It is well-known that there is no elliptic curve defined over the field Q of rational numbers having good reduction at every finite place. On the other hand, several examples of such curves over quadratic fields are known.In this paper, we will show nonexistence theorems of elliptic curves having good reduction everywhere over certain real quadratic fields. Ishii obtained a similar result in [4]. While he handled real quadratic fields in which 2 remains prime, we will prove theorems for quadratic fields in which 2 is unramified. Thus our results are a generalization and a refinement of his result.Throughout this paper, we always assume that our elliptic curves have a global minimal model. Equivalently, we consider elliptic curves that have a model with a unit discriminant. This is always the case if the class number of the field of definition is prime to six (see [9, Corollary to Theorem 1]). Here we should mention a result of Stroeker. He showed, in his paper [12], that if an elliptic curve defined over an imaginary quadratic field has good reduction everywhere, then the curve does not admit a global minimal model.

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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We study the variation of the reduction type of elliptic curves under base change. A complete description of the variation is given when the base field is the p-adic field and the base change is of small degree.

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