Reduction of elliptic curves over certain real quadratic number fields

Kida, Masanari

doi:10.1090/s0025-5718-99-01129-1

Cited by 14 publications

(13 citation statements)

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“…This is true only if E has a K-rational point of order 2. Thus we showed that This extends results of Pinch [13] and Kida [7]. An elliptic curve E over K is called admissible if the following conditions are satisified:…”

Section: A List Of Pairs Not Satisfying ( †)supporting

confidence: 84%

The nonexistence of certain representations of the absolute Galois group of quadratic fields

Sengun

2008

Proc. Amer. Math. Soc.

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Abstract. For a quadratic field K, we investigate continuous mod p representations of Gal(K/K) that are unramified away from {p, ∞}. We prove that for certain (K, p) there are no such irreducible representations. We also list some imaginary quadratic fields for which such irreducible representations exist. As an application, we look at elliptic curves with good reduction away from 2 over quadratic fields.

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Section: A List Of Pairs Not Satisfying ( †)supporting

confidence: 84%

The nonexistence of certain representations of the absolute Galois group of quadratic fields

Sengun

2008

Proc. Amer. Math. Soc.

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“…For the values of m above such that the class number of Q m p is prime to 6, that is, for m 2, 3, 5, 6, 7, 13, 14, 17, 21, 41, 47, 73, 94, 97, the same results have already been obtained in [2], [4], [5], [10]. It is worth remarking that we use the class field theory only, whereas the authors of [2], [4] and [5] used Serres results on Galois representation theory ( [11]) or the ramification theory in Kummer extensions in addition.…”

Section: Appendixsupporting

confidence: 68%

“…We also deal with many other fields in Appendix, where we use similar arguments given in section 3.1 in order to simplify the arguments in our previous papers [2], [4], [5], [10].…”

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confidence: 99%

Determination of elliptic curves with everywhere good reduction over real quadratic fields

Kagawa

1999

Arch. Math.

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All elliptic curves having everywhere good reduction over Q 29 p are determined by studying the fields of 2-and 3-division points. As a byproduct of the argument, the elliptic curves over some real quadratic fields are determined. Though part of the result are already obtained in [2], [4], [5], [10], the proof given in the present paper is simpler. 1. Introduction. Let d be the discriminant of a real quadratic field and c d the associated Dirichlet character. Let S d S 2 G 0 d; c d be the space of cuspforms of Neben-type of weight two. When S d has a 2-dimensional Q-simple factor, Shimura [14] constructed a certain abelian surface A defined over Q from the factor and showed that A splits over the field Q d p as B Â B H , where B is an elliptic curve defined over Q d p and B H is the conjugate of B. It is known that the curve B, which we call Shimuras elliptic curve over Q d p , has everywhere good reduction over Q d p , and is isogenous over Q d p to B H . Conversely, it is conjectured by Pinch ([10]) that any elliptic curve with such properties should be isogenous over Q d p to Shimuras elliptic curve. By Shimura [14], S d is f0g for d 5; 13; 17, and 2-dimensional and Q-simple for d 29; 37; 41. Hence, assuming Pinchs conjecture, there are no such curves when d 5; 13; 17, and there is only one isogeny class of such curves when d 29; 37; 41. In [3], [4], it was proved that this conclusion is true without the conjecture for all these d except d 29 (see also [2], [10]). In this paper, we prove that it is also true for d 29 by determining all elliptic curves with everywhere good reduction over k Q 29 p . In his paper [9], Nakamura has proved the conjecture of Serre given in [11], p. 184, which states that Shimuras elliptic curve over k is isogenous over k to

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“…This result is further generalized to certain quadratic fields and some other fields (see [9] and the references there).…”

Section: Introductionmentioning

confidence: 72%

Good reduction of elliptic curves over imaginary quadratic fields

Kida¹

2001

Journal De Théorie Des Nombres De Bordeaux

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Reduction of elliptic curves over certain real quadratic number fields

Cited by 14 publications

References 5 publications

The nonexistence of certain representations of the absolute Galois group of quadratic fields

The nonexistence of certain representations of the absolute Galois group of quadratic fields

Determination of elliptic curves with everywhere good reduction over real quadratic fields

Good reduction of elliptic curves over imaginary quadratic fields

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