Lie weights for CM curves and unipotent Selmer dimensions

Abstract

We prove an equidistribution result for Galois characters in the graded Lie components $L_n$ associated to the $\mathbf{Q}_p$-fundamental group of a projective hyperbolic curve $X$ over a number field $k$ with geometrically CM Jacobian. That is, we consider the $p$-adic completion of a fixed character lattice, where we define measures $\mu_n$ counting normalized character multiplicities in $L_n$, and show that $(\mu_n)_n$ weakly converges to the Haar measure. The key idea to prove convergence is to write the Fourier coefficients of $\mu_n$ in terms of traces of operators on $L_n$ induced by symplectic operators on the Tate module, and control them with a known symplectic character formula, followed by Weyl's criterion for compact abelian groups. Our main arithmetic application is to combine equidistribution with the Iwasawa-theoretic method of Coates--Kim and a compactness argument for moving analytic loci, giving a dimension inequality for unipotent Selmer schemes: $\dim_{\mathbf{Q}_p}H^1_f(G_{k,T},U_n)<\dim_{\mathbf{Q}_p}\prod_{v\mid p}H^1_f(G_v,U_n)$ for all sufficiently large $n,$ with the quotient having limit superior at most $1/2.$ This answers a particular case of a conjecture by Kim, from which the finiteness of $X(\mathbf{Q})$ can be recovered when $k=\mathbf{Q}.$