On reducible ℓ-torsion representations of Picard curves
Abstract
Let C/ℚ be a Picard curve and let ℓ≡1pmod3 be a prime such that the λ-torsion representation of the Jacobian of C is reducible for a prime λ of ℚ(ζ3) above ℓ. We describe the constraints imposed by complex conjugation on the $2$-dimensional subquotient ρ, in particular, showing that if it is absolutely irreducible, then up to twist it descends to ℚ. We then give criteria for ruling out the irreducible case, using Dickson's classification of subgroups of GL2(𝔽ℓ), under the natural hypothesis that the characteristic polynomials of all elements in the image of ρ split. We also develop algorithms to compute the set of $1$-dimensional Jordan--Hölder constitutents of the ℓ-torsion representation over ℚ(ζ3), and Frobenius characteristic polynomials of the $2$-dimensional subquotient ρ. We apply these methods to $381$ Picard curves with possibly reducible $7$-torsion representations, and to five corresponding $13$-torsion examples. Under the reducibility and splitting hypotheses, we rule out an irreducible $2$-dimensional subquotient in all of the examples considered, showing that the semisimplification of the ℓ-torsion representation is a direct sum of $6$ characters.
Related
- Reducible Holonomy in Closed Torsion Geometries
- On the modularity of residually reducible Galois representations of rank 2
- Rational curves on toroidal compactifications of Picard modular varieties
- Some instances of modularity for residually reducible representations in the mixed case
- Torsion points in families of elliptic curves
BibTeX
@misc{chen2026reducible,
title = {{On reducible $\ell$-torsion representations of Picard curves}},
author = {Xiyao Chen and Shiva Chidambaram and Andrew Coursin and Erik Reed and Leo Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24751},
eprint = {2610.24751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24751},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On reducible ℓ-torsion representations of Picard curves
AU - Xiyao Chen
AU - Shiva Chidambaram
AU - Andrew Coursin
AU - Erik Reed
AU - Leo Yang
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24751
UR - https://arxiv.org/abs/2610.24751
AB - Let C/ℚ be a Picard curve and let ℓ≡1pmod3 be a prime such that the λ-torsion representation of the Jacobian of C is reducible for a prime λ of ℚ(ζ_3) above ℓ. We describe the constraints imposed by complex conjugation on the $2$-dimensional subquotient ρ, in particular, showing that if it is absolutely irreducible, then up to twist it descends to ℚ. We then give criteria for ruling out the irreducible case, using Dickson's classification of subgroups of GL_2(𝔽_ℓ), under the natural hypothesis that the characteristic polynomials of all elements in the image of ρ split. We also develop algorithms to compute the set of $1$-dimensional Jordan--Hölder constitutents of the ℓ-torsion representation over ℚ(ζ_3), and Frobenius characteristic polynomials of the $2$-dimensional subquotient ρ. We apply these methods to $381$ Picard curves with possibly reducible $7$-torsion representations, and to five corresponding $13$-torsion examples. Under the reducibility and splitting hypotheses, we rule out an irreducible $2$-dimensional subquotient in all of the examples considered, showing that the semisimplification of the ℓ-torsion representation is a direct sum of $6$ characters.
ER -