Rigorous computation of the endomorphism ring of a Jacobian

Costa, Edgar; Mascot, Nicolas; Sijsling, Jeroen; Voight, John

doi:10.1090/mcom/3373

Rigorous computation of the endomorphism ring of a Jacobian
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by Edgar Costa, Nicolas Mascot, Jeroen Sijsling and John Voight;

Math. Comp. 88 (2019), 1303-1339

DOI: https://doi.org/10.1090/mcom/3373

Published electronically: September 10, 2018

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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 282985
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) Proc. Sympos. Pure Math., Vol. XX, Amer. Math. Soc., Providence, RI, 1971, pp. 53–64. MR 314847
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