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

Technical Summary: Ribet Points, Geometric Divisibility Sequences, and Order of Reductions on Semiabelian Varieties

Problem Statement
The paper addresses a longstanding open problem regarding the behavior of geometric divisibility sequences attached to Zariski-dense points on irreducible commutative algebraic groups of dimension at least two. Specifically, it investigates Silverman's Conjecture 1.2, which posits that for a group scheme $G/\mathbb{Z}$ with a generic fiber $G_{\mathbb{Q}}$ that is an irreducible commutative algebraic group of dimension $\ge 2$ with no unipotent part, and for a point $P \in G(\mathbb{Z})$ with a Zariski-dense cyclic orbit, the equality $D_{nP} = D_P$ (where $D_P$ is the denominator ideal) holds for infinitely many positive integers $n$.

This conjecture stands in logical tension with the Silverman–Cheon–Hahn Theorem (Theorem 1.1), which establishes that for elliptic curves (dimension 1), every sufficiently large integer $n$ appears as the exact order of reduction for some prime. While the dimension-one case is complete, the behavior in higher dimensions, particularly for semiabelian varieties, was unknown. The paper seeks to determine if the analogue of the Silverman–Cheon–Hahn theorem holds for higher-dimensional semiabelian varieties or if the exact recurrence predicted by Silverman's conjecture occurs.

Methodology
The author constructs unconditional examples of geometrically nonsplit semiabelian varieties over number fields where the geometric divisibility sequence returns to its initial value infinitely often. The methodology relies on the arithmetic properties of Ribet points on extensions of abelian varieties by the multiplicative group $\mathbb{G}_m$.

1. Geometric Setup: The paper considers a semiabelian variety $G_q$ defined by an extension $1 \to \mathbb{G}_m \to G_q \to A \to 0$, represented by a point $q \in A^\vee(K)$.
2. Ribet Sections: Utilizing the normalized Poincaré biextension, the author defines a Ribet point $R_\beta(q) \in G_q(K)$ associated with a homomorphism $\beta: A^\vee \to A$. The antisymmetric part $\delta = \beta - \hat{\beta}$ plays a crucial role.
3. Torsion Translation: The core construction involves translating the Ribet point by a fixed torsion point $t$ in the toric kernel: $P = R_\beta(q) + \iota(t)$.
4. Arithmetic Amplification: A key technical lemma (Lemma 3.3) demonstrates that if the order of the Ribet point is quadratically bounded by the order of its projection (a property derived from the Weil pairing), then translating by a torsion point forces a specific divisibility condition on the order of the resulting point $P$.
5. Local-to-Global Argument: By combining the finite-group amplification with the properties of denominator ideals on Néron models, the author proves that for a specific integer $N_{\delta, t}$, the reduction order $d_v(P)$ is divisible by $N_{\delta, t}$ for all but finitely many places $v$.
6. Descent: To obtain examples over $\mathbb{Q}$, the paper employs a quadratic descent technique (Theorem 4.3). This involves constructing the extension over a quadratic field $L$ where the necessary antisymmetric endomorphisms exist, and then descending the extension and the point to $\mathbb{Q}$ by pairing the Galois action on the endomorphism with the sign character of a norm-one torus.

Key Contributions and Results

• Unconditional Examples: The paper provides the first unconditional examples of geometrically nonsplit semiabelian varieties over number fields where the geometric divisibility sequence returns to its initial value infinitely often.
• Theorem 1.4 (General Construction): For a positive-dimensional abelian variety $A$, a homomorphism $\beta$ such that $\delta = \beta - \hat{\beta}$ is an isogeny, and a torsion point $t$, the point $P = R_\beta(q) + \iota(t)$ has the property that a specific integer $N_{\delta, t}$ divides the order of reduction $d_v(P)$ for almost all places. Consequently, if $(n, Q) = 1$ for a squarefree integer $Q$ divisible by the radical of $N_{\delta, t}$, then the denominator ideal satisfies $\mathfrak{d}_{\mathbb{N}}(nP) = \mathfrak{d}_{\mathbb{N}}(P)$.
• Theorem 1.5 (Surface over $\mathbb{Q}$): The author explicitly constructs a geometrically nonsplit semiabelian surface $G_{surf}/\mathbb{Q}$ (an extension of an elliptic curve $E: y^2 = x^3 - 2$ by the norm-one torus $R^1_{L/\mathbb{Q}}\mathbb{G}_m$ with $L=\mathbb{Q}(\sqrt{-3})$) and a point $P_{surf} \in G_{surf}(\mathbb{Q})$ with a Zariski-dense cyclic orbit. For this point, the eventual reduction divisor is $U(P_{surf}) = 2$. This implies that for all $n$ coprime to a fixed integer, $D_{nP_{surf}} = D_{P_{surf}}$.
• Theorem 4.9 (Threefold over $\mathbb{Q}$): A geometrically nonsplit semiabelian threefold over $\mathbb{Q}$ is constructed using the elliptic curve $E: y^2 + y = x^3 + x^2 - 2x$ (conductor 389). The resulting point $P$ satisfies $D_P = 1$ and $U(P) = 2$, confirming the conjecture's conditions for a threefold.
• Dimensional Generality: Theorem 4.16 extends these results to every dimension $d \ge 2$ over $\mathbb{Q}$ by taking products with split tori, showing that the phenomenon is not restricted to low dimensions.
• Density Results: The paper establishes that the set of indices $n$ for which $D_{nP} = D_P$ has a lower natural density of at least $1 - 1/N_{\delta, t}$. Conversely, the set of exact reduction orders (primitive divisor indices) has an upper density of at most $1/N_{\delta, t}$.

Significance and Claims
The paper demonstrates that the analogue of the Silverman–Cheon–Hahn theorem fails sharply in the category of semiabelian varieties, while simultaneously constructing explicit instances where Silverman's Conjecture 1.2 holds. Specifically:

1. Failure of Exact Order Realization: Unlike the elliptic curve case, there exist dense points on semiabelian varieties where the set of exact reduction orders is sparse (density $\le 1/N$), meaning many integers never appear as exact orders of reduction.
2. Validation of Silverman's Conjecture for Specific Instances: The constructed points satisfy the conditions of Silverman's Conjecture 1.2, proving that the "exact return" phenomenon ($D_{nP} = D_P$) occurs infinitely often, and in fact, for a set of indices with positive density. The paper notes that while the conjecture remains open in its full generality over number fields, these examples provide the first unconditional verification for geometrically nonsplit semiabelian varieties.
3. Role of Ribet Points: The work highlights that Ribet sections (specifically their torsion-shifted versions) act as the obstruction to the existence of primitive divisors for all large $n$. This contrasts with split semiabelian varieties (products of abelian varieties and tori), where Perucca's theorem implies that the component number of the orbit closure dictates the universal divisor, which is 1 for connected dense points.
4. Geometric Nonsplitting: The phenomenon is intrinsically linked to geometrically nonsplit extensions. The paper notes that for split products, the component number of the orbit closure equals the greatest integer dividing almost every reduction order, preventing the construction of such "missing order" sets. The nonsplit nature of the extension, combined with the specific arithmetic of the Ribet point, allows the eventual reduction divisor $U(P)$ to be strictly greater than 1 (e.g., $U(P)=2$), even when the orbit closure is connected.

The paper concludes that the "missing datum" (the universal divisor of reduction orders) is genuinely mixed, carried by the extension class, the biextension lift, and the torsion translation, and that this behavior is absent in pure abelian varieties or split products.