The group law on an elliptic curve, and multiplication by $n$

Choosing an origin identifies a genus-$1$ curve with its $\Pic^0$; collinearity is addition, and $[n]$ has degree $n^2$.Suggest a slogan

Let $E$ be a curve of genus $1$ with a chosen point $p_0$. Then $p \mapsto \OO_E(p - p_0)$ is a bijection $E \to \Pic^0(E)$, and the group structure it pulls back makes $E$ a group variety: inversion and addition are morphisms, so $E$ is a group scheme over $k$ with identity $p_0$. Under the embedding by $\abs{3p_0}$ as a plane cubic, three points sum to zero exactly when they are collinear.

(Multiplication by $n$)

$[n] \colon E \to E$ is a finite morphism of degree $n^2$. If $n$ is prime to $\operatorname{char} k$ then $E[n] \cong (\ZZ/n)^2$. If $n = p = \operatorname{char} k$ then $E[p]$ is $\ZZ/p$ or trivial, according to the Hasse invariant; the trivial case is called supersingular.

The bijection is the whole proof and is pure Riemann–Roch: given $D$ of degree zero, $D + p_0$ has degree $1 > 2g-2 = 0$, so $\ell(D+p_0) = 1$ and there is a unique effective divisor $x$ with $x \sim D + p_0$, that is $D \sim x - p_0$. Surjectivity and injectivity both fall out of this uniqueness.

The chord-and-tangent construction is then a translation of the same statement: $p + q + r \sim 3p_0$ exactly when $p, q, r$ are cut out by a line, since $\abs{3p_0}$ is the hyperplane system. So the geometric group law is not an extra structure, it is the linear equivalence written in coordinates.

Two consequences: a morphism of elliptic curves carrying base point to base point is automatically a group homomorphism, and $\operatorname{End}(E, p_0)$ is a ring containing $\ZZ$. When it is strictly larger, $E$ has complex multiplication, which over $\CC$ means $\tau$ generates an imaginary quadratic field.

Why is an elliptic curve a group?

Is an elliptic curve a group scheme?

What is the kernel of multiplication by $n$?

Dependencies

T-MWDVLRiemann--Roch and Serre duality on a curveuses
T-U5QSYThe Jacobian, Abel's theorem, and the Abel--Jacobi mapuses

Backlinks

T-CRVCMComplex multiplication, and $\operatorname{End}(E,p_0)$ as an order in an imaginary quadratic fieldrelated-to
T-CRVUNIFEvery elliptic curve over $\CC$ is a torus $\CC/\Lambda$related-to
T-CRVJACFUNThe Jacobian represents $\Pic^0(X/-)$, and is a smooth proper group scheme of dimension $g$related-to
T-CRVMORDELLMordell's theorem, and $E(\QQ)$ as a finitely generated abelian groupuses
T-CRVJINVThe $j$-invariant classifies elliptic curvesuses
D-CRVHASSEThe Hasse invariant, and ordinary versus supersingularuses

What links to this

Elliptic curves