p-adic equidistribution of CM points

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Abstract

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(C) and in the reductions of X modulo primes different from p are well understood. We study the equidistribution problem in the Berkovich analytification Xanp of XQp.
We partition the set of CM points of sufficiently high conductor in XQp into finitely many explicit \emph{basins} BV, indexed by the irreducible components V of the mod-p reduction of the canonical model of X. We prove that a sequence zn of local Galois orbits of CM points with p-conductor going to infinity has a limit in Xanp if and only if it is eventually supported in a single basin BV. If so, the limit is the unique point of Xanp 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.
Original languageEnglish
StatePublished - 2019

Keywords

  • math.NT
  • math.AG
  • 11G15, 14K22

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