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A refined non-vanishing of the $p$-adic logarithm of a rational point on an abelian variety

Ashay Burungale, Christopher Skinner, Xin Wan

Abstract

Inspired by a beautiful formula of Bertolini, Darmon, and Prasanna -- the oft-termed BDP formula -- we address questions about the non-vanishing of non-torsion points under $p$-adic logarithms of abelian varieties. We largely consider situations most applicable to ${\mathrm GL}_2$-type abelian varieties associated with Hilbert modular newforms and Heegner points. Not surprisingly, the main tool employed is the $p$-adic analytic subgroup theorem.

A refined non-vanishing of the $p$-adic logarithm of a rational point on an abelian variety

Abstract

Inspired by a beautiful formula of Bertolini, Darmon, and Prasanna -- the oft-termed BDP formula -- we address questions about the non-vanishing of non-torsion points under -adic logarithms of abelian varieties. We largely consider situations most applicable to -type abelian varieties associated with Hilbert modular newforms and Heegner points. Not surprisingly, the main tool employed is the -adic analytic subgroup theorem.
Paper Structure (22 sections, 12 theorems, 41 equations)

This paper contains 22 sections, 12 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. The BDP formula
  3. The question
  4. The case of general $B_f$
  5. A representative result
  6. Some additional context
  7. Main results: statements
  8. The set-up
  9. Trace fields
  10. Main results
  11. Applications to ${\mathrm{GL}}_2$-type abelian varieties
  12. An Example
  13. Towards the $p$-part of the BSD formula for ${\mathrm{GL}}_2$-type abelian varieties
  14. The $p$-adic analytic subgroup theorem
  15. Proofs of the main results
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $A/\overline{{\mathbb {Q}}}$ be an abelian variety. Suppose there is a field $F\subset {\mathrm{End}}^0_{\overline{{\mathbb {Q}}}}(A)$ such that For any non-torsion $x\in A(\overline{{\mathbb {Q}}})$, we have for all $0\neq \omega\in \Omega_\sigma$ and all $\sigma\in \Sigma_F$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • ...and 14 more