1. Let k be a number field. It is a fascinating problem to determine the elliptic curves with everywhere good reduction over k. It is well known that there is no such curve over the field of rational numbers. When k is an imaginary quadratic field, Stroeker [Str] showed that such a curve does not admit a global minimal model, and also that there is no such curve over k provided that the class number of k is prime to 6. Hence the problem is essentially solved in this case. It is natural that we next turn to the case where k is a real quadratic field. Another reason we are interested in this case is related to Shimura’s elliptic curves obtained in the following way. Let N be a positive fundamental discriminant and let χN be the associated Dirichlet character. When the space S2(Γ0(N), χN ) of cuspforms of Neben-type of weight two has a 2dimensional Q-simple factor, Shimura [Shim] constructed an abelian surface A defined over Q. Over the real quadratic field k = Q( √ N), A splits as B×B′, where B is an elliptic curve defined over k and B′ is the conjugate of B. We call B Shimura’s elliptic curve over k. It is known that B is isogenous to B′ over k ([Shim]), and that B has everywhere good reduction over k (cf. [Ca], [DR], [KM]). Conversely, an elliptic curve E over a real quadratic field k with the properties stated above is conjectured by Pinch [Pi1] to be isogenous over k to Shimura’s elliptic curve. For related topics concerning modularity of elliptic curves over number fields, see [Ha1], [HHM]. Hence the case of a real quadratic field is especially interesting. In this case, the following is known:

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Determination of elliptic curves with everywhere good reduction over ℚ(√37)

Potential Good Reduction of Elliptic Curves

Elliptic curves over real quadratic fields with everywhere good reduction and a non-trivial 3-division point

We give a criterion for cubic fields over which there exist no elliptic curves with good reduction everywhere, and we construct a certain infinite family of cubic fields over which there exist elliptic curves with good reduction everywhere.

Elliptic curves with good reduction everywhere over cubic fields

We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quadratic fields Q(m) for m≤200. These results of computations give best-possible data including structures of Mordell-Weil groups over some real quadratic fields via two-descent. We also prove similar results for the case of certain cubic fields. Especially, we give the first example of elliptic curve having everywhere good reduction over a pure cubic field using our method.

/pdf/on-elliptic-curves-with-everywhere-good-reduction-over-1y744svfiy.pdf

On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields

An elliptic curve $E$ defined over $\bar{\mathbf{Q}}$ is called a $\mathbf{Q}$-curve, if $E$ and $E^\sigma$ are isogenous over $\bar{\mathbf{Q}}$ for any $\sigma$ in $\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})$. For a real quadratic field $K$ and a prime number $p$, we consider a $\mathbf{Q}$-curve $E$ with the following properties: 1) $E$ is defined over $K$, 2) $E$ has everywhere good reduction over $K$, 3) there exists a $p$-isogeny between $E$ and its conjugate $E^\sigma$. In this paper, a method to construct such a $\mathbf{Q}$-curve $E$ for some $p$ will be given.

https://projecteuclid.org/download/pdf_1/euclid.tjm/1270041995

Atsuki Umegaki

Papers

A Construction of Everywhere Good $\mathbf{Q}$-Curves with $p$-Isogeny