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Bounding Selmer Groups of Superelliptic Jacobians via Class Groups
Abstract
Let K be a number field containing a primitive p-th root of unity ζp. Let f(x)∈ K[x] be a monic integral polynomial, and let f0 denote its radical. Let C/K be the superelliptic curve defined by yp=f(x), and let J be its Jacobian variety. The variety J admits multiplication by ζp over K; in particular, the endomorphism induced by Π:=1−ζp gives an isogeny of J over K. Let SelΠ(J) denote the Selmer group associated to Π. Under suitable hypotheses, we obtain bounds for SelΠ(J) in terms of the p-torsion subgroup of the class group of L:=K[x]/(f0). Several examples illustrating the results are also discussed.
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BibTeX
@misc{wang2026bounding,
title = {{Bounding Selmer Groups of Superelliptic Jacobians via Class Groups}},
author = {Pengfei Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02173},
eprint = {2609.02173},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02173},
}
RIS
TY - UNPB
TI - Bounding Selmer Groups of Superelliptic Jacobians via Class Groups
AU - Pengfei Wang
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02173
UR - https://arxiv.org/abs/2609.02173
AB - Let K be a number field containing a primitive p-th root of unity ζ_p. Let f(x)∈ K[x] be a monic integral polynomial, and let f_0 denote its radical. Let C/K be the superelliptic curve defined by y^p=f(x), and let J be its Jacobian variety. The variety J admits multiplication by ζ_p over K; in particular, the endomorphism induced by Π:=1−ζ_p gives an isogeny of J over K. Let Sel_Π(J) denote the Selmer group associated to Π. Under suitable hypotheses, we obtain bounds for Sel_Π(J) in terms of the p-torsion subgroup of the class group of L:=K[x]/(f_0). Several examples illustrating the results are also discussed.
ER -