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Arithmetic of the moduli of semistable elliptic surfaces

  • Changho Han
  • Jun-Yong ParkEmail author
Article
  • 63 Downloads

Abstract

We prove a new sharp asymptotic with the lower order term of zeroth order on \({\mathcal {Z}}_{{\mathbb {F}}_q(t)}(\mathcal {B})\) for counting the semistable elliptic curves over \({\mathbb {F}}_q(t)\) by the bounded height of discriminant \(\Delta (X)\). The precise count is acquired by considering the moduli of nonsingular semistable elliptic fibrations over \(\mathbb {P}^{1}\), also known as semistable elliptic surfaces, with 12n nodal singular fibers and a distinguished section. We establish a bijection of K-points between the moduli functor of semistable elliptic surfaces and the stack of morphisms \({\mathcal {L}}_{1,12n} \cong {\mathrm {Hom}}_n(\mathbb {P}^{1}, \overline{{\mathcal {M}}}_{1,1})\) where \(\overline{{\mathcal {M}}}_{1,1}\) is the Deligne–Mumford stack of stable elliptic curves and K is any field of characteristic \(\ne 2,3\). For \({\mathrm {char}}(K)=0\), we show that the class of \({\mathrm {Hom}}_n(\mathbb {P}^1,\mathcal {P}(a,b))\) in the Grothendieck ring of K–stacks, where \(\mathcal {P}(a,b)\) is a 1-dimensional (ab) weighted projective stack, is equal to \({\mathbb {L}}^{(a+b)n+1}-{\mathbb {L}}^{(a+b)n-1}\). Consequently, we find that the motive of the moduli \({\mathcal {L}}_{1,12n}\) is \({\mathbb {L}}^{10n + 1}-{\mathbb {L}}^{10n - 1}\) and the cardinality of the set of weighted \({\mathbb {F}}_q\)-points to be \(\#_q({\mathcal {L}}_{1,12n}) = q^{10n + 1}-q^{10n - 1}\). In the end, we formulate an analogous heuristic on \({\mathcal {Z}}_{{\mathbb {Q}}}(\mathcal {B})\) for counting the semistable elliptic curves over \({\mathbb {Q}}\) by the bounded height of discriminant \(\Delta \) through the global fields analogy.

Notes

Acknowledgements

Jun-Yong Park would like to express sincere gratitude to his doctoral advisor Craig Westerland for his guidance. We would like to thank Leo Herr and Jonathan Wise for their interest and a lynch pin idea of looking at the weighted projective embeddings. We would also like to thank Denis Auroux, Kenneth Ascher, Dori Bejleri, Jordan S. Ellenberg, Joe Harris, Minhyong Kim, Barry Mazur, Bjorn Poonen, András Stipsicz, and Jesse Wolfson for helpful conversations and inspirations. Finally, we thank the referee for in depth review and significant insights throughout the revision guidelines. Jun-Yong Park was supported by IBS-R003-D1, Institute for Basic Science in Korea. Changho Han acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) [PGSD3-487436-2016].

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Center for Geometry and PhysicsInstitute for Basic Science (IBS)PohangKorea

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