Determine the exact eventual reduction divisor U(P) for the geometrically nonsplit semiabelian threefolds constructed by translating normalized Ribet points by torsion of nonsquarefree order, and in particular establish whether an arbitrary prescribed nonsquarefree integer can occur exactly as U(P).

Background

The paper proves that every integer M>1 can be forced to divide almost every reduction order d_v(P) for a suitable Zariski-dense point on a geometrically nonsplit semiabelian threefold. For squarefree M, it further proves exact realization U(P)=M, where U(P) is the greatest integer dividing almost every reduction order.

For a general nonsquarefree M, the construction yields only the bounds M | U(P) | H(M), with H(M)=∏ℓ ℓ{2rℓ−1} when M=∏ℓ ℓ{rℓ}. The unresolved issue is whether cancellation in the local factor ω_v ζ̄_v{m_v} can be controlled sufficiently to determine U(P) exactly and realize arbitrary prime-power exponents.

References

Determining $U(P{\mathrm{forc}}_M)$ exactly for nonsquarefree $M$ requires finer control of cancellation in $\omega_v\overline\zeta_v{\,m_v}$. The theorem proves that every integer is a universal forced divisor, but exact realization of arbitrary prime powers remains open.

— Ribet Points, geometric divisibility sequence and order of reductions on semiabelian varieties  (2608.19742 - Nguyen-Dang, 20 Aug 2026) in Remark following Theorem 4.20, Section 4, subsection “Primitive-divisor and densities”