Shimura subgroup

In mathematics, the Shimura subgroup Σ(N) is a subgroup of the Jacobian of the modular curve X₀(N) of level N, given by the kernel of the natural map to the Jacobian of X₁(N). It is named after Goro Shimura. There is a similar subgroup Σ(N,D) associated to Shimura curves of quaternion algebras.

References

• Ling, San; Oesterlé, Joseph (1991), "The Shimura subgroup of J₀(N)", Astérisque (196): 171–203, ISSN 0303-1179, MR 1141458 
• Mazur, Barry (1977), "Modular curves and the Eisenstein ideal", Publications Mathématiques de l'IHÉS (47): 33–186, ISSN 1618-1913, MR 488287 
• Ribet, Kenneth A. (1984), "Congruence relations between modular forms", Proceedings of the International Congress of Mathematicians, Vol. 1 (Warsaw, 1983), Warszawa: PWN, pp. 503–514, MR 804706 
• Ribet, Kenneth A. (1988), "On the component groups and the Shimura subgroup of J₀(N)", Séminaire de Théorie des Nombres, 1987--1988 (Talence, 1987--1988), Talence: Univ. Bordeaux I, pp. Exp. No. 6, 10, MR 993107