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For a line bundle on an elliptic curve over a number field, the geometric cube identity remains exact as a line-bundle identity. The Weil height machine translates it into the displayed bounded-error identity for a chosen logarithmic height, uniformly on the rational triples. This distinction is essential: arbitrary naive heights do not obey an exact quadratic identity. Dividing the identity for $2^{n}P,2^{n}Q,2^{n}R$ by $4^n$ produces an exact identity for the canonical height of an elliptic curve.

With zero degree assigned to the zero element of the homomorphism group of elliptic curves, the degree parallelogram law makes this symmetric pairing additive in both arguments. Its diagonal is the degree of an isogeny; it is positive definite on every finite-dimensional rational span after extension to the real numbers. The integral polarization $2B(\varphi ,\psi )$ is integer-valued, while $B$ can be half-integral. For instance, if an elliptic curve has an automorphism $\omega$ with ${\omega }^2+\omega +1=0$, then $1+\omega =-{\omega }^2$ has degree one, so $B(1,\omega )=-1/2$.

For an elliptic curve over a number field, its nonnegative canonical height of an elliptic curve has finite bounded subsets by the Northcott theorem and its bounded difference from half the naive logarithmic height. By the Weak Mordell-Weil theorem, choose representatives $R_i$ modulo $2E(K)$ and put $H={\max }_i\hat{h}(R_i)$. If $P=2Q+R_i$, the Cauchy-Schwarz inequality for the height pairing gives $\hat{h}(Q)\le (\hat{h}(P)+H)/2$. Iterating brings every point into the finite set of height at most $H+1$. This set and the $R_i$ generate $E(K)$, proving the Mordell-Weil theorem without assuming finite generation when constructing the height.

On $E\times E$ for a complex elliptic curve, the two factor curves and the diagonal have intersection matrix $\left(\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right)$, of determinant two. Their cohomology classes are therefore independent. The degree-one image of the external product of Chow classes has cohomology contained in the span of the two factor classes. The cycle class map thus shows that the diagonal is outside that image, proving failure of surjectivity.

An elliptic curve with complex multiplication is called a CM elliptic curve. Distinguish geometric endomorphisms from endomorphisms defined over the base field. In characteristic zero, adjoining the imaginary quadratic field selected by the tangent action makes all geometric endomorphisms defined over the base. Good split primes have ordinary reduction; good inert primes have supersingular reduction.

An abelian variety of dimension $g$ has complex multiplication when its rational endomorphism ring contains a commutative semisimple $\mathbb{Q}$-algebra of dimension $2g$. For an elliptic curve in characteristic zero, this means its geometric endomorphism ring of an elliptic curve is an order in an imaginary quadratic field. The resulting rank-one Tate module representations connect torsion points to class field theory.

Every complex elliptic curve, as a complex Lie group, is the quotient of $\mathbb{C}$ by a rank-two lattice. The coordinate functions are built from the Weierstrass elliptic function and its derivative. Holomorphic group homomorphisms lift to scalar multiplication on $\mathbb{C}$, and their endomorphism condition is $aL\subseteq L$. This turns complex multiplication into a statement about lattice stabilizers.

For distinct finite point classes on a complex elliptic curve, ordinary column determinants test whether a line meets the curve at those three points. Repeated columns instead vanish for a purely linear-algebraic reason. To prescribe a tangent intersection at $u$, replace the second equal column by its derivative; to prescribe a triple intersection, use $c(u),c^{\prime }(u),c^{\prime \prime }(u)$. These confluent determinants vanish exactly when the corresponding degree-three intersection divisor has point sum zero, away from the affine chart's pole. A local homogeneous lift supplies the version at the origin.

These elliptic curves encode the congruent number problem through rational right triangles.

For a positive square-free integer $D$, rational points on $E_D:D{y}^2=x^3-x$ with $2P\ne O$ correspond to rational right triangles of area $D$. The point $(x,y)$ gives side lengths

For $d\in {\mathbb{Q}}^{\times }$, the smooth projective curve $X^3+Y^3+d{Z}^3=0$, with identity $O=(1:-1:0)$, is a genus-one elliptic curve. Its partial derivatives have no common projective zero. The invertible linear projective transformationsends it to ${\mathcal{Y}}^2\mathcal{Z}={\mathcal{X}}^3-432d^2{\mathcal{Z}}^3$, taking $O$ to $(0:1:0)$. Indeed $(X+Y{)}^3+3(X+Y)(X-Y{)}^2=4(X^3+Y^3)=-4d{Z}^3$. In affine coordinates $u=X/Z,v=Y/Z$, this gives $x=-12d/(u+v)$ and $y=-36d(u-v)/(u+v)$, with inverse $u=(y-36d)/(6x)$, $v=(-y-36d)/(6x)$. Thus diagonal sums of two cubes are naturally studied through elliptic-curve addition formula.

On the product of an elliptic curve with itself, let $D_{+}$ and $D_{-}$ be the diagonal and the graph of negation. The principal divisor of $x(P)-x(Q)$ is $D_{+}+D_{-}-2(\{O\}\times E)-2(E\times \{O\})$: equality of the coordinates means $Q=\pm P$, while $x$ has a double pole at $O$. Pulling back along $(\varphi ,\psi )$ and taking degrees proves the parallelogram law for the degree of an isogeny when $\varphi ,\psi$ are nonzero and $\varphi \ne \pm \psi$. The exceptional cases follow from $\mathrm{deg}[2]=4$. The same argument uses the quotient by negation in characteristic two. Thus degree, with degree zero assigned to the zero map, is a positive quadratic form on $\mathrm{Hom}(E_1,E_2)$.

The elliptic curves $E_1:y^2=x^3-x$ and $E_2:y^2=x^3-x+1$ over ${\mathbb{F}}_5$ both have eight rational points. The first has four rational 2-torsion points and the second has two, so their point groups are not isomorphic. They are nevertheless degree-two isogenous via $X=x+2/(x-1)$, $Y=y(1-2/(x-1{)}^2)$. The map sends $O$ and $(1,0)$ to $O$; its pole divisor for $X$ is $2O+2(1,0)$, giving degree two. Their j-invariant of an elliptic curve values are respectively three and one, so the curves are not geometrically isomorphic either.

A primitive effective nef class of square zero on a complex K3 surface defines a basepoint-free pencil with connected genus-one fibres. Its general fibre is a smooth elliptic curve. A section is a smooth rational curve $\Gamma$ with $E\cdot \Gamma =1$ and ${\Gamma }^2=-2$.

A finite étale morphism is both a finite morphism and an étale morphism. Affine-locally it is represented by a finite locally free algebra with zero module of Kähler differentials. Its fibres are finite products of finite separable field extensions. For an elliptic curve with good reduction at a prime not dividing $m$, multiplication by $m$ on its smooth proper group model is finite étale of degree $m^2$: its differential is multiplication by the unit $m$. A division torsor of a section consequently gives an unramified local extension, the good-prime input to the Weak Mordell-Weil theorem.

For an elliptic curve over a local field with integral Minimal Weierstrass equation, let $E_0(K)$ consist of points with nonsingular reduction and let $E_1(K)$ be the kernel of $E_0(K)\to {\tilde{E}}_{\mathrm{ns}}(k)$. This definition also applies at a prime of bad reduction of an elliptic curve. The point at infinity is always smooth. With $t=-x/y$ and $w=-1/y$, the Weierstrass equation of an elliptic curve becomesThe unit linear coefficient in $w$ gives a unique integral formal power series $w(t)=t^3+O(t^4)$. Then $x=t/w$ and $y=-1/w$, so the nonidentity points of $E_1(K)$ have $v(x)=-2r$ and $v(y)=-3r$ for $r=v(t)>0$. The local parameter $t$ bijects $E_1(K)$ with ${\mathfrak{m}}_K$ and turns its addition into the formal group of an elliptic curve. At good reduction this agrees with the kernel of reduction of an elliptic curve.

For an elliptic curve over ${\mathbb{F}}_q$, the Frobenius isogeny has characteristic polynomial $T^2-tT+q$. Taking traces of its powers gives this recurrence. If $\alpha ,\beta$ are the roots, then $t_n={\alpha }^n+{\beta }^n$ and $\#E({\mathbb{F}}_{q^n})=q^n+1-t_n$. The Hasse bound implies $|\alpha |=|\beta |=\sqrt{q}$, including the repeated-root case.

For a projective variety over the complex numbers, analytification gives an equivalence between algebraic and analytic coherent sheaves, preserves their cohomology, and is fully faithful on their morphisms. In particular, a holomorphic isomorphism between algebraic line bundles is algebraic. This allows a complex-torus proof of a line-bundle identity to establish the corresponding algebraic identity on a projective elliptic curve.

A continuous one-cocycle for a Galois group acting on an abelian module $M$ is a map satisfying the displayed identity. Cocycles modulo maps $\sigma \mapsto \sigma u-u$ form $H^1(K,M)$ in Galois cohomology. For an elliptic curve, choosing $mQ=P$ gives the cocycle $\sigma \mapsto \sigma Q-Q$ in $E[m]$, which represents the Kummer map of an elliptic curve.

For an elliptic curve $E/K$, regard $E(K^{\mathrm{sep}})$ as a discrete Galois module for the absolute Galois group $G_K$. Its continuous Galois cohomology has $H^0(K,E)=E(K)$; $H^1(K,E)$ classifies torsors under $E$. For an integer $n$ invertible in $K$, the Kummer exact sequence of an elliptic curve gives $0\to E(K)/nE(K)\to H^1(K,E[n])\to H^1(K,E)[n]\to 0$. If $[n]Q=P$, the first map sends $P$ to $\sigma \mapsto \sigma Q-Q$. Changing $Q$ changes the cocycle by a coboundary; this class vanishes exactly when $P$ has a division point over $K$.

A discrete Galois module over $K$ is an abelian group with a continuous action of the absolute Galois group $G_K$. Continuity means that each element has an open stabilizer. Examples are the algebraic points of an elliptic curve, its finite torsion module $E[m]$, and the group of roots of unity ${\mu }_m$. Galois cohomology takes continuous cocycles with values in such a module.