On Iwasawa theory of abelian varieties over $\mathbb{Z}_p^2$-extension with applications to Diophantine stability and integally Diophantine extensions

1. Problem Statement and Context

The paper investigates the behavior of the Mordell-Weil rank of abelian varieties (specifically elliptic curves) over infinite towers of number fields, known as $\mathbb{Z}_p^2$-extensions. The central questions addressed are:

• Mazur's Growth Conjecture: How many $\mathbb{Z}_p$-subextensions within a $\mathbb{Z}_p^2$-extension allow for unbounded growth of the Mordell-Weil rank?
• Diophantine Stability: Under what conditions does an abelian variety $A$ satisfy $A(L) = A(F)$ for an extension $L/F$?
• Integrally Diophantine Extensions: When is the ring of integers of an extension $L$ Diophantine over the ring of integers of the base field $F$?

The study focuses on two distinct reduction scenarios at a fixed odd prime $p$:

1. Potentially Good Ordinary Reduction: The standard setting for Iwasawa theory.
2. Good Supersingular Reduction: A more complex case requiring "signed" Selmer groups (following Kobayashi and Kim).

2. Methodology

The author employs a combination of algebraic Iwasawa theory, control theorems, and arithmetic geometry.

• Algebraic Framework:

  • The paper utilizes the structure theory of finitely generated torsion modules over the Iwasawa algebra $\Lambda = \mathbb{Z}_p[[G]]$, where $G \cong \mathbb{Z}_p^2$.
  • It identifies subgroups $H_{a,b} \subset G$ corresponding to $\mathbb{Z}_p$-extensions $F_{a,b}$.
  • A key algebraic tool is Proposition 2.2, which provides an upper bound on the number of subgroups $H_{a,b}$ for which a torsion module $M$ fails to be torsion over the quotient algebra. This bound is expressed via the order of vanishing of the characteristic polynomial.

• Arithmetic Tools:

  • Mazur's Control Theorem: Used to relate Selmer groups over finite layers to those over infinite extensions, ensuring finite kernels and cokernels under specific ramification conditions.
  • Signed Selmer Groups: For the supersingular case, the author adapts the theory of Kobayashi and Kim, defining "plus/minus" norm groups and signed Selmer groups to handle the lack of ordinary reduction.
  • Shlapentokh's Theorem: A crucial link between rank stability and Diophantine properties. It states that if $L_1/L_2$ is an extension where the rank of an abelian variety remains constant and positive, then $L_1/L_2$ is integrally Diophantine.

• Assumptions:

  • The primary assumption is Mazur's Conjecture: The Pontryagin dual of the Selmer group $X(A/F_{cyc})$ is a torsion module over the cyclotomic Iwasawa algebra $\mathbb{Z}_p[[\Gamma]]$.
  • For elliptic curves over $\mathbb{Q}$, this is often unconditional due to deep results by Kato and Rubin.

3. Key Contributions and Results

A. Bounding Unbounded Rank Growth (The Ordinary Case)

Theorem 3.3 (Main Theorem):
Let $A$ be an abelian variety with potentially good ordinary reduction over a number field $F$. Let $F_\infty$ be a $\mathbb{Z}_p^2$-extension containing the cyclotomic $\mathbb{Z}_p$-extension $F_{cyc}$. Assuming $X(A/F_{cyc})$ is torsion, the set $\mathcal{M}(A/F_\infty)$ of $\mathbb{Z}_p$-subextensions where the Mordell-Weil rank is unbounded is finite.

• Bound: $|\mathcal{M}(A/F_\infty)| \leq \text{ord}_U(\text{char}_{\mathbb{Z}_p[[\Gamma]]} X(A/F_{cyc}))$.
• Improvement: This generalizes previous results (e.g., Gajek-Leonard et al.) from elliptic curves to abelian varieties and removes the restrictive assumption that the Selmer group dual must be a direct sum of cyclic modules.

B. Diophantine Stability and Integrally Diophantine Extensions

The paper applies the finiteness of unbounded rank extensions to prove the abundance of stable extensions.

• Theorem 5.1: If $F$ has at least one complex prime and $A$ satisfies the ordinary reduction and torsion assumptions, the set $D_n(A/F)$ (pairs of fields in a $\mathbb{Z}_p$-tower with degree $p^n$ where $A$ is Diophantine stable) is uncountable.
• Theorem 5.5: If additionally $\text{rank}_{\mathbb{Z}} A(F) > 0$, the set $I_n(F)$ (pairs where the extension is integrally Diophantine) is uncountable.
• Corollaries:
  • Corollary 5.2: Unconditional result for elliptic curves over $\mathbb{Q}$ with good ordinary reduction over finite abelian imaginary extensions.
  • Corollary 5.6: Unconditional result for integrally Diophantine extensions over finite abelian imaginary extensions of $\mathbb{Q}$.

C. The Supersingular Case

The author extends the analysis to elliptic curves with good supersingular reduction ($a_p = 0$) over imaginary quadratic fields.

• Theorem 7.8: An analog of Theorem 3.3 for signed Selmer groups. The number of subextensions with unbounded rank is bounded by the sum of the orders of vanishing of the characteristic polynomials of the signed Selmer groups over the cyclotomic extension.
• Theorem 7.10: An unconditional result stating that for such curves, the set of Diophantine stable extensions $D_n(E/F)$ is uncountable, provided specific conditions on the conductor or the finiteness of the $p$-primary Selmer group over the base field are met.

D. Refinement of Mazur's Growth Conjecture

The paper refines the understanding of the Mazur Growth Conjecture (predicting bounded rank growth in all but one specific $\mathbb{Z}_p$-extension).

• Proposition 4.1 & 4.3: Under specific conditions (e.g., $ord_U(\text{char}) = 1$), the Mordell-Weil rank is bounded in every $\mathbb{Z}_p$-extension of the base field except the anti-cyclotomic one. This improves upon prior work by removing structural assumptions on the Selmer module.

4. Significance and Implications

1. Generalization to Abelian Varieties: The results move beyond elliptic curves to general abelian varieties, broadening the scope of Iwasawa theoretic applications.
2. Removal of Structural Assumptions: A major technical achievement is the removal of the assumption that the Pontryagin dual of the Selmer group must be a direct sum of cyclic modules. This makes the results applicable to a wider class of arithmetic objects where the module structure is more complex.
3. Diophantine Applications: The paper provides a robust mechanism to generate uncountably many integrally Diophantine extensions. This supports the Relative Denef-Lipshitz Conjecture (that every finite extension of number fields is integrally Diophantine) by providing a vast supply of examples, even if the specific base field extension cannot always be explicitly constructed.
4. Unconditional Results: By leveraging deep theorems of Kato, Rubin, and Gross-Zagier, the author derives unconditional results for elliptic curves over $\mathbb{Q}$, bypassing the need for unproven conjectures in specific cases.
5. Supersingular Theory: The successful adaptation of these methods to the supersingular case using signed Selmer groups demonstrates the flexibility of the Iwasawa theoretic framework in handling non-ordinary reduction.

In summary, Meng Fai Lim's paper establishes a rigorous upper bound on the number of "bad" $\mathbb{Z}_p$-extensions (where rank grows unboundedly) and uses this to prove the existence of a vast, uncountable family of extensions that are Diophantine stable and integrally Diophantine, significantly advancing the understanding of the interplay between Iwasawa theory and Diophantine geometry.