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.