Diagonal Forms of Higher Degree Over Function Fields of p-adic Curves

Pumpl�n, S.; Pumpluen, S.

Abstract

We investigate diagonal forms of degree d over the function field F of a smooth projective p-adic curve: if a form is isotropic over the completion of F with respect to each discrete valuation of F , then it is isotropic over certain fields F_U , F_P and F_p. These fields appear naturally when applying the methodology of patching; F is the inverse limit of the finite inverse system of fields {F_U , F_P , F_p}. Our observations complement some known bounds on the higher u-invariant of diagonal forms of degree d. We only consider diagonal forms of degree d over fields of characteristic not dividing d!.

Citation

Pumplün, S., & Pumpluen, S. (2020). Diagonal Forms of Higher Degree Over Function Fields of p-adic Curves. International Journal of Number Theory, 16(1), 161-172. https://doi.org/10.1142/S1793042120500098

• Journal Article Type: Article
• Acceptance Date: Jul 1, 2019
• Online Publication Date: Sep 5, 2019
• Publication Date: Feb 1, 2020
• Deposit Date: Sep 11, 2019
• Publicly Available Date: Sep 6, 2020
• Journal: International Journal of Number Theory
• Print ISSN: 1793-0421
• Electronic ISSN: 1793-7310
• Publisher: World Scientific
• Peer Reviewed: Peer Reviewed
• Volume: 16
• Issue: 1
• Pages: 161-172
• DOI: https://doi.org/10.1142/S1793042120500098
• Keywords: Forms of higher degree; Diagonal forms; Function fields; p-adic curves; u- invariant mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/2248895
• Publisher URL: https://www.worldscientific.com/doi/abs/10.1142/S1793042120500098?journalCode=ijnt
• Additional Information: Electronic version of an article published as Diagonal forms of higher degree over function fields of p-adic curves; S. Pumplün (School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK); International Journal of Number Theory 2020 16:01, 161-172 https://doi.org/10.1142/S1793042120500098 © World Scientific Publishing Company] https://www.worldscientific.com/worldscinet/ijnt