Rational points on algebraic curves in infinite towers of number fields

TL;DR

This paper investigates the growth of rational points on algebraic curves over infinite towers of number fields, providing explicit conditions for stability and generalizing existing theorems in Iwasawa theory.

Contribution

It offers explicit growth results for rational points in infinite towers and generalizes Imai's theorem to pro-$p$ extensions.

Findings

Explicit growth formulas for ivisors of points on curves

Conditions for stable rational points across towers

Generalization of Imai's theorem to broader extensions

Abstract

We study a natural question in the Iwasawa theory of algebraic curves of genus $>1$. Fix a prime number $p$. Let $X$ be a smooth, projective, geometrically irreducible curve defined over a number field $K$ of genus $g>1$, such that the Jacobian of $X$ has good ordinary reduction at the primes above $p$. Fix an odd prime $p$ and for any integer $n>1$, let $K_n^{(p)}$ denote the degree-$p^n$ extension of $K$ contained in $K(\mu_{p^{\infty}})$. We prove explicit results for the growth of $\#X(K_n^{(p)})$ as $n\rightarrow \infty$. When the Jacobian of $X$ has rank zero and the associated adelic Galois representation has big image, we prove an explicit condition under which $X(K_{n}^{(p)})=X(K)$ for all $n$. This condition is illustrated through examples. We also prove a generalization of Imai's theorem that applies to abelian varieties over arbitrary pro-$p$ extensions.