Lie weights for CM curves and unipotent Selmer dimensions
Abstract
We prove an equidistribution result for Galois characters in the graded Lie components Ln associated to the Qp-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 μn counting normalized character multiplicities in Ln, and show that (μn)n weakly converges to the Haar measure. The key idea to prove convergence is to write the Fourier coefficients of μn in terms of traces of operators on Ln 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: dimQpH1f(Gk,T,Un)< dimQp∏v| pH1f(Gv,Un) 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(Q) can be recovered when k=Q.
Related
- Arithmetic statistics of isogeny Selmer groups associated to hyperelliptic curves
- Experiments on $3$-isogeny Selmer groups of elliptic curves with a $3$-torsion point
- Selmer groups of CM-twists of elliptic curves
- Refined conjectures on Fitting ideals of BDP Selmer groups
- On Conjectures of μ-Invariant for Selmer Groups at p=2
BibTeX
@misc{martin2026weights,
title = {{Lie weights for CM curves and unipotent Selmer dimensions}},
author = {Ander Martin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12013},
eprint = {2610.12013},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12013},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lie weights for CM curves and unipotent Selmer dimensions
AU - Ander Martin
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.12013
UR - https://arxiv.org/abs/2610.12013
AB - We prove an equidistribution result for Galois characters in the graded Lie components L_n associated to the 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 μ_n counting normalized character multiplicities in L_n, and show that (μ_n)_n weakly converges to the Haar measure. The key idea to prove convergence is to write the Fourier coefficients of μ_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_Q_pH^1_f(G_k,T,U_n)< dim_Q_p∏_v| pH^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(Q) can be recovered when k=Q.
ER -