Abstract
Consider hypersurfaces of fixed degree d in a fixed projective space Pn over ℚ. We present a conjecture about the fraction of these that have rational points, and present evidence for the conjecture, including a proof that a positive fraction of the hypersurfaces have points over every completion of ℚ, provided that n, d ≥ 2 and (n, d) ≠ (2, 2). Generalizations to number fields are discussed. One of our proofs uses a result of Colliot-Thélène, proved in an appendix, that there is no Brauer-Manin obstruction to the Hasse principle for smooth complete intersections of dimension ≥ 3 in projective space over number fields. Colliot-Thélène’s proof, in turn, uses a consequence of the Weak Lefschetz Theorem proved in an appendix by Katz.
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Poonen, B., Voloch, J.F. (2004). Random Diophantine Equations. In: Poonen, B., Tschinkel, Y. (eds) Arithmetic of Higher-Dimensional Algebraic Varieties. Progress in Mathematics, vol 226. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8170-8_11
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DOI: https://doi.org/10.1007/978-0-8176-8170-8_11
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