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 $K(\mu _{p^{\infty }})$. 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.