Everywhere local solubility for hypersurfaces in products of projective spaces


We prove that a positive proportion of hypersurfaces in products of projective spaces over \({\mathbb {Q}}\) are everywhere locally soluble, for almost all multidegrees and dimensions, as a generalization of a theorem of Poonen and Voloch [25]. We also study the specific case of genus 1 curves in \({\mathbb {P}}^1 \times {\mathbb {P}}^1\) defined over \({\mathbb {Q}}\), represented as bidegree (2, 2)-forms, and show that the proportion of everywhere locally soluble such curves is approximately \(87.4\%\). As in the case of plane cubics [2], the proportion of these curves in \({\mathbb {P}}^1 \times {\mathbb {P}}^1\) soluble over \({\mathbb {Q}}_p\) is a rational function of p for each finite prime p. Finally, we include some experimental data on the Hasse principle for these curves.

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Much of this work was completed during a workshop organized by Alexander Betts, Tim Dokchitser, Vladimir Dokchitser and Celine Maistret, a trimester organized by David Harari, Emmanuel Peyre, and Alexei Skorobogatov, and a workshop organized by Michael Stoll; we thank the organizers as well as Baskerville Hall, Institut Henri Poincaré, and Franken-Akademie Schloss Schney, respectively, for their hospitality during those periods. We also thank Bhargav Bhatt, John Cremona, David Harari, Max Lieblich, Daniel Loughran, Bjorn Poonen, and the anonymous referee for helpful conversations and comments. WH was partially supported by NSF grants DMS-1701437 and DMS-1844763 and the Sloan Foundation. JP was partially supported by NSF grant DMS-1902199.

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Fisher, T., Ho, W. & Park, J. Everywhere local solubility for hypersurfaces in products of projective spaces. Res. number theory 7, 6 (2021). https://doi.org/10.1007/s40993-020-00223-z

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