Supersingular curves of genus four in characteristic two

Dragutinović, Dušan

doi:10.1090/proc/16792

Supersingular curves of genus four in characteristic two
HTML articles powered by AMS MathViewer

by Dušan Dragutinović

Proc. Amer. Math. Soc.

DOI: https://doi.org/10.1090/proc/16792

Published electronically: April 12, 2024

HTML | PDF | Request permission

Abstract:

We describe the intersection of the Torelli locus $j(\mathcal {M}_4^{ct}) = \mathcal {J}_4$ with Newton and Ekedahl-Oort strata related to the supersingular locus in characteristic 2. We show that the locus of supersingular Jacobians $\mathcal {S}_4\cap \mathcal {J}_4$ in characteristic 2 is pure of dimension three. One way to obtain that result uses an analysis of the data of smooth genus four curves and principally polarized abelian fourfolds defined over $\mathbb {F}_2$, and another involves a more careful study of some relevant Ekedahl-Oort loci.

References

Jeffrey D. Achter and Everett W. Howe, Hasse-Witt and Cartier-Manin matrices: a warning and a request, Arithmetic geometry: computation and applications, Contemp. Math., vol. 722, Amer. Math. Soc., [Providence], RI, [2019] ©2019, pp. 1–18. MR 3896846, DOI 10.1090/conm/722/14534
J. Bergström, V. Karemaker, and S. Marseglia, Polarizations of abelian varieties over finite fields via canonical liftings, Int. Math. Res. Not. IMRN 4 (2021), 3194–3248. Repository: https://github.com/stmar89/PolsAbVarFpCanLift (last commit: Nov 16, 2021)
Tommaso Giorgio Centeleghe and Jakob Stix, Categories of abelian varieties over finite fields, I: Abelian varieties over $\Bbb {F}_p$, Algebra Number Theory 9 (2015), no. 1, 225–265. MR 3317765, DOI 10.2140/ant.2015.9.225
Ching-Li Chai and Frans Oort, Monodromy and irreducibility of leaves, Ann. of Math. (2) 173 (2011), no. 3, 1359–1396. MR 2800716, DOI 10.4007/annals.2011.173.3.3
A. J. de Jong and F. Oort, Purity of the stratification by Newton polygons, J. Amer. Math. Soc. 13 (2000), no. 1, 209–241. MR 1703336, DOI 10.1090/S0894-0347-99-00322-7
Taylor Dupuy, Kiran Kedlaya, David Roe, and Christelle Vincent, Isogeny classes of abelian varieties over finite fields in the LMFDB, Arithmetic geometry, number theory, and computation, Simons Symp., Springer, Cham, [2021] ©2021, pp. 375–448. MR 4427971, DOI 10.1007/978-3-030-80914-0_{1}3
Dragutinović, D., Supersingular curves of genera four and five in characteristic two, Master’s thesis, Utrecht University, 2021.
Torsten Ekedahl and Gerard van der Geer, Cycle classes of the E-O stratification on the moduli of abelian varieties, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser Boston, Boston, MA, 2009, pp. 567–636. MR 2641181, DOI 10.1007/978-0-8176-4745-2_{1}3
Arsen Elkin and Rachel Pries, Hyperelliptic curves with $a$-number 1 in small characteristic, Albanian J. Math. 1 (2007), no. 4, 245–252. MR 2367217
Arsen Elkin and Rachel Pries, Ekedahl-Oort strata of hyperelliptic curves in characteristic 2, Algebra Number Theory 7 (2013), no. 3, 507–532. MR 3095219, DOI 10.2140/ant.2013.7.507
Carel Faber and Gerard van der Geer, Complete subvarieties of moduli spaces and the Prym map, J. Reine Angew. Math. 573 (2004), 117–137. MR 2084584, DOI 10.1515/crll.2004.056
E. W. Howe, Deducing information about curves over finite fields from their Weil polynomials, arXiv:2110.04221, 2021.
T. Ibukiyama, V. Karemaker, and C.-F. Yu, The Gauss problem for central leaves, arXiv:2205.13180, 2022.
Momonari Kudo, Shushi Harashita, and Hayato Senda, The existence of supersingular curves of genus 4 in arbitrary characteristic, Res. Number Theory 6 (2020), no. 4, Paper No. 44, 17. MR 4170348, DOI 10.1007/s40993-020-00217-x
The LMFDB Collaboration, The L-functions and modular forms database, https://www.lmfdb.org, 2022.
Ke-Zheng Li and Frans Oort, Moduli of supersingular abelian varieties, Lecture Notes in Mathematics, vol. 1680, Springer-Verlag, Berlin, 1998. MR 1611305, DOI 10.1007/BFb0095931
Ju. I. Manin, Theory of commutative formal groups over fields of finite characteristic, Uspehi Mat. Nauk 18 (1963), no. 6(114), 3–90 (Russian). MR 157972
S. Marseglia, https://github.com/stmar89/AbVarFq (last commit: Dec 10, 2021)
Stefano Marseglia, Computing square-free polarized abelian varieties over finite fields, Math. Comp. 90 (2021), no. 328, 953–971. MR 4194169, DOI 10.1090/mcom/3594
J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 167–212. MR 861976
Frans Oort, Hyperelliptic supersingular curves, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 247–284. MR 1085262, DOI 10.1007/978-1-4612-0457-2_{1}1
Frans Oort, Moduli of abelian varieties in mixed and in positive characteristic, Handbook of moduli. Vol. III, Adv. Lect. Math. (ALM), vol. 26, Int. Press, Somerville, MA, 2013, pp. 75–134. MR 3135436
Rachel Pries, A short guide to $p$-torsion of abelian varieties in characteristic $p$, Computational arithmetic geometry, Contemp. Math., vol. 463, Amer. Math. Soc., Providence, RI, 2008, pp. 121–129. MR 2459994, DOI 10.1090/conm/463/09051
Rachel Pries, The $p$-torsion of curves with large $p$-rank, Int. J. Number Theory 5 (2009), no. 6, 1103–1116. MR 2569747, DOI 10.1142/S1793042109002560
Rachel Pries, Current results on Newton polygons of curves, Open problems in arithmetic algebraic geometry, Adv. Lect. Math. (ALM), vol. 46, Int. Press, Somerville, MA, [2019] ©2019, pp. 179–207. MR 3971184
David Savitt, The maximum number of points on a curve of genus 4 over ${\Bbb F}_8$ is 25, Canad. J. Math. 55 (2003), no. 2, 331–352. With an appendix by Kristin Lauter. MR 1969795, DOI 10.4153/CJM-2003-015-7
Karl-Otto Stöhr and José Felipe Voloch, A formula for the Cartier operator on plane algebraic curves, J. Reine Angew. Math. 377 (1987), 49–64. MR 887399, DOI 10.1515/crll.1987.377.49
Gerard van der Geer, Cycles on the moduli space of abelian varieties, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg, Braunschweig, 1999, pp. 65–89. MR 1722539, DOI 10.1007/978-3-322-90172-9_{4}
X. Xarles, A census of all genus 4 curves over the field with 2 elements, arXiv:2007.07822, 2020. Repository: https://github.com/XavierXarles/Censusforgenus4curvesoverF2
Zijian Zhou, A bound on the genus of a curve with Cartier operator of small rank, Rend. Circ. Mat. Palermo (2) 68 (2019), no. 3, 569–577. MR 4148766, DOI 10.1007/s12215-018-0379-1
Zijian Zhou, Ekedahl-Oort strata on the moduli space of curves of genus four, Rocky Mountain J. Math. 50 (2020), no. 2, 747–761. MR 4104409, DOI 10.1216/rmj.2020.50.747