TY - JOUR

T1 - Everywhere local solubility for hypersurfaces in products of projective spaces

AU - Fisher, Tom

AU - Ho, Wei

AU - Park, Jennifer

N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

PY - 2021/3

Y1 - 2021/3

N2 - We prove that a positive proportion of hypersurfaces in products of projective spaces over 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 P1× P1 defined over 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 P1× P1 soluble over Qp is a rational function of p for each finite prime p. Finally, we include some experimental data on the Hasse principle for these curves.

AB - We prove that a positive proportion of hypersurfaces in products of projective spaces over 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 P1× P1 defined over 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 P1× P1 soluble over Qp 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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U2 - 10.1007/s40993-020-00223-z

DO - 10.1007/s40993-020-00223-z

M3 - Article

AN - SCOPUS:85098762814

SN - 2363-9555

VL - 7

JO - Research in Number Theory

JF - Research in Number Theory

IS - 1

M1 - 6

ER -