Reduction of elliptic curves over certain real quadratic number fields

Kida, Masanari

doi:10.1090/S0025-5718-99-01129-1

Reduction of elliptic curves over certain real quadratic number fields
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Math. Comp. 68 (1999), 1679-1685 Request permission

Abstract:

The main result of this paper is that an elliptic curve having good reduction everywhere over a real quadratic field has a $2$-rational point under certain hypotheses (primarily on class numbers of related fields). It extends the earlier case in which no ramification at $2$ is allowed. Small fields satisfying the hypotheses are then found, and in four cases the non-existence of such elliptic curves can be shown, while in three others all such curves have been classified.

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