Rational points on algebraic curves in infinite towers of number fields

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 $K(μ_{p^{\infty}})$. 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.