Computing systems of Hecke eigenvalues associated to Hilbert modular forms

Greenberg, Matthew; Voight, John

doi:10.1090/S0025-5718-2010-02423-8

Computing systems of Hecke eigenvalues associated to Hilbert modular forms

Authors: Matthew Greenberg and John Voight
Journal: Math. Comp. 80 (2011), 1071-1092
MSC (2010): Primary 11F46, 11G18
DOI: https://doi.org/10.1090/S0025-5718-2010-02423-8
Published electronically: September 30, 2010
MathSciNet review: 2772112
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Abstract: We utilize effective algorithms for computing in the cohomology of a Shimura curve together with the Jacquet-Langlands correspondence to compute systems of Hecke eigenvalues associated to Hilbert modular forms over a totally real field .

1. 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, https://doi.org/10.1006/jsco.1996.0125
2. Henri Carayol, Sur la mauvaise réduction des courbes de Shimura, Compositio Math. 59 (1986), no. 2, 151–230 (French). MR 860139
3. J. E. Cremona, Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compositio Math. 51 (1984), no. 3, 275–324. MR 743014
4. J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
5. John Cremona, The elliptic curve database for conductors to 130000, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 11–29. MR 2282912, https://doi.org/10.1007/11792086_2
6. J. E. Cremona and M. P. Lingham, Finding all elliptic curves with good reduction outside a given set of primes, Experiment. Math. 16 (2007), no. 3, 303–312. MR 2367320
7. Henri Darmon, Integration on ℋ_{𝓅}×ℋ and arithmetic applications, Ann. of Math. (2) 154 (2001), no. 3, 589–639. MR 1884617, https://doi.org/10.2307/3062142
8. Henri Darmon and Samit Dasgupta, Elliptic units for real quadratic fields, Ann. of Math. (2) 163 (2006), no. 1, 301–346. MR 2195136, https://doi.org/10.4007/annals.2006.163.301
9. Henri Darmon and Robert Pollack, Efficient calculation of Stark-Heegner points via overconvergent modular symbols, Israel J. Math. 153 (2006), 319–354. MR 2254648, https://doi.org/10.1007/BF02771789
10. Pierre Deligne, Travaux de Shimura, Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, Springer, Berlin, 1971, pp. 123–165. Lecture Notes in Math., Vol. 244 (French). MR 0498581
11. Lassina Dembélé, An algorithm for modular elliptic curves over real quadratic fields, Experiment. Math. 17 (2008), no. 4, 427–438. MR 2484426
12. Lassina Dembélé, Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms, Math. Comp. 76 (2007), no. 258, 1039–1057. MR 2291849, https://doi.org/10.1090/S0025-5718-06-01914-4
13. Lassina Dembélé and Steve Donnelly, Computing Hilbert modular forms over fields with nontrivial class group, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 5011, Springer, Berlin, 2008, pp. 371–386. MR 2467859, https://doi.org/10.1007/978-3-540-79456-1_25
14. L. Dieulefait, L. Guerberoff, A. Pacetti, Proving modularity for a given elliptic curve over an imaginary quadratic field, Math. Comp. 79 (2010), 1145-1170.
15. S. Donnelly and J. Voight, Tables of Hilbert modular forms and elliptic curves over totally real fields, in preparation.
16. M. Eichler, Grenzkreisgruppen und kettenbruchartige Algorithmen, Acta Arith. 11 (1965), 169–180 (German). MR 228436, https://doi.org/10.4064/aa-11-2-169-180
17. Noam D. Elkies, Shimura curve computations, Algorithmic number theory (Portland, OR, 1998) Lecture Notes in Comput. Sci., vol. 1423, Springer, Berlin, 1998, pp. 1–47. MR 1726059, https://doi.org/10.1007/BFb0054850
18. Eberhard Freitag, Hilbert modular forms, Springer-Verlag, Berlin, 1990. MR 1050763
19. Matthew Greenberg, Stark-Heegner points and the cohomology of quaternionic Shimura varieties, Duke Math. J. 147 (2009), no. 3, 541–575. MR 2510743, https://doi.org/10.1215/00127094-2009-017
20. Haruzo Hida, On abelian varieties with complex multiplication as factors of the Jacobians of Shimura curves, Amer. J. Math. 103 (1981), no. 4, 727–776. MR 623136, https://doi.org/10.2307/2374049
21. Haruzo Hida, On 𝑝-adic Hecke algebras for 𝐺𝐿₂ over totally real fields, Ann. of Math. (2) 128 (1988), no. 2, 295–384. MR 960949, https://doi.org/10.2307/1971444
22. Haruzo Hida, 𝑝-adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer-Verlag, New York, 2004. MR 2055355
23. Haruzo Hida, Hilbert modular forms and Iwasawa theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2243770
24. H. Jacquet and R. P. Langlands, Automorphic forms on 𝐺𝐿(2), Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654
25. M. Kirschmer and J. Voight, Algorithmic enumeration of ideal classes for quaternion orders, submitted.
26. Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66 (Russian). MR 0314846
27. Yozô Matsushima and Goro Shimura, On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes, Ann. of Math. (2) 78 (1963), 417–449. MR 155340, https://doi.org/10.2307/1970534
28. Takeshi Saito, Hilbert modular forms and 𝑝-adic Hodge theory, Compos. Math. 145 (2009), no. 5, 1081–1113. MR 2551990, https://doi.org/10.1112/S0010437X09004175
29. Matthias Schütt, On the modularity of three Calabi-Yau threefolds with bad reduction at 11, Canad. Math. Bull. 49 (2006), no. 2, 296–312. MR 2226253, https://doi.org/10.4153/CMB-2006-031-9
30. Goro Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679. MR 507462
31. Goro Shimura, Construction of class fields and zeta functions of algebraic curves, Ann. of Math. (2) 85 (1967), 58–159. MR 204426, https://doi.org/10.2307/1970526
32. Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kanô Memorial Lectures, 1. MR 1291394
33. C. M. Skinner and A. J. Wiles, Residually reducible representations and modular forms, Inst. Hautes Études Sci. Publ. Math. 89 (1999), 5–126 (2000). MR 1793414
34. William Stein, Modular forms, a computational approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, RI, 2007. With an appendix by Paul E. Gunnells. MR 2289048
35. William A. Stein and Mark Watkins, A database of elliptic curves—first report, Algorithmic number theory (Sydney, 2002) Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 267–275. MR 2041090, https://doi.org/10.1007/3-540-45455-1_22
36. Richard Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), no. 2, 265–280. MR 1016264, https://doi.org/10.1007/BF01388853
37. Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949
38. John Voight, Computing fundamental domains for Fuchsian groups, J. Théor. Nombres Bordeaux 21 (2009), no. 2, 469–491 (English, with English and French summaries). MR 2541438
39. John Voight, Shimura curves of genus at most two, Math. Comp. 78 (2009), no. 266, 1155–1172. MR 2476577, https://doi.org/10.1090/S0025-5718-08-02163-7
40. J. Voight, Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms, submitted.