Lower bound for the canonical on Abelian varieties over totally $p$-adic extensions

Technical Summary: Lower Bound for the Canonical Height on Abelian Varieties over Totally $p$-adic Extensions

Problem Statement
The paper addresses the problem of establishing absolute lower bounds for the canonical (Néron–Tate) height of non-torsion points on abelian varieties defined over number fields, specifically when these points lie in infinite algebraic extensions. The central focus is the Bogomolov property, which asserts that the set of non-torsion points in an infinite extension $K$ has a strictly positive lower bound on their height:
$$\liminf_{P \in A(K) \setminus A_{\text{tor}}(K)} \hat{h}(P) > 0.$$
While this property is known for totally real fields (Zhang), maximal abelian extensions (Baker–Silverman), and certain totally $v$-adic fields in the totally degenerate case (Gubler), a general lower bound for totally $p$-adic extensions in the good reduction case for higher-dimensional abelian varieties remained an open question. Specifically, the author addresses a question posed by V. Dimitrov regarding the Bogomolov property for totally $p$-adic points of abelian varieties with good reduction at $p$.

Methodology
The proof employs a geometric adaptation of techniques from transcendental number theory, specifically the auxiliary function method combined with zero estimates. The core strategy involves the following steps:

1. Reduction to Simple Varieties: Using Poincaré's complete reducibility theorem, the problem is reduced to the case where the abelian variety $A$ is geometrically simple and the line bundle $L$ is very ample.
2. Construction of an Auxiliary Section: The author constructs a non-zero auxiliary section $F$ in the space of global sections $\Gamma(A \times A, M \otimes L)$ (where $M = L \boxtimes L$) using Siegel's Lemma. This section is designed to vanish to a high order $T_0$ at the origin of the abelian variety. The coefficients of $F$ are chosen to have small height.
3. Height Estimates and Liouville's Inequality: For a point $P \in A(K)$, the paper analyzes the value of $F$ and its derivatives at $P$. If $P$ reduces to the identity modulo many places of a specific norm $q$, the local values of the derivatives are small. Conversely, if the global height of $P$ is small, Liouville's inequality provides a lower bound on the product of these local values.
4. Zero Estimates (Philippon's Theorem): To prevent the auxiliary function $F$ from vanishing on too many points of small height, the author applies Philippon's zero estimate. This theorem bounds the number of zeros of a section vanishing to a certain order along a subvariety. Crucially, because $A$ is assumed simple, any proper algebraic subgroup is trivial, forcing the zero set to be finite and controlled.
5. Parameter Selection: The proof involves a delicate balancing of parameters: the degree $L$ of the section, the order of vanishing $T_0$, and the order of the first non-vanishing derivative $T_f$. These are optimized to derive a contradiction if the set of points with height below a certain threshold is infinite.

Key Results
The paper establishes the following main theorems:

• Theorem 1.1 (Totally $p$-adic Extensions): There exists a finite set of primes $\mathcal{P}$ such that for any asymptotically positive extension $K/\mathbb{Q}$ with $\psi_{p^f} > 0$ for $p \notin \mathcal{P}$, the canonical height satisfies:
  $$\liminf_{P \in A(K)} \hat{h}(P) \geq C(A, K) \frac{\psi_{p^f} \log(p^f)}{(p^f)^{2g}}.$$
  Here, $\psi_q$ measures the density of prime ideals of norm $q$ in the extension.

• Corollary 1.2 (Maximal Totally $p$-adic Extension): For the maximal totally $p$-adic extension $\mathbb{Q}(p)$, for all but finitely many primes $p$:
  $$\liminf_{P \in A(\mathbb{Q}(p))} \hat{h}(P) \geq C(A) \frac{\log p}{p^{2g}}.$$
  This confirms the Bogomolov property for $A(\mathbb{Q}(p))$ in the good reduction case.

• Theorem 1.3 (General Asymptotically Positive Extensions): For a general infinite extension $K/K$ that is asymptotically positive, the lower bound is a sum over all relevant prime powers:
  $$\liminf_{P \in A(K)} \hat{h}(P) \geq C'(A, K) \sum_{q=p^k, p \notin \mathcal{P}} \psi_q \frac{\log q}{q^{2g}}.$$

Significance and Claims
The paper claims to provide the first instance where the Bogomolov property is established for totally $p$-adic points of an abelian variety in the good reduction case. Previous results by Gubler covered the totally degenerate case, and results by Chambert-Loir established the property only for a Zariski dense open subset in the good reduction case.

The author emphasizes that the result is significant because:

1. It answers a specific question raised by V. Dimitrov.
2. It extends the understanding of the Bogomolov property to higher-dimensional abelian varieties ($g > 1$) in non-Galois settings (asymptotically positive extensions).
3. It provides an explicit lower bound that depends on the local behavior of the extension (via $\psi_q$) and the dimension of the variety, combining contributions from all places in the bound.

The work builds upon the foundational equidistribution theorems of Szpiro-Ullmo-Zhang and the auxiliary function techniques of David and Hindry, adapting them to the specific arithmetic constraints of totally $p$-adic fields.