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Reduction of bielliptic surfaces

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  12 February 2025

Teppei Takamatsu Open the ORCID record for Teppei Takamatsu [Opens in a new window]
Teppei Takamatsu*
Affiliation:
Department of Mathematics (Hakubi Center), Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
*
e-mail: teppeitakamatsu.math@gmail.com
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Abstract

A bielliptic surface (or hyperelliptic surface) is a smooth surface with a numerically trivial canonical divisor such that the Albanese morphism is an elliptic fibration. In the first part of this article, we study the structure of bielliptic surfaces over a field of characteristic different from $2$ and $3$, in order to prove the Shafarevich conjecture for bielliptic surfaces with rational points. Furthermore, we demonstrate that the Shafarevich conjecture does not generally hold for bielliptic surfaces without rational points. In particular, this article completes the study of the Shafarevich conjecture for minimal surfaces of Kodaira dimension $0$. In the second part of this article, we study a Néron model of a bielliptic surface. We establish the potential existence of a Néron model for a bielliptic surface when the residual characteristic is not equal to $2$ or $3$.

Keywords

Bielliptic surfacesShafarevich conjectureNéron model

MSC classification

Primary: 11G35: Varieties over global fields
Secondary: 11G25: Varieties over finite and local fields
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 18
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work was supported by JSPS KAKENHI Grant number JP19J22795, JP22J00962, JP22KJ1780

References

André, Y., On the Shafarevich and Tate conjectures for hyper-Kähler varieties . Math. Ann. 305(1996), no. 2, 205–248.Google Scholar
Bădescu, L., Algebraic surfaces. Universitext, Springer-Verlag, New York, NY, 2001. Translated from the 1981 Romanian original by Vladimir Maşek and revised by the author.Google Scholar
Bombieri, E. and Mumford, D., Enriques’ classification of surfaces in char. $p$ . II. In: Complex analysis and algebraic geometry, (eds. Baily, W. L. and Shioda, T.), Iwanami Shoten Publishers, Tokyo, 1977, pp. 23–42.Google Scholar
Bosch, S., Lütkebohmert, W., and Raynaud, M., Néron models. volume 21 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1990.Google Scholar
Chiarellotto, B. and Lazda, C., Combinatorial degenerations of surfaces and Calabi-Yau threefolds . Algebra Number Theory 10(2016), no. 10, 2235–2266.Google Scholar
Creutz, B. and Viray, B., Degree and the Brauer-Manin obstruction . Algebra Number Theory 12(2018), no. 10, 2445–2470. With an appendix by Alexei N. Skorobogatov.Google Scholar
Edixhoven, B., Van der Geer, G., and Moonen, B., Abelian varieties, 2014. Preprint, http://van-der-geer.nl/~gerard/AV.pdf.Google Scholar
Faltings, G., Wüstholz, G., Grunewald, F., Schappacher, N., and Stuhler, U., Rational points. 3rd ed., Aspects of Mathematics, E6. Friedr. Vieweg & Sohn, Braunschweig, 1992. Papers from the seminar held at the Max-Planck-Institut für Mathematik, Bonn/Wuppertal, 1983/1984, With an appendix by Wüstholz.Google Scholar
Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S. L., Nitsure, N., and Vistoli, A., Fundamental algebraic geometry, volume 123 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2005. Grothendieck’s FGA explained.Google Scholar
Fu, L., Li, Z., Takamatsu, T., and Zou, H., Unpolarized shafarevich conjectures for hyper-käahler varieties. Preprint, 2022. arXiv:2203.10391. to appear in Algebr. Geom.Google Scholar
Gabber, O., Liu, Q., and Lorenzini, D., Hypersurfaces in projective schemes and a moving lemma . Duke Math. J. 164(2015), no. 7, 1187–1270.Google Scholar
Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV . Inst. Hautes Études Sci. Publ. Math. 361(1967), no. 32, pp. 5–361.Google Scholar
Ito, T., Kanemitsu, A., Takamatsu, T., and Tanaka, Y., Arithmetic finiteness of mukai varieties of genus 7. Preprint, 2024. arXiv:2409.20046.Google Scholar
Javanpeykar, A., Néron models and the arithmetic Shafarevich conjecture for certain canonically polarized surfaces . Bull. Lond. Math. Soc. 47(2015), no. 1, 55–64.Google Scholar
Javanpeykar, A. and Loughran, D., Good reduction of algebraic groups and flag varieties . Arch. Math. (Basel) 104(2015), no. 2, 133–143.Google Scholar
Javanpeykar, A. and Loughran, D., Complete intersections: Moduli, Torelli, and good reduction . Math. Ann. 368(2017), nos. 3–4, 1191–1225.Google Scholar
Javanpeykar, A. and Loughran, D., Good reduction of Fano threefolds and sextic surfaces . Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18(2018), no. 2, 509–535.Google Scholar
Knutson, D., Algebraic spaces, Lecture Notes in Mathematics, 203, Springer-Verlag, Berlin, 1971.Google Scholar
Kolyvagin, V. A., Finiteness of E(Q) and for a sha(e) for a subclass of Weil curves . Izv. Akad. Nauk SSSR Ser. Mat. 52(1988), no. 3, 522–540. 670–671.Google Scholar
Kolyvagin, V. A., On the Mordell-Weil group and the Shafarevich-Tate group of modular elliptic curves . In: Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), (eds. Satake, I.), Mathematical Society of Japan, Tokyo, 1991, pp. 429–436.Google Scholar
Krämer, T. and Maculan, M., Arithmetic finiteness of very irregular varieties. Preprint, 2023. arXiv:2310.08485.Google Scholar
Lang, S. and Néron, A., Rational points of abelian varieties over function fields . Amer. J. Math. 81(1959), 95–118.Google Scholar
Lawrence, B. and Sawin, W., The shafarevich conjecture for hypersurfaces in abelian varieties. Preprint, 2020. arXiv:2004.09046.Google Scholar
Liedtke, C. and Matsumoto, Y., Good reduction of K3 surfaces . Compos. Math. 154(2018), no. 1, 1–35.Google Scholar
Liu, Q., Algebraic geometry and arithmetic curves, volume 6 of Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné, Oxford Science Publications.Google Scholar
Liu, Q. and Tong, J., Néron models of algebraic curves . Trans. Amer. Math. Soc. 368(2016), no. 10, 7019–7043.Google Scholar
Mazur, B., Arithmetic on curves . Bull. Amer. Math. Soc. (N.S.) 14(1986), no. 2, 207–259.Google Scholar
Nagamachi, I. and Takamatsu, T., The Shafarevich conjecture and some extension theorems for proper hyperbolic polycurves . Math. Res. Lett. 29(2022), no. 2, 541–557.Google Scholar
Néron, A., Problèmes arithmétiques et géométriques rattachés à la notion de rang d’une courbe algébrique dans un corps . Bull. Soc. Math. France 80(1952), 101–166.Google Scholar
Néron, A., Modèles minimaux des variétés abéliennes sur les corps locaux et globaux . Inst. Hautes Études Sci. Publ. Math. 128(1964), no. 21, pp. 5–128.Google Scholar
Ogg, A. P., Abelian curves of $2$ -power conductor . Proc. Cambridge Philos. Soc. 62(1966), 143–148.Google Scholar
Scholl, A. J., A finiteness theorem for del Pezzo surfaces over algebraic number fields . J. London Math. Soc. (2) 32(1985), no. 1, 31–40.Google Scholar
She, Y., The unpolarized shafarevich conjecture for k3 surfaces. Preprint, 2017. arXiv:1705.09038.Google Scholar
Takamatsu, T., On a cohomological generalization of the Shafarevich conjecture for K3 surfaces . Algebra Number Theory 14(2020), no. 9, 2505–2531.Google Scholar
Takamatsu, T., On the Shafarevich conjecture for Enriques surfaces . Math. Z. 298(2021), nos. 1–2, 489–495.Google Scholar