On Birch and Swinnerton-Dyer's cubic surfaces

DOI10.1007/978-3-319-74998-3_10zbMATH Open1445.11056arXiv1510.03769OpenAlexW2620323057MaRDI QIDQ1716106

Mckenzie West

Publication date: 29 January 2019

Abstract: In a 1975 paper of Birch and Swinnerton-Dyer, a number of explicit norm form cubic surfaces are shown to fail the Hasse Principle. They make a correspondence between this failure and the Brauer--Manin obstruction, recently discovered by Manin. We generalize their work, making use of modern computer algebra software to show that a larger set of cubic surfaces have a Brauer--Manin obstruction to the Hasse principle, thus verifying the Colliot-Th'el`ene--Sansuc conjecture for infinitely many cubic surfaces.

Full work available at URL: https://arxiv.org/abs/1510.03769

zbMATH Keywords

surfaceBrauer-Manin obstructioncubic

Mathematics Subject Classification ID

(L)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture (11G40) Varieties over global fields (11G35)

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