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Abstract We prove an inequality between the conductor and the discriminant for all hyperelliptic curves defined over discretely valued fields $$K$$ with perfect residue field of characteristic not $$2$$. Specifically, if such a curve is given by $$y^{2} = f(x)$$ with $$f(x) \in \mathcal{O}_{K}[x]$$, and if $$\mathcal{X}$$ is its minimal regular model over $$\mathcal{O}_{K}$$, then the negative of the Artin conductor of $$\mathcal{X}$$ (and thus also the number of irreducible components of the special fiber of $$\mathcal{X}$$) is bounded above by the valuation of $$\operatorname{disc}(f)$$. There are no restrictions on genus of the curve or on the ramification of the splitting field of $$f$$. This generalizes earlier work of Ogg, Saito, Liu, and the second author.
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Bounding cohomology classes over semiglobal fields
We provide a uniform bound for the index of cohomology classes over semiglobal fields (i.e., over one-variable function fields over a complete discretely valued field K). The bound is given in terms of the analogous data for the residue field of K and its finitely generated extensions of transcendence degree at most one. We also obtain analogous bounds for collections of cohomology classes. Our results provide recursive formulas for function fields over higher rank complete discretely valued fields, and explicit bounds in some cases when the information on the residue field is known. In the process, we prove a splitting result for cohomology classes of degree 3 in the context of surfaces over finite fields. ∗
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- PAR ID:
- 10516723
- Publisher / Repository:
- The Hebrew University of Jerusalem
- Date Published:
- Journal Name:
- Israel Journal of Mathematics
- Volume:
- 257
- Issue:
- 2
- ISSN:
- 0021-2172
- Page Range / eLocation ID:
- 353 to 387
- Subject(s) / Keyword(s):
- Galois cohomology, splitting, arithmetic curves, local-global principles, finite group schemes, semi-global fields, patching, étale algebras
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s11856-023-2549-x
