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Nonsplit conics in the reduction of an arithmetic curve

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Abstract

For a function field in one variable F/K and a discrete valuation v of K with perfect residue field k, we bound the number of discrete valuations on F extending v whose residue fields are non-ruled function fields in one variable over k. Assuming that K is relatively algebraically closed in F, we find that the number of non-ruled residually transcendental extensions of v to F is bounded by \({\mathfrak {g}}+1\) where \({\mathfrak {g}}\) is the genus of F/K. An application to sums of squares in function fields of curves over \({\mathbb {R}}(\!(t)\!)\) is outlined.

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References

  1. Becher, K.J., Grimm, D., Van Geel, J.: Sums of squares in algebraic function fields over a complete discretely valued field. Pac. J. 267, 257–276 (2014)

    MathSciNet  Google Scholar 

  2. Becher, K.J., Gupta, P.: A ruled residue theorem for function fields of conics. J. Pure Appl. Algebra 225, 106638 (2021)

    Article  MathSciNet  Google Scholar 

  3. Becher, K.J., Gupta, P., Chandra Mishra, S.: A ruled residue theorem for function fields of elliptic curves. J. Pure Appl. Algebra (2023). https://doi.org/10.1016/j.jpaa.2023.107492

    Article  Google Scholar 

  4. Becher, K.J., Van Geel, J.: Sums of squares in function fields of hyperelliptic curves. Math. Z. 261, 829–844 (2009)

    Article  MathSciNet  Google Scholar 

  5. Becker, E.: Hereditarily-Pythagorean fields and orderings of higher level. Monogr. Math. 29, 1 (1978)

    MathSciNet  Google Scholar 

  6. Deuring, M.: Lectures on the Theory of Algebraic Functions of One Variable. Lecture Notes in Mathematics, vol. 314. Springer, Berlin (1973)

    Book  Google Scholar 

  7. Dutta, A.: A ruled residue theorem for algebraic function fields of curves of prime degree. J. Pure Appl. Algebra 227, 107453 (2023)

    Article  MathSciNet  Google Scholar 

  8. Elman, R., Karpenko, N., Merkurjev, A.: The Algebraic and Geometric Theory of Quadratic Forms. AMS Colloquium Publ., vol. 56. Amer. Math. Soc., Providence (2008)

    Google Scholar 

  9. Green, B., Matignon, M., Pop, F.: On valued function fields, I. Manuscr. Math. 65, 357–376 (1989)

    Article  MathSciNet  Google Scholar 

  10. Liu, Q.: Algebraic Geometry and Arithmetic Curves. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford (2002)

    Book  Google Scholar 

  11. Manzano-Flores, G.: Sums of squares in function fields over Henselian discretely valued fields. Preprint (2023). arXiv:2203.14357

  12. Mathieu, H.: Das Verhalten des Geschlechts bei Konstantenreduktionen algebraischer Funktionenkörper. Arch. Math. (Basel) 20, 597–611 (1969)

    Article  MathSciNet  Google Scholar 

  13. Matignon, M.: Genre et genre résiduel des corps de fonctions valués. Manuscri. Math. 58, 179–214 (1987)

    Article  Google Scholar 

  14. Mumford, D., Oda, T.: Algebraic Geometry II, Hindustan Book Agency Series, Volume 73 of Texts and Readings in Mathematics. Hindustan Book Agency, New York (2015)

    Google Scholar 

  15. Nagata, M.: A theorem on valuation rings and its applications. Nagoya Math. J. 29, 85–91 (1967)

    Article  MathSciNet  Google Scholar 

  16. Ohm, J.: The ruled residue theorem for simple transcendental extensions of valued fields. Proc. Am. Math. Soc. 89, 16–18 (1983)

    Article  MathSciNet  Google Scholar 

  17. Tikhonov, S.V., Van Geel, J., Yanchevskiĭ, V.I.: Pythagoras numbers of function fields of hyperelliptic curves with good reduction. Manuscr. Math. 119, 305–322 (2006)

    Article  MathSciNet  Google Scholar 

  18. Stacks Project. http://stacks.math.columbia.edu

  19. Stichtenoth, H.: Algebraic Function Fields and Codes. Graduate Texts in Mathematics, vol. 254, 2nd edn. Springer, Berlin (2009)

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Antwerp, Middelheimlaan 1, 2020, Antwerp, Belgium

    Karim Johannes Becher

  2. Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Av. Libertador Bernardo O’ Higgins 3363, Santiago, Chile

    David Grimm

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  1. Karim Johannes Becher
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  2. David Grimm
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Correspondence to David Grimm.

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This work was supported by the FWO Odysseus Programme (Project G0E6114N, Explicit Methods in Quadratic Form Theory), funded by the Fonds Wetenschappelijk Onderzoek—Vlaanderen, by ANID (proyecto FONDECYT 11150956) and by the Universidad de Santiago de Chile (USACH proyecto DICYT, Codigo 041933 G).

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Becher, K.J., Grimm, D. Nonsplit conics in the reduction of an arithmetic curve. Math. Z. 306, 12 (2024). https://doi.org/10.1007/s00209-023-03395-3

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