This website requires JavaScript.

p-adic equidistribution of CM points.

Daniel Disegni
arXiv: Number Theory
Apr 2019
摘要
Let $X$ be a modular curve and consider a sequence of Galois orbits of CM points in $X$, whose $p$-conductors tend to infinity. Its equidistribution properties in $X({\bf C})$ and in the reductions of $X$ modulo primes different from $p$ are well understood. We study the equidistribution problem in the Berkovich analytification $X_{p}^{\rm an}$ of $X_{{\bf Q}_{p}}$. We partition the set of CM points of sufficiently high conductor in $X_{{\bf Q}_{p}}$ into finitely many \emph{basins} $B_{V}$, indexed by the irreducible components $V $ of the mod-$p$ reduction of the canonical model of $X$. We prove that a sequence $z_{n}$ of local Galois orbits of CM points with $p$-conductor going to infinity has a limit in $X_{p}^{\rm an}$ if and only if it is eventually supported in a single basin $B_{V}$. If so, the limit is the unique point of $X_{p}^{\rm an}$ whose mod-$p$ reduction is the generic point of $V$. The result is proved in the more general setting of Shimura curves over totally real fields. The proof combines Gross's theory of quasicanonical liftings with a new formula for the intersection numbers of CM curves and vertical components in a Lubin--Tate space.
展开全部
图表提取

暂无人提供速读十问回答

论文十问由沈向洋博士提出,鼓励大家带着这十个问题去阅读论文,用有用的信息构建认知模型。写出自己的十问回答,还有机会在当前页面展示哦。

Q1论文试图解决什么问题?
Q2这是否是一个新的问题?
Q3这篇文章要验证一个什么科学假设?