# minimal model for an elliptic curve

Let $K$ be a local field^{}, complete^{} with respect to a discrete valuation^{} $\nu $ (for example, $K$ could be ${\mathbb{Q}}_{p}$, the field of http://planetmath.org/node/PAdicIntegers$\mathrm{p}$-adic numbers, which is complete with respect to the http://planetmath.org/node/PAdicValuation$\mathrm{p}$-adic valuation^{}).

Let $E/K$ be an elliptic curve^{} defined over $K$ given by a Weierstrass equation

$${y}^{2}+{a}_{1}xy+{a}_{3}y={x}^{3}+{a}_{2}{x}^{2}+{a}_{4}x+{a}_{6}$$ |

where ${a}_{1},{a}_{2},{a}_{3},{a}_{4},{a}_{6}$ are constants in $K$. By a suitable change of variables, we may assume that $\nu ({a}_{i})\ge 0$. As it is pointed out in http://planetmath.org/node/WeierstrassEquationOfAnEllipticCurvethis entry, any other Weierstrass equation for $E$ is obtained by a change of variables of the form

$$x={u}^{2}{x}^{\prime}+r,y={u}^{3}{y}^{\prime}+s{u}^{2}{x}^{\prime}+t$$ |

with $u,r,s,t\in K$ and $u\ne 0$. Moreover, by Proposition^{} 2 in the same entry, the discriminants^{} of both equations satisfy $\mathrm{\Delta}={u}^{12}{\mathrm{\Delta}}^{\prime}$, so they only differ by a $12$th power of a non-zero number in $K$. Let us define a set:

$$S=\{\nu (\mathrm{\Delta}):\mathrm{\Delta}\text{is the discriminant of a Weierstrass eq. for}E\text{and}\nu (\mathrm{\Delta})\ge 0\}$$ |

Since $\nu $ is a discrete valuation, the set $S$ is a set of non-negative integers, therefore it has a minimum value $m\in S$. Moreover, by the remark above, $m$ satisfies $$ and $m$ is the unique number $t\in S$ with $$.

###### Definition.

Let $E\mathrm{/}K$ be an elliptic curve over a local field $K$, complete with respect to a discrete valuation $\nu $. A Weierstrass equation for $E$ with discriminant $\mathrm{\Delta}$ is said to be a minimal model for $E$ (at $\nu $) if $\nu \mathit{}\mathrm{(}\mathrm{\Delta}\mathrm{)}\mathrm{=}m$, the minimum of the set $S$ above.

It follows from the discussion above that every elliptic curve over a local field $K$ has a minimal model over $K$.

###### Definition.

Let $F$ be a number field^{} and let $\nu $ be an infinite^{} or finite place (archimedean or non-archimedean prime) of $F$. Let $E\mathrm{/}F$ be an elliptic curve over $F$. A given Weierstrass model for $E\mathrm{/}F$ is said to be minimal^{} at $\nu $ if the same model is minimal over ${F}_{\nu}$, the completion of $F$ at $\nu $. A Weierstrass equation for $E\mathrm{/}F$ is said to be minimal if it is minimal at $\nu $ for all places $\nu $ of $F$.

It can be shown that all elliptic curves over $\mathbb{Q}$ have a global minimal model. However, this is not true over general number fields. There exist elliptic curves over a number field $F$ which do not have a global minimal model (i.e. any given model is not minimal at $\nu $ for every $\nu $).

Title | minimal model for an elliptic curve |
---|---|

Canonical name | MinimalModelForAnEllipticCurve |

Date of creation | 2013-03-22 15:48:03 |

Last modified on | 2013-03-22 15:48:03 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 4 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 14H52 |

Classification | msc 11G05 |

Classification | msc 11G07 |

Synonym | minimal equation |

Defines | minimal model |