## Rigorous computation of the endomorphism ring of a Jacobian

HTML articles powered by AMS MathViewer

- by
Edgar Costa, Nicolas Mascot, Jeroen Sijsling and John Voight
**HTML**| PDF - Math. Comp.
**88**(2019), 1303-1339 Request permission

## Abstract:

We describe several improvements and generalizations to algorithms for the rigorous computation of the endomorphism ring of the Jacobian of a curve defined over a number field.## References

- Wieb Bosma, John Cannon, and Catherine Playoust,
*The Magma algebra system. I. The user language*, J. Symbolic Comput.**24**(1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR**1484478**, DOI 10.1006/jsco.1996.0125 - Andrew R. Booker, Jeroen Sijsling, Andrew V. Sutherland, John Voight, and Dan Yasaki,
*A database of genus-2 curves over the rational numbers*, LMS J. Comput. Math.**19**(2016), no. suppl. A, 235–254. MR**3540958**, DOI 10.1112/S146115701600019X - C. Cunningham and L. Dembélé,
*Lifts of Hilbert modular forms and application to modularity of abelian varieties*, arXiv:1705.03054, 2017. - François Charles,
*On the Picard number of K3 surfaces over number fields*, Algebra Number Theory**8**(2014), no. 1, 1–17. MR**3207577**, DOI 10.2140/ant.2014.8.1 - E.Costa, N. Mascot, and J. Sijsling,
*Rigorous computation of the endomorphism ring of a Jacobian*, https://github.com/edgarcosta/endomorphisms/, 2017. - Kamal Khuri-Makdisi,
*Linear algebra algorithms for divisors on an algebraic curve*, Math. Comp.**73**(2004), no. 245, 333–357. MR**2034126**, DOI 10.1090/S0025-5718-03-01567-9 - Abhinav Kumar and Ronen E. Mukamel,
*Real multiplication through explicit correspondences*, LMS J. Comput. Math.**19**(2016), no. suppl. A, 29–42. MR**3540944**, DOI 10.1112/S1461157016000188 - D. Liang,
*Explicit equations of non-hyperelliptic genus 3 curves with real multiplication by $\mathbb {Q} (\zeta _7 + \zeta _7^{-1})$*, Ph.D. thesis, Louisiana State University, 2014. - A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász,
*Factoring polynomials with rational coefficients*, Math. Ann.**261**(1982), no. 4, 515–534. MR**682664**, DOI 10.1007/BF01457454 - Qing Liu, Dino Lorenzini, and Michel Raynaud,
*On the Brauer group of a surface*, Invent. Math.**159**(2005), no. 3, 673–676. [See corrigendum in: 3858404]. MR**2125738**, DOI 10.1007/s00222-004-0403-2 - The LMFDB Collaboration,
*The l-functions and modular forms database*, http://www.lmfdb.org, 2016. [Online; accessed 21 July 2016]. - D. Lombardo,
*Computing the geometric endomorphism ring of a genus 2 Jacobian*, arXiv:1610.09674, 2016. - Nicolas Mascot,
*Computing modular Galois representations*, Rend. Circ. Mat. Palermo (2)**62**(2013), no. 3, 451–476. MR**3118315**, DOI 10.1007/s12215-013-0136-4 - J. S. Milne,
*On a conjecture of Artin and Tate*, Ann. of Math. (2)**102**(1975), no. 3, 517–533. MR**414558**, DOI 10.2307/1971042 - J. S. Milne,
*On a conjecture of Artin and Tate*, Ann. of Math. (2)**102**(1975), no. 3, 517–533. MR**414558**, DOI 10.2307/1971042 - P. Molin and C. Neurohr,
*Computing period matrices and the Abel-Jacobi map of superelliptic curves*, arXiv:1707.07249, 2017. - P. Molin,
*Numerical integration and L functions computations*, Theses, Université Sciences et Technologies - Bordeaux I, October 2010. - David Mumford,
*Abelian varieties*, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR**0282985** - Frans Oort,
*Endomorphism algebras of abelian varieties*, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 469–502. MR**977774** - Bjorn Poonen,
*Computational aspects of curves of genus at least $2$*, Algorithmic number theory (Talence, 1996) Lecture Notes in Comput. Sci., vol. 1122, Springer, Berlin, 1996, pp. 283–306. MR**1446520**, DOI 10.1007/3-540-61581-4_{6}3 - Maria Petkova and Hironori Shiga,
*A new interpretation of the Shimura curve with discriminant 6 in terms of Picard modular forms*, Arch. Math. (Basel)**96**(2011), no. 4, 335–348. MR**2794089**, DOI 10.1007/s00013-011-0235-4 - I. Reiner,
*Maximal orders*, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original; With a foreword by M. J. Taylor. MR**1972204** - Christophe Ritzenthaler and Matthieu Romagny,
*On the Prym variety of genus 3 covers of elliptic curves*, arXiv:1612.07033, 2016. - J. Sijsling,
*arithmetic-geometric_mean; a package for calculating period matrices via the arithmetic-geometric mean*, https://github.com/JRSijsling/arithmetic-geometric\_mean/, 2016. - B. Smith,
*Explicit endomorphisms and correspondences*, Ph.D. thesis, University of Sydney, 2005. - John Tate,
*Endomorphisms of abelian varieties over finite fields*, Invent. Math.**2**(1966), 134–144. MR**206004**, DOI 10.1007/BF01404549 - Paul van Wamelen,
*Examples of genus two CM curves defined over the rationals*, Math. Comp.**68**(1999), no. 225, 307–320. MR**1609658**, DOI 10.1090/S0025-5718-99-01020-0 - Paul van Wamelen,
*Proving that a genus $2$ curve has complex multiplication*, Math. Comp.**68**(1999), no. 228, 1663–1677. MR**1648415**, DOI 10.1090/S0025-5718-99-01101-1 - Paul van Wamelen,
*Poonen’s question concerning isogenies between Smart’s genus $2$ curves*, Math. Comp.**69**(2000), no. 232, 1685–1697. MR**1677415**, DOI 10.1090/S0025-5718-99-01179-5 - Paul B. van Wamelen,
*Computing with the analytic Jacobian of a genus 2 curve*, Discovering mathematics with Magma, Algorithms Comput. Math., vol. 19, Springer, Berlin, 2006, pp. 117–135. MR**2278925**, DOI 10.1007/978-3-540-37634-7_{5} - William C. Waterhouse,
*Abelian varieties over finite fields*, Ann. Sci. École Norm. Sup. (4)**2**(1969), 521–560. MR**265369** - W. C. Waterhouse and J. S. Milne,
*Abelian varieties over finite fields*, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 53–64. MR**0314847** - David Zywina,
*The splitting of reductions of an abelian variety*, Int. Math. Res. Not. IMRN**18**(2014), 5042–5083. MR**3264675**, DOI 10.1093/imrn/rnt113

## Additional Information

**Edgar Costa**- Affiliation: Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, New Hampshire 03755
- Address at time of publication: Department of Mathematics, 77 Massachusetts Ave., Bldg. 2-252B, Cambridge, Massachusetts 02139
- MR Author ID: 1041071
- ORCID: 0000-0003-1367-7785
- Email: edgarc@mit.edu
**Nicolas Mascot**- Affiliation: Department of Mathematics, University of Warwick, Coventry CV4 7AL, United Kingdom
- Address at time of publication: Department of Mathematics, Faculty of Arts and Sciences, American University of Beirut, P.O. Box 11-0236, Riad El Solh, Beirut 1107 2020, Lebanon
- MR Author ID: 1040021
- Email: nm116@aub.edu.lb
**Jeroen Sijsling**- Affiliation: Universität Ulm, Institut für Reine Mathematik, D-89068 Ulm, Germany
- MR Author ID: 974789
- ORCID: 0000-0002-0632-9910
- Email: jeroen.sijsling@uni-ulm.de
**John Voight**- Affiliation: Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, New Hampshire 03755
- MR Author ID: 727424
- ORCID: 0000-0001-7494-8732
- Email: jvoight@gmail.com
- Received by editor(s): May 30, 2017
- Received by editor(s) in revised form: January 14, 2018, and April 4, 2018
- Published electronically: September 10, 2018
- Additional Notes: The second author was supported by the EPSRC Programme Grant EP/K034383/1 “LMF: L-Functions and Modular Forms”.

The third author was supported by the Juniorprofessuren-Programm “Endomorphismen algebraischer Kurven” (7635.521(16)) from the Science Ministry of Baden–Württemberg.

The fourth author was supported by an NSF CAREER Award (DMS-1151047) and a Simons Collaboration Grant (550029). - © Copyright 2018 American Mathematical Society
- Journal: Math. Comp.
**88**(2019), 1303-1339 - MSC (2010): Primary 11G10, 11Y99, 14H40, 14K15, 14Q05
- DOI: https://doi.org/10.1090/mcom/3373
- MathSciNet review: 3904148