Ribet点、几何整除序列与半阿贝尔簇上的约化阶
Ribet Points, geometric divisibility sequence and order of reductions on semiabelian varieties
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中文总结 AI 辅助
本文在数域几何非分裂半阿贝尔簇上构造了Silverman猜想所述现象的无条件例子，推导了相关整数整除关系，证明存在无平方因子整数满足分母理想不变的条件。
中文摘要 AI 辅助
Silverman猜想：对于特征零代数闭域上维数至少为2且无幂么部分的不可换代数群上的扎里斯基稠密点，其对应的几何整除序列会无穷多次回到初始值。本文首次在数域上的几何非分裂半阿贝尔簇上构造了该现象的无条件例子。具体而言，设$A/K$为数域$K$上的正维阿贝尔簇，$q\in A^\vee(K)$对应扩张$1\longrightarrow\mathbf G_m\xrightarrow{\iota}G_q\xrightarrow{\pi}A \longrightarrow0$，$R_\beta(q)\in G_q(K)$为与同态$\beta:A^\vee\to A$关联的正规化Ribet点。假设$\delta$是同源，且由$\delta q$生成的循环子群在$A$中扎里斯基稠密，$t\in\mathbf G_m(K)$为挠点，则$G_q$几何非分裂，且$P=R_\beta(q)+\iota(t)$具有扎里斯基稠密的循环轨道。令$\delta=\beta-\widehat{\beta}$，$e_\delta$表示$\ker\delta$的指数，$h=\operatorname{ord}(t)$，定义$N_{\delta,t}:= \prod_{\ell} \ell^{ \max\left\{ 0, \left\lceil \frac{v_\ell(h)-2v_\ell(e_\delta)}{2} \right\rceil \right\}}$。若$N_{\delta,t}>1$，则对几乎所有有限位$v$，$N_{\delta,t}\mid d_v(P)$，其中$d_v(P)$表示$P$的约化阶。因此存在无平方因子整数$Q>1$，使得当$(n,Q)=1$时，$\mathfrak d_{\mathcal N}(nP) = \mathfrak d_{\mathcal N}(P)$，其中$\mathfrak d_{\mathcal N}$表示Néron lft-模型$\mathcal N$上的全分母理想。
英文摘要
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 \[ 1\longrightarrow\mathbf G_m\xrightarrowιG_q\xrightarrowπA \longrightarrow0 \] be the extension represented by $q\in A^\vee(K)$, and let $R_β(q)\in G_q(K)$ be the normalized Ribet point associated with a homomorphism $β:A^\vee\to A$. We set $δ:=β-\widehatβ$ and assume that $δ$ is an isogeny and that $\mathbf Z(δq)$ is Zariski dense in $A$. For a torsion point $t\in\mathbf G_m(K)_{\mathrm{tors}}$, 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 $N_{δ,t}\mid \mathrm{ord}(\overline P_v)$ 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 Néron 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.
发表机构
- Morningside Center of Mathematics, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
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