Abstract
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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References
Bright, M.J., Browning, T.D., Loughran, D.: Failures of weak approximation in families. Compos. Math. 152(7), 1435–1475 (2016). MR3530447
Bhargava, M., Cremona, J.E., Fisher, T.A.: The proportion of plane cubic curves over \({\mathbb{Q}}\) that everywhere locally have a point. Int. J. Number Theory 12(4), 1077–1092 (2016)
Bhargava, M., Cremona, J.E., Fisher, T.A.: The proportion of genus one curves over \({\mathbb{Q}}\) defined by a binary quartic that everywhere locally have a point, (2020). to appear in International J. Number Theory, arXiv:2004.12085
Bhargava, M., Cremona, J.E., Fisher, T.A., Jones, N.G., Keating, J.P.: What is the probability that a random integral quadratic form in n variables has an integral zero? Int. Math. Res. Not. IMRN 12, 3828–3848 (2016). MR3544620
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language, 1997, pp. 235–265. Computational algebra and number theory (London, 1993). MR1484478
Bhargava, M., Ho, W.: Coregular spaces and genus one curves. Camb. J. Math. 4(1), 1–119 (2016). MR3472915
Bhargava, M., Ho, W.: The Hasse principle for some genus one curves in \({\mathbb{P}}^1 \times {\mathbb{P}}^{1}\), (2020). In preparation
Bhargava, M., Ho, W.: On average sizes of Selmer groups and ranks in families of elliptic curves having marked points, (2020). Preprint
Bhargava, M.: A positive proportion of plane cubics fail the Hasse principle, (2014). arXiv:1402.1131
Browning, T., Le Boudec, P., Sawin, W.: The Hasse principle for random Fano hypersurfaces, (2020). arXiv:2006.02356
Bober, J.W.: Conditionally bounding analytic ranks of elliptic curves, ANTS X—Proceedings of the Tenth Algorithmic Number Theory Symposium, pp. 135–144 (2013). MR3207411
Campana, F.: Connexité rationnelle des variétés de Fano. Ann. Sci. École Norm. Sup. (4) 25(5), 539–545 (1992). MR1191735
Colliot-Thélène, J.-L.: Points rationnels sur les fibrations, Higher dimensional varieties and rational points (Budapest, 2001), pp. 171–221 (2003). MR2011747
Fisher, T.A., Ho, W., Park, J.: Data for “Everywhere local solubility for hypersurfaces in products of projective spaces”, (2019). Available at https://www.dpmms.cam.ac.uk/~taf1000/papers/probs22.html
Fisher, T.A.: The proportion of plane cubic curves with a rational point, (2015). Oberwolfach report, Explicit methods in number theory
Fisher, T.A.: On binary quartics and the Cassels-Tate pairing, (2016). http://www.dpmms.cam.ac.uk/~taf1000/papers/bq-ctp.html
Fisher, T.A., Radičević, L.: Some minimisation algorithms in arithmetic invariant theory. J. Théor. Nombres Bordeaux 30(3), 801–828 (2018). MR3938627
Graber, T., Harris, J., Starr, J.: Families of rationally connected varieties. J. Amer. Math. Soc. 16(1), 57–67 (2003). MR1937199
Hartshorne, R.: Ample subvarieties of algebraic varieties. Lecture Notes in Mathematics, vol. 156. Springer, Berlin (1970)
Iskovskih, V.A.: Rational surfaces with a pencil of rational curves. Mat. Sb. (N.S.) 74(116), 608–638 (1967). MR0220734
Iskovskih, V.A.: A counterexample to the Hasse principle for systems of two quadratic forms in five variables. Mat. Zametki 10, 253–257 (1971). MR0286743
Kollár, J., Miyaoka, Y., Mori, S.: Rational connectedness and boundedness of Fano manifolds. J. Differ. Geom. 36(3), 765–779 (1992). MR1189503
Monsky, P.: Generalizing the Birch-Stephens theorem. I. Modular curves. Math. Z. 221(3), 415–420 (1996). MR1381589
Poonen, B., Stoll, M.: The Cassels-Tate pairing on polarized abelian varieties. Ann. Math. (2) 150(3), 1109–1149 (1999). MR1740984
Poonen, B., Voloch, J.F.: Random Diophantine equations, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), (2004), pp. 175–184. With appendices by Jean-Louis Colliot-Thélène and Nicholas M. Katz. MR2029869
The Sage Developers, Sagemath, the Sage Mathematics Software System (Version 8.7), (2019). https://www.sagemath.org
Schindler, D.: Manin’s conjecture for certain biprojective hypersurfaces. J. Reine Angew. Math. 714, 209–250 (2016). MR3491888
Serre, J.-P.: Spécialisation des éléments de \(\text{ Br}_{2}({\mathbf{Q}}(T_{1}, \cdots, T_{n}))\). C. R. Acad. Sci. Paris Sér. I Math. 311(7), 397–402 (1990). MR1075658
Stamminger, S.: Explicit 8-descent on elliptic curves, PhD thesis, Bremen (2005)
Acknowledgements
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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DOI: https://doi.org/10.1007/s40993-020-00223-z