What do the following terms mean in the context of elliptic (and Tate) curves?

  1. Stable reduction

  2. Bad primes, bad reduction

  • $\begingroup$ Where did you learn about elliptic curves that these weren't defined? $\endgroup$ – Matt Nov 20 '12 at 20:12
  • $\begingroup$ I have Milne's "Elliptic curve" but I did not start to read it, and I cannot look at it right now because there is no electricity. $\endgroup$ – user48900 Nov 20 '12 at 20:16
  • $\begingroup$ If you haven't started reading it, will you be able to understand the answer(s) to your question, i.e., do you have a prior knowledge of elliptic curves? $\endgroup$ – Keenan Kidwell Nov 20 '12 at 20:18
  • $\begingroup$ I know they are algebraic curves of genus 1. $\endgroup$ – user48900 Nov 20 '12 at 20:21
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    $\begingroup$ @user48900: I'm intrigued that you couldn't look at Milne's book due to lack of electricity but at the same time you were able to post a comment on the internet. $\endgroup$ – KCd Nov 20 '12 at 21:20

If $E$ is an elliptic curve over a finite extension of $\mathbf{Q}_p$, say, for some $p$, then, concretely, $E$ has good reduction over $\mathbf{Q}_p$ if there exists a Weierstrass equation for $E$ with coefficients in $\mathbf{Z}_p$ such that, when we reduce the coefficients modulo the maximal ideal of $\mathbf{Z}_p$, the resulting cubic equation over $\mathbf{F}_p$ defines an elliptic curve over $\mathbf{F}_p$, i.e., the equation is ``nonsingular," meaning the discriminant doesn't vanish. This amounts to finding an equation for the curve over $\mathbf{Z}_p$ whose discriminant is not divisible by $p$, i.e., whose discriminant is a unit. If no such equation can be found, $E$ has bad reduction.

The term stable reduction has to do with the nature of the curve over $\mathbf{F}_p$ one gets by reducing the coefficients of a "minimal Weierstrass equation" for $E$ with coefficients in $\mathbf{Z}_p$ (an equation for which the valuation of the discriminant is minimized). In case this curve is not an elliptic curve (bad reduction), the reduction type is classified as either additive or multiplicative according as the group of of $\overline{\mathbf{F}}_p$-rational points of the reduced curve which are ``nonsingular" is isomorphic to $\overline{\mathbf{F}}_p$ or $\overline{\mathbf{F}}_p^\times$ (as a group). We say $E$ has stable reduction if it has good reduction, semi-stable reduction if it has bad multiplicative, and unstable reduction if it has bad additive reduction. One reason for these terms is that, if we extend the base field from $\mathbf{Q}_p$ to a finite extension, the first two kinds of reduction stay the same, but the third can become one of the other two (over the extended base field). If you take a big enough extension of the base field, $E$ will eventually acquire good or bad multiplicative reduction.

If $E$ is an elliptic curve over $\mathbf{Q}$, then it can be viewed as an elliptic curve over $\mathbf{Q}_p$ for all $p$, and we say the reduction of $E$ is good, bad, additive, multiplicative, etc. at a particular $p$ if, when we view $E$ over $\mathbf{Q}_p$, it has the corresponding behavior. We then say $p$ is a good or bad prime for $E$ accordingly. Also, $E$ is called semi-stable if it has good or bad multiplicative reduction at all primes $p$.

There are much more precise (sophisticated) definitions for reduction, additive, multiplictive, etc., using the language of Neron models, which I can say something about if you're interested.

EDIT: The Neron model $\mathscr{E}$ of an elliptic curve over $\mathbf{Q}_p$ is the (unique up to unique isomorphism) smooth group scheme over $\mathbf{Z}_p$ whose generic fiber is $E$ and which satisfies the Neron mapping property for smooth $\mathbf{Z}_p$-schemes, meaning that for any smooth $\mathbf{Z}_p$-scheme $S$, any $\mathbf{Q}_p$-morphism from the generic fiber of $S$ to $E$ extends uniquely to a $\mathbf{Z}_p$-morphism from $S$ to $\mathscr{E}$. In particular, $\mathscr{E}(\mathbf{Z}_p)=E(\mathbf{Q}_p)$ via the canonical map. The Neron model can be connected to certain other concrete Weierstrass models of $E$, but its definition is completely canonically and doesn't involve any choice of equations (it is the smooth locus of the minimal regular model of $E$ over $\mathbf{Z}_p$). The existence of $\mathscr{E}$ allows us to canonically associate to $E$ a smooth group scheme over $\mathbf{F}_p$, namely the special fiber $\mathscr{E}_s$ of $\mathscr{E}$. While this is a smooth group scheme, it might not be proper, but it is an algebraic group over $\mathbf{F}_p$, so the open and closed identity component $\mathscr{E}_s^0$ of $\mathscr{E}_s$ is a subgroup scheme. This group scheme is either an elliptic curve (in which case it equals $\mathscr{E}_s$, which is proper), meaning good reduction, or a $1$-dimensional torus, i.e., a $1$-dimensional $\mathbf{F}_p$-group which becomes isomorphic to $\mathbf{G}_m$ after a finite extension, meaning multiplicative reduction (split or non-split according as the torus is split or non-split), or isomorphic to $\mathbf{G}_a$, the additive group, meaning additive reduction.

If you write down a Weierstrass equation with coefficients in $\mathbf{Z}_p$ (or any discrete valuation ring) with unit discriminant, then the associated closed subscheme of $\mathbf{P}^2_{\mathbf{Z}_p}$ is an abelian scheme of relative dimension $1$, i.e., an elliptic curve over $\mathbf{Z}_p$, and is necessarily the Neron model of its generic fiber.

This all basically extends to elliptic curves over the function field of a Dedekind scheme.

  • $\begingroup$ I would like to see the definitions of reduction using the language of Neron models, if you have time to write something. Thanks! $\endgroup$ – user27126 Nov 20 '12 at 20:39
  • $\begingroup$ "If no such equation can be found, E has bad reduction. " How can I tell that I 'cannot find' such an equation? $\endgroup$ – Dan Apr 8 '16 at 12:06

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