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the  -invariant classifies elliptic curves up to isomorphism
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(Theorem)
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In this entry, an isomorphism over $K$ should be understood in the sense of the entry isomorphism of varieties.
Theorem 1 Let $K$ be a field, and let $\overline{K}$ be a fixed algebraic closure of $K$
- Two elliptic curves $E_1$ and $E_2$ are isomorphic (over $\overline{K}$ if and only if they have the same $j$ invariant, i.e. $j(E_1)=j(E_2)$
- Let $j_0\in \overline{K}$ be fixed. There exists an elliptic curve $E$ defined over the field $K(j_0)$ such that $j(E)=j_0$
Proof. For part $2$
- For $j_0=0$ the curve $E_0\colon y^2+y=x^3$ satisfies $j(E)=0$
- For $j_0=1728$ the curve $E_{1728} \colon y^2=x^3+x$ satisfies $j(E_{1728})=1728$
- If $j_0\neq 0, 1728$ consider the elliptic curve: $$E=E_{j_0} \colon y^2+xy=x^3-\frac{36}{j_0-1728}x-\frac{1}{j_0-1728}.$$ It satisfies $j(E)=j_0$ and it is defined over $K(j_0)$
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"the  -invariant classifies elliptic curves up to isomorphism" is owned by alozano.
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Cross-references: curve, elliptic curves, algebraic closure, fixed, field, isomorphism of varieties, isomorphism
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This is version 2 of the  -invariant classifies elliptic curves up to isomorphism, born on 2005-03-01, modified 2005-03-01.
Object id is 6839, canonical name is JInvariantClassifiesEllipticCurvesUpToIsomorphism.
Accessed 2263 times total.
Classification:
| AMS MSC: | 14H52 (Algebraic geometry :: Curves :: Elliptic curves) | | | 11G05 (Number theory :: Arithmetic algebraic geometry :: Elliptic curves over global fields) |
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Pending Errata and Addenda
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