Minimal Models for Elliptic Curves

In the 1960’s, the efforts of many mathematicians (Kodaira, Néron, Raynaud, Tate, Lichtenbaum, Shafarevich, Lipman, and Deligne-Mumford) led to a very elegant theory of preferred integral models for both (positive-genus) curves and abelian varieties. This work was largely inspired by the theory of minimal models for smooth proper algebraic surfaces over algebraically closed fields [2]. There are some very special integral models, called minimal regular proper models for curves and Néron models for abelian varieties. Excellent references on these topics are [3] and [11], as well as [4] and [1], and we will provide an overview of the main definitions and results below. Elliptic curves occupy a special place in these theories, as they straddle the worlds of both curves and abelian varieties. Thus, an elliptic curve over the fraction field K of a discrete valuation ring R has both a Néron model and a minimal regular proper model over R. Moreover, it has an abstract minimal Weierstrass model over R that is unique up to unique R-isomorphism. It is natural to ask how the preferred models for elliptic curves are related to each other. A tricky aspect is that minimal Weierstrass models are (usually) defined in a manner that is a bit too explicit and is lacking in an abstract universal property, whereas both Néron models of abelian varieties and minimal regular proper models of smooth (positive-genus) proper curves are characterized by simple abstract universal properties. The key aspects of the story (including all necessary background on regular models of curves) are presented in [11] in complete detail, so these notes may be viewed as a complement to the discussion in [11] (I will generally refer to [1], [3], and [4] for results that are also proved in [11] because the former are the references from which I learned about these matters, before [11] was written; the reader may well find that [11] is more useful and/or more understandable than these notes). We begin in §2 with a brief summary of the theory of Weierstrass models of elliptic curves. The main point is to formulate the theory in a manner that eliminates the appearance of Weierstrass equations; this is accomplished by using Serre duality over the residue field, generalizing the use of Riemann–Roch to free the theory of elliptic curves from the curse of Weierstrass equations. In §3, we provide an overview of the theory of integral models of smooth curves, with an emphasis on minimal regular proper models. The theory of minimal Weierstrass models is addressed in §4, where we give an abstract criterion for minimality of a Weierstrass model; this criterion (which was brought to my attention by James Parson, and is technically much simpler than my earlier viewpoint on these matters) is expressed in terms of R-rational maps. In §5 we switch to the category of abelian varieties and we present the basic definitions and existence theorem concerning Néron models, and we deduce some relations between Weierstrass models and Néron models. The basic properties of relative dualizing sheaves and arithmetic intersection theory are summarized in §6 and §7, and we apply these notions in §8 to establish some additional conceptual characterizations of minimal Weierstrass models (e.g., in terms of rational singularities). In particular, we provide a conceptual explanation for why Tate’s algorithm does not require the computation of normalizations. In these notes, a singularity on a curve over a field k is a point not in the k-smooth locus, rather than a point in the non-regular locus. The reason for this distinction is that there are examples of Weierstrass cubics over k that are regular and not smooth (these can only exist when k has characteristic 2 or 3; see the discussion following Theorem 5.5), and we want to consider these as singular.