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Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers I

  • Andrew Obus
  • Padmavathi Srinivasan
  • Connor Stewart
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Abstract

We prove conductor-discriminant inequalities for all β„€/n-covers of β„™1 defined over discretely valued fields K with excellent valuation ring π’ͺK and perfect residue field of characteristic not dividing n, modulo some calculations appearing in work of the third author (arXiv:2609.20585). Specifically, when such a curve X is given by yn = f(x) with f(x) ∈π’ͺK[x] and n|deg(f), and if 𝒳 is its minimal regular model over π’ͺK, then the negative of the Artin conductor of 𝒳 is bounded above by (nβˆ’1)vK(disc(rad(f))). This is a direct generalization of previous work of the first two authors on hyperelliptic curves, which in turn generalized work of Ogg, Saito, Liu, and the second author. When f is monic, this strengthens a result of Kohls stating that the conductor exponent of the Jacobian of such a curve is bounded above by (nβˆ’1)vK(disc(rad(f))).

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Cite

BibTeXRIS
BibTeX
@misc{obus2026conductordiscriminant,
  title = {{Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers I}},
  author = {Andrew Obus and Padmavathi Srinivasan and Connor Stewart},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20553},
  eprint = {2609.20553},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20553},
}
RIS
TY  - UNPB
TI  - Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers I
AU  - Andrew Obus
AU  - Padmavathi Srinivasan
AU  - Connor Stewart
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.20553
UR  - https://arxiv.org/abs/2609.20553
AB  - We prove conductor-discriminant inequalities for all β„€/n-covers of β„™^1 defined over discretely valued fields K with excellent valuation ring π’ͺ_K and perfect residue field of characteristic not dividing n, modulo some calculations appearing in work of the third author (arXiv:2609.20585). Specifically, when such a curve X is given by y^n = f(x) with f(x) ∈π’ͺ_K[x] and n|deg(f), and if 𝒳 is its minimal regular model over π’ͺ_K, then the negative of the Artin conductor of 𝒳 is bounded above by (nβˆ’1)v_K(disc(rad(f))). This is a direct generalization of previous work of the first two authors on hyperelliptic curves, which in turn generalized work of Ogg, Saito, Liu, and the second author. When f is monic, this strengthens a result of Kohls stating that the conductor exponent of the Jacobian of such a curve is bounded above by (nβˆ’1)v_K(disc(rad(f))).
ER  -