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Symplectic extension graphs and Ribet's lemma
Abstract
Let p be an odd prime, and let ρ: G→ GSp2n(Zp) be an irreducible symplectic residually multiplicity free representation. We define the symplectic extension graph ΓρGSp associated with ρ to be the directed graph whose vertices are the residual factors of ρ and a directed edge between two vertices corresponds to a nonsplit extension between the vertices realized by ρ. Using the Euclidean Bruhat--Tits building associated with GSp2n(Qp), under some assumptions, we prove the existence of certain paths in ΓρGSp and show that ΓρGSp is a disjoint union of nonempty strongly connected subgraphs. Moreover, we provide examples to show that the result we obtain is optimal.
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BibTeX
@misc{sudarshan2026symplectic,
title = {{Symplectic extension graphs and Ribet's lemma}},
author = {Aniruddha Sudarshan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19434},
eprint = {2610.19434},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19434},
}
RIS
TY - UNPB
TI - Symplectic extension graphs and Ribet's lemma
AU - Aniruddha Sudarshan
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19434
UR - https://arxiv.org/abs/2610.19434
AB - Let p be an odd prime, and let ρ: G→ GSp_2n(Z_p) be an irreducible symplectic residually multiplicity free representation. We define the symplectic extension graph Γ_ρ^GSp associated with ρ to be the directed graph whose vertices are the residual factors of ρ and a directed edge between two vertices corresponds to a nonsplit extension between the vertices realized by ρ. Using the Euclidean Bruhat--Tits building associated with GSp_2n(Q_p), under some assumptions, we prove the existence of certain paths in Γ_ρ^GSp and show that Γ_ρ^GSp is a disjoint union of nonempty strongly connected subgraphs. Moreover, we provide examples to show that the result we obtain is optimal.
ER -