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On the growth of Tate-Shafarevich groups of $p$-supersingular abelian varieties of ${\rm GL}_2$-type over $\mathbb{Z}_p$-extensions of number fields

Erman Isik, Antonio Lei

Abstract

We study the boundedness of the Mordell-Weil rank and the growth of the $v$-primary part of the Tate-Shafarevich group of $p$-supersingular abelian varieties of ${\rm GL}_2$-type with real multiplication over $\mathbb{Z}_p$-extensions of number fields, where $v$ is a prime lying above $p$. Building on the work of Iovita-Pollack in the case of elliptic curves, under precise ramification and splitting conditions on $p$, we construct explicit systems of local points using the theory of Lubin-Tate formal groups. We then define signed Coleman maps, which in turn allow us to formulate and analyse signed Selmer groups. Assuming these Selmer groups are cotorsion, we prove that the Mordell-Weil groups are bounded over any subextensions of the $\mathbb{Z}_p$-extension and provide an asymptotic formula for the growth of the $v$-primary part of the Tate-Shafarevich groups. Our results extend those of Kobayashi, Pollack, and Sprung on $p$-supersingular elliptic curves.

On the growth of Tate-Shafarevich groups of $p$-supersingular abelian varieties of ${\rm GL}_2$-type over $\mathbb{Z}_p$-extensions of number fields

Abstract

We study the boundedness of the Mordell-Weil rank and the growth of the -primary part of the Tate-Shafarevich group of -supersingular abelian varieties of -type with real multiplication over -extensions of number fields, where is a prime lying above . Building on the work of Iovita-Pollack in the case of elliptic curves, under precise ramification and splitting conditions on , we construct explicit systems of local points using the theory of Lubin-Tate formal groups. We then define signed Coleman maps, which in turn allow us to formulate and analyse signed Selmer groups. Assuming these Selmer groups are cotorsion, we prove that the Mordell-Weil groups are bounded over any subextensions of the -extension and provide an asymptotic formula for the growth of the -primary part of the Tate-Shafarevich groups. Our results extend those of Kobayashi, Pollack, and Sprung on -supersingular elliptic curves.

Paper Structure

This paper contains 13 sections, 30 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Statement of the main theorem
  3. Organization
  4. Outlook
  5. Lubin--Tate extensions and formal groups attached to abelian varieties
  6. Construction of local points over Lubin--Tate extensions
  7. Signed Coleman maps
  8. Signed Selmer groups
  9. Definition and basic properties of the Kobayashi rank
  10. Proof of Theorem \ref{['maintheorem']}
  11. Calculating Kobayashi ranks of certain auxiliary modules
  12. Final steps of the proof of Theorem \ref{['maintheorem']}
  13. Honda-type of formal groups

Key Result

Theorem A

Let $A$ be an abelian variety of ${\rm GL}_2$-type over ${\mathbb Q}$ associated to a weight-two newform with trivial nebentype. Let $K$ be a number field and $p$ be an odd prime satisfying Hypothesis splittingtype. Furthermore, assume that the signed Selmer groups ${\rm Sel}^{\mathbf s}_v(A/K_\inft

Theorems & Definitions (77)

  • Theorem A
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • ...and 67 more