A bound on the primes of bad reduction for CM curves of genus 3

Kılıçer, Pınar; Lauter, Kristin; García, Elisa Lorenzo; Newton, Rachel; Ozman, Ekin; Streng, Marco

doi:10.1090/proc/14975

A bound on the primes of bad reduction for CM curves of genus $3$
HTML articles powered by AMS MathViewer

by Pınar Kılıçer, Kristin Lauter, Elisa Lorenzo García, Rachel Newton, Ekin Ozman and Marco Streng

Proc. Amer. Math. Soc. 148 (2020), 2843-2861

DOI: https://doi.org/10.1090/proc/14975

Published electronically: March 30, 2020

PDF | Request permission

Abstract:

We give bounds on the primes of geometric bad reduction for curves of genus $3$ of primitive complex multiplication (CM) type in terms of the CM orders. In the case of elliptic curves, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least $2$, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus $2$.

In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are important for algorithmic construction of curves with given characteristic polynomials over finite fields.

References

The Stacks Project Authors. The Stacks project. http://stacks.math.columbia.edu, 2016.
Jennifer S. Balakrishnan, Sorina Ionica, Kristin Lauter, and Christelle Vincent, Constructing genus-3 hyperelliptic Jacobians with CM, LMS J. Comput. Math. 19 (2016), no. suppl. A, 283–300. MR 3540961, DOI 10.1112/S1461157016000322
Irene Bouw, Jenny Cooley, Kristin Lauter, Elisa Lorenzo García, Michelle Manes, Rachel Newton, and Ekin Ozman, Bad reduction of genus three curves with complex multiplication, Women in numbers Europe, Assoc. Women Math. Ser., vol. 2, Springer, Cham, 2015, pp. 109–151. MR 3596603, DOI 10.1007/978-3-319-17987-2_{5}
Irene Bouw, Angelos Koutsianas, Jeroen Sijsling, and Stefan Wewers. Conductor and discriminant of Picard curves. Preprint, arXiv:1902.09624, 2019.
Florian Bouyer and Marco Streng, Examples of CM curves of genus two defined over the reflex field, LMS J. Comput. Math. 18 (2015), no. 1, 507–538. MR 3376741, DOI 10.1112/S1461157015000121
Reinier Bröker, Kristin Lauter, and Marco Streng, Abelian surfaces admitting an $(l,l)$-endomorphism, J. Algebra 394 (2013), 374–396. MR 3092726, DOI 10.1016/j.jalgebra.2013.07.011
Michel Demazure, Fibrés tangents, algèbres de Lie, Schémas en Groupes (Sém. Géométrie Algébrique, Inst. Hautes Études Sci., 1963) Inst. Hautes Études Sci., Paris, 1963, pp. Fasc. 1, Exposé 2, 40 (French). MR 0212023
P. Gaudry, T. Houtmann, D. Kohel, C. Ritzenthaler, and A. Weng, The 2-adic CM method for genus 2 curves with application to cryptography, Advances in cryptology—ASIACRYPT 2006, Lecture Notes in Comput. Sci., vol. 4284, Springer, Berlin, 2006, pp. 114–129. MR 2444631, DOI 10.1007/11935230_{8}
Eyal Z. Goren and Kristin E. Lauter, Class invariants for quartic CM fields, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 2, 457–480 (English, with English and French summaries). MR 2310947, DOI 10.5802/aif.2264
Eyal Z. Goren and Kristin E. Lauter, Genus 2 curves with complex multiplication, Int. Math. Res. Not. IMRN 5 (2012), 1068–1142. MR 2899960, DOI 10.1093/imrn/rnr052
Philipp Habegger and Fabien Pazuki, Bad reduction of genus 2 curves with CM jacobian varieties, Compos. Math. 153 (2017), no. 12, 2534–2576. MR 3705297, DOI 10.1112/S0010437X17007424
Pınar Kılıçer, Hugo Labrande, Reynald Lercier, Christophe Ritzenthaler, Jeroen Sijsling, and Marco Streng, Plane quartics over $\Bbb {Q}$ with complex multiplication, Acta Arith. 185 (2018), no. 2, 127–156. MR 3856187, DOI 10.4064/aa170227-16-3
Pınar Kılıçer, Elisa Lorenzo García, and Marco Streng. Primes dividing invariants of CM Picard curves. Canadian Journal of Mathematics, (Published online), 2018. https://doi.org/10.4153/S0008414X18000111.
Kenji Koike and Annegret Weng, Construction of CM Picard curves, Math. Comp. 74 (2005), no. 249, 499–518. MR 2085904, DOI 10.1090/S0025-5718-04-01656-4
Joan-Carles Lario and Anna Somoza. A note on Picard curves of CM-type. arXiv:1611.02582, 2016.
Kristin Lauter and Bianca Viray, An arithmetic intersection formula for denominators of Igusa class polynomials, Amer. J. Math. 137 (2015), no. 2, 497–533. MR 3337802, DOI 10.1353/ajm.2015.0010
Reynald Lercier, Qing Liu, Elisa Lorenzo García, and Christophe Ritzenthaler. Reduction type of smooth quartics. Preprint, arXiv:1803.05816, 2018.
Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008. With appendices by C. P. Ramanujam and Yuri Manin; Corrected reprint of the second (1974) edition. MR 2514037
Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722
Tetsuji Shioda, On the graded ring of invariants of binary octavics, Amer. J. Math. 89 (1967), 1022–1046. MR 220738, DOI 10.2307/2373415
Carl Ludwig Siegel, Lectures on the geometry of numbers, Springer-Verlag, Berlin, 1989. Notes by B. Friedman; Rewritten by Komaravolu Chandrasekharan with the assistance of Rudolf Suter; With a preface by Chandrasekharan. MR 1020761, DOI 10.1007/978-3-662-08287-4
Marco Streng. RECIP – REpository of Complex multIPlication SageMath code. http://pub.math.leidenuniv.nl/~strengtc/recip/.
Marco Streng, Computing Igusa class polynomials, Math. Comp. 83 (2014), no. 285, 275–309. MR 3120590, DOI 10.1090/S0025-5718-2013-02712-3
Annegret Weng, A class of hyperelliptic CM-curves of genus three, J. Ramanujan Math. Soc. 16 (2001), no. 4, 339–372. MR 1877806