Torsion points of elliptic curves with bad reduction at some primes II
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let K be a number field and fix a prime number p. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set BK,p of primes of K satisfying that any elliptic curve over K with BK,p-reduction has no p-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with BK,p- reduction and a p-torsion point. The action of the absolute Galois group on the p-torsion subgroup of E gives its associated Galois representation ΡE,p modulo p. We also study the irreducibility and surjectivity of ΡE,p for semistable elliptic curves with BK,p-reduction.
Abstract
Let K be a number field and fix a prime number p. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set BK,p of primes of K satisfying that any elliptic curve over K with BK,p-reduction has no p-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with BK,p- reduction and a p-torsion point. The action of the absolute Galois group on the p-torsion subgroup of E gives its associated Galois representation ΡE,p modulo p. We also study the irreducibility and surjectivity of ΡE,p for semistable elliptic curves with BK,p-reduction.
참고문헌 (14)
- D. Jeon, "Families of elliptic curves over quartic number fields with prescribed torsion subgroups", Math. Comp, 80(276), pp.2395~2410 (2011)
- S. Kamienny, "Torsion points on elliptic curves and q-coefficients of modular forms", Invent. Math, 109(2), pp.221~229 (1992)
- M. Kenku, "Torsion points on elliptic curves defined over quadratic fields", Nagoya Math. J, 109, pp.125~149 (1988)
- A. Kraus, "Courbes elliptiques semi-stables et corps quadratiques", J. Number Theory, 60(2), pp.245~253 (1996)
- D. Kubert, "Universal bounds on the torsion of elliptic curves", Proc. London Math. Soc, 33(2), pp.193~237 (1976)
- B. Mazur, "Modular curves and the Eisenstein ideal", Inst. Hautes ´Etudes Sci. Publ. Math, 47, pp.33~186 (1977)
- B. Mazur, "Rational isogenies of prime degree", Invent. Math, 44(2), pp.129~162 (1978)
- J. Neukirch, "Algebraic Number Theory, Grudlehren der Mathematischen Wisenshaften" (1999), Springer-Verlag
- Oort, "Group schemes of prime order", Ann. Sci. ´Ecole Norm, 3, pp.1~21 (1970)
- "Bordeaux, PARI/GP"[URL]
- J.-P. Serre, "Propri´et´es galoisiennes des points d’ortre fini des courbes elliptiques", Invent. Math, 15(4), pp.259~331 (1972)
- J. H. Silverman, "The Arithmetic of Elliptic Curves, Graduate Texts in Math" (1994), Springer-Verlag
- L. C. Washington, "Introduction to Cyclotomic Fields, Graduate Texts in Math" (1982), Springer-Verlag
- M. Yasuda, "Torsion points of elliptic curves with bad reduction at some primes", Commentarii Math. Univ. St. Pauli