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Summary

This study explores p-adic formal Manin-Mumford results for p-divisible formal groups, showing uniform bounds for certain points. Counter-examples to a full p-adic formal Mordell-Lang result are also presented.

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Area of Science:

  • Number Theory
  • Algebraic Geometry
  • p-adic Analysis

Background:

  • The Mordell-Lang conjecture and Manin-Mumford conjecture are foundational in number theory.
  • p-adic formal groups and their properties are crucial in modern arithmetic geometry.
  • Previous work established p-adic formal Manin-Mumford results for specific cases.

Purpose of the Study:

  • To generalize the author's p-adic formal Manin-Mumford results for n-dimensional p-divisible formal groups.
  • To investigate the behavior of finitely generated subgroups within these formal groups concerning closed subschemes.
  • To explore the implications for p-adic deformations and automorphic forms.

Main Methods:

  • Generalization of p-adic formal Manin-Mumford theorems.
  • Analysis of finitely generated subgroups and closed subschemes in p-divisible formal groups.
  • Construction of counter-examples to a full p-adic formal Mordell-Lang result.
  • Study of Zariski-density in p-adic deformations.

Main Results:

  • Uniform bounds are established for minimal orders n of points satisfying specific conditions, provided the subscheme does not contain a formal subgroup translate of positive dimension.
  • Counter-examples are provided, demonstrating limitations to a full p-adic formal Mordell-Lang result.
  • Consequences for the Zariski-density of automorphic objects in p-adic deformations are outlined.

Conclusions:

  • The research refines understanding of p-adic formal Manin-Mumford type problems and their connection to the Mordell-Lang conjecture.
  • The findings highlight the complexities and nuances of p-adic formal Mordell-Lang statements.
  • The study contributes to the theory of automorphic forms and p-adic deformations.