Modularity conjecture for q-curves and QM-curves

International Journal of Mathematics

Volumn 10, Issue 8, 1999, Pages 1011-1036

  Hasegawa, Yuji ^a   Hashimoto, Ki Ichiro ^a   Momose, Fumiyuki ^b  

^a WASEDA UNIVERSITY   (Japan)

^b CHUO UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033270336     PISSN: 0129167X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0129167X99000434     Document Type: Article

References (58)

1. 0(m), Math. Ann. 185 (1970), 134-160.
2. H. Carayol, Sur la mauvaise réduction des courbes de Shimura, Compositio Math. 59 (1986), 151-230.
3. v), Funktional. Anal. i Prilozhen. 9 (1975), 95-96.
4. W. Chi, Twists of central simple algebras and endomorphism algebras of some Abelian varieties, Math. Ann. 276 (1987), 615-632.
5. P. Deligne, Formes modulaires et représentations ℓ-adiques, Séminaire Bourbaki 355°, in "Séminaire Bourbaki Vol. 1968/69 Exposés 347-363", Lecture Notes in Math. 179, Springer-Verlag, 1971, pp. 139-172.
6. P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, in Modular Functions of One Variable II, eds. P. Deligne and W. Kuyk, Lecture Notes in Math. 349, Springer-Verlag, 1973, pp. 143-316.
7. P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. scient. Éc. Norm. Sup. 7 (1974), 507-530.
8. F. Diamond, The refined conjecture of Serre, in Conference on Elliptic Curves and Modular Forms, Hong Kong, December 18-21, 1993, Int. Press, 1995, pp. 22-37.
9. _, On deformation rings and Hecke rings, Ann. Math. 144 (1996), pp. 137-166.
10. S. J. Edixhoven, The weight in Serre's conjectures on modular forms, Invent. Math. 109 (1992), 563-594.
11. N. Elkies, A remark on elliptic K-curves, preprint.
12. G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahl-körpern, Invent. Math. 73 (1983), 349-366; Erratum: 75 (1984), 381.
13. B. Gross, Arithmetic on Elliptic Curves with Complex Multiplication, Lecture Notes in Math. 776, Springer-Verlag, 1980.
14. A. Grothendieck, Groupes de Monodromie en Géométrie Algébrique I, Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7I), Lecture Notes in Math. 288, Springer-Verlag, 1972.
15. 0(N) with genus 2, Proc. Japan Acad. Ser. A 71 (1995), 235-239.
16. _, On some examples of modular QM-Abelian surfaces, Proc. Japan Acad. Ser. A 72 (1996), 23-27.
17. _, Q-curves over quadratic fields, Manuscripta Math. 94 (1997), 347-364.
18. K. Hashimoto, Explicit form of quaternion modular embeddings, Osaka J. Math. 32 (1995), 533-546.
19. 2-type, Manuscripta Math. 98 (1999), 165-182.
20. _, On Brumer's family of RM-curves of genus two, preprint.
21. K. Hashimoto and N. Murabayashi, Shimura curves as intersections of Humbert surfaces and defining equations of QM-curves of genus two, Tôhoku Math. J. 47 (1995), 271-296.
22. B. Jordan, On the diophantine arithmetic of Shimura curves, Dissertation of Harvard Univ., Cambridge, 1981.
23. M. Koike, Congruences between cuspforms and linear representations of the Galois group, Nagoya Math. J. 64 (1976), 63-85.
24. W. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285-315.
25. B. Mazur, Deforming Galois representations, in Galois groups over Q (Proc. of a Workshop Held March 23-27, 1987), eds. Y. Ihara, K. Ribet and J.-P. Serre, Math. Sci. Res. Inst. Publ. 16, Springer-Verlag, 1989, pp. 385-437.
26. J. S. Milne, On the arithmetic of Abelian varieties, Invent. Math. 17 (1972), 177-190.
27. B. Mitchell, Theory of Categories, Pure and Applied Math. XVII, Academic Press, 1965.
28. 2 and Hecke operators, Ann. Math. 94 (1971), 174-189.
29. F. Momose, On the ℓ-adic representations attached to modular forms, J. Fac. Sci. Univ. Tokyo 28 (1981), 89-109.
30. _, Galois action on some ideal section points of the Abelian variety associated with a modular form and its application, Nagoya Math. J. 91 (1983), 19-36.
31. D. Mumford, Abelian Varieties, 2nd ed., Oxford Univ. Press, 1974.
32. D. Mumford, J. Fogarty and F. Kirman, Geometric Invariant Theory, 3rd enlarged edition, Ergebnisse der Mathematik und ihre Grenzgebiete, Bd. 34, Springer-Verlag, 1994.
33. F. Oort and J. Tate, Group schemes of prime order. Ann. scient. Éc. Norm. Sup. 3 (1970), 1-21.
34. E. Pyle, Abelian varieties over Q with large endomorphism algebras and their simple components over Q̄, Dissertation of the Univ. of California at Berkeley, 1995.
35. M. Raynaud, Schémas en groupes de type (p, . . . , p), Bull. Soc. Math. France 102 (1974), 241-280.
36. K. Ribet, Endomorphisms of semi-stable Abelian varieties over number fields, Ann. Math. 101 (1975), 555-562.
37. _, Galois representations attached to eigenforms with Nebentypus, in Modular Functions of One Variable V, Lecture Notes in Math. 601, eds. J.-P. Serre and D. B. Zagier, Springer-Verlag, 1977, pp. 17-52.
38. _, Twists of modular forms and endomorphisms of Abelian varieties, Math. Ann. 253 (1980), 43-62.
39. _, Endomorphism algebras of Abelian varieties attached to newforms of weight 2, in Séminaire de Théorie des Nombres, Paris 1979-1980, ed. M.-J. Bertin, Progress Math. 12, Birkhäuser, (1981), pp. 263-276.
40. _, Abelian varieties over Q and modular forms, in 1992 Proc. KAIST Mathematics Workshop, Korea Advanced Inst. of Science and Tech. (1992), pp. 53-79.
41. _, Fields of definition of Abelian varieties with real multiplication, Contemporary Math. 174 (1994), 107-118.
42. _, Report on mod ℓ representations of Gal(Q̄/Q), in Proc. Symposia in Pure Mathematics, Vol. 55 (2) (1994), 639-676.
43. J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331.
44. _, Modular forms of weight one and Galois representations, in Algebraic Number Fields, Academic Press (1977), pp. 193-268.
45. _, Sur les représentations modulaires de degré 2 de Gal(Q̄/Q), Duke Math. J. 54 (1987), 179-230.
46. J.-P. Serre and J. Tate, Good reduction of Abelian varieties, Ann. Math. 88 (1968), 492-517.
47. G. Shimura, On the zeta-functions of the algebraic curves uniformized by certain automorphic functions, J. Math. Soc. Japan 13 (1961), 275-331.
48. _, On analytic families of polarized Abelian varieties and automorphic functions, Ann. Math. 78 (1963), 149-192.
49. _, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. Math. 91 (1970), 144-222.
50. _, On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J. 43 (1971), 199-208.
51. _, Class fields over real quadratic fields and Hecke operators, Ann. Math. 95 (1972), 130-190.
52. _, On the factors of the Jacobian varieties of a modular function field, J. Math. Soc. Japan 25 (1973), 523-544.
53. _, On the real points of an arithmetic quotient of a bounded symmetric domain, Math. Ann. 215 (1975), 135-164.
54. _, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton Univ. Press, 1971.
55. J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, in Modular Functions of One Variable IV, eds. B. Birch and W. Kuyk, Lecture Notes in Math. 476, Springer-Verlag, 1975, pp. 33-52.
56. R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), 553-572.
57. A. Weil, Adeles and algebraic groups, Progress Math. 23, Birkhäuser, 1982.
58. A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. Math. 141 (1995), 443-551.