发表机构
School of Mathematics, University of Edinburgh（爱丁堡大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对几何CM雅可比的亏格≥2的Q上光滑射影曲线，证明了Bloch-Kato Selmer概形的维数不等式，回答了Kim猜想的特殊情形，核心方法基于Iwasawa理论与特征重数界估。
AI 中文摘要
设$X/\boldsymbol{\text{Q}}$为亏格$g\boldsymbol{\text{≥}}2$的光滑射影几何整曲线，带有点$b\boldsymbol{\text{∈}}X(\boldsymbol{\text{Q}})$，且其雅可比簇为几何复乘（CM）型。取一个奇素数$p$作为好约化素数，记$U$为$X$的平展幂单$\boldsymbol{\text{Q}}_p$-基本群。我们证明，对所有足够大的$n$，Bloch--Kato Selmer概形满足$\boldsymbol{\text{Q}}_p$-维数不等式$\boldsymbol{\text{dim}}_{\boldsymbol{\text{Q}}_p} H^1_f(G_T, U_n) < \boldsymbol{\text{dim}}_{\boldsymbol{\text{Q}}_p}H^1_f(G_p, U_n)$，且二者比值的上极限至多为$1/2$。其中$U_n$是下中心序列商，$T$是包含$p$和坏约化素数的有限集合。这一结果回答了Kim提出的一个猜想的特殊情形。我们遵循Coates--Kim的Iwasawa理论方法，将问题归约为对$n$次的某些特征重数进行界估。随后我们定义了一个扭曲特征格$B$及其$p$-adic完备化$B_p$。我们的核心思路是，将归一化重数视为$\boldsymbol{\text{μ}}_n(\boldsymbol{\text{\braceleft}} x \boldsymbol{\text{∈}} B_p: f(x,n)\boldsymbol{\text{=}}0\boldsymbol{\text{\braceright}})$，其中$f$是来自Iwasawa理论的$p$-adic解析函数，且测度序列$(\boldsymbol{\text{μ}}_n)_n$弱收敛于$B_p$上的Haar概率测度$\boldsymbol{\text{μ}}$，后者是均匀测度的$p$-adic类比。我们通过考虑$B_p$的特征$\boldsymbol{\text{η}}$，并将$\boldsymbol{\text{∫}}_{B_p} \boldsymbol{\text{η}} \boldsymbol{\text{ d}}\boldsymbol{\text{μ}}_n$与辛算子$h_\boldsymbol{\text{η}}$诱导的算子迹及特征公式进行比较来证明收敛性。对$p$-adic轨迹序列的紧性论证给出了我们的主要Iwasawa估计，再结合Poitou-Tate对偶、整体欧拉特征公式及Hodge滤子估计，最终得到该维数不等式。
英文摘要
Let $X/\mathbb{Q}$ be a smooth, projective, geometrically integral curve of genus $g \geq 2$ with a point $ b \in X(\mathbb{Q})$ and geometrically CM Jacobian. Fix an odd prime $p$ of good reduction, and let $U$ be the étale pro-unipotent $\mathbb{Q}_p$-fundamental group of $X$. We prove, for all $n \gg 0,$ the inequality $\dim_{\mathbb{Q}_p} H^1_f(G_T, U_n) < \dim_{\mathbb{Q}_p}H^1_f(G_p, U_n)$ of Bloch--Kato Selmer schemes, with their ratio having limit superior at most $1/2$. Here $U_n$ is the lower central series quotient, and $T$ a finite set containing $p$ and the primes of bad reduction. This answers a particular case of a conjecture by Kim. Following the Iwasawa-theoretic method of Coates--Kim, we reduce the problem to bounding certain character multiplicities at degree $n$. We then define a twisted character lattice $B$ and its $p$-adic completion $B_p.$ Our main idea is to regard normalized multiplicities as $μ_n(\lbrace x \in B_p: f(x,n)=0\rbrace)$ for a $p$-adic analytic function $f$ coming from Iwasawa theory and a sequence of measures $(μ_n)_n$ weakly converging to the Haar probability measure $μ$ on $B_p,$ which is the $p$-adic analogue of the uniform measure. We prove convergence by considering characters $η$ of $B_p$ and comparing $\int_{B_p} η\ dμ_n$ to traces of operators induced by symplectic operators $h_η$, followed by a character formula. A compactness argument for sequences of $p$-adic loci gives our main Iwasawa estimate. Poitou-Tate duality, the global Euler characteristic formula, and Hodge filtration estimates then give the dimension inequality.
Comments15 pages
