Unipotent Selmer dimensions for CM curves via Iwasawa loci

Abstract

Let $X/\mathbb{Q}$ be a smooth, projective, geometrically integral curve of genus $g \geq 2$ with a point $ b \in X(\mathbb{Q})$ and geometrically CM Jacobian. Fix an odd prime $p$ of good reduction, and let $U$ be the \'etale pro-unipotent $\mathbb{Q}_p$-fundamental group of $X$. We prove, for all $n \gg 0,$ the inequality $\dim_{\mathbb{Q}_p} H^1_f(G_T, U_n)<\dim_{\mathbb{Q}_p}H^1_f(G_p, U_n)$ of Bloch--Kato Selmer schemes, with their ratio having limit superior at most $1/2$. Here $U_n$ is the lower central series quotient, and $T$ a finite set containing $p$ and the primes of bad reduction. This answers a particular case of a conjecture by Kim. Following the Iwasawa-theoretic method of Coates--Kim, we reduce the problem to bounding certain character multiplicities at degree $n$. We then define a twisted character lattice $B$ and its $p$-adic completion $B_p.$ Our main idea is to regard normalized multiplicities as $\mu_n(\lbrace x \in B_p: f(x,n)=0\rbrace)$ for a $p$-adic analytic function $f$ coming from Iwasawa theory and a sequence of measures $(\mu_n)_n$ weakly converging to the Haar probability measure $\mu$ on $B_p,$ which is the $p$-adic analogue of the uniform measure. We prove convergence by considering characters $\eta$ of $B_p$ and comparing $\int_{B_p} \eta \ d\mu_n$ to traces of operators induced by symplectic operators $h_\eta$, followed by a character formula. A compactness argument for sequences of $p$-adic loci gives our main Iwasawa estimate. Poitou-Tate duality, the global Euler characteristic formula, and Hodge filtration estimates then give the dimension inequality.