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The 3-torsion of an elliptic curve $E$ is the kernel of multiplication by three:with the field of points specified. If the characteristic is not three, the geometric group over an algebraic closure is $(\mathbb{Z}/3\mathbb{Z}{)}^2$ and has nine points. The subgroup of points rational over the original field can be smaller.

For $E:y^2=x^3+1$ over $\mathbb{Q}$, the point $P=(0,1)$ has order three. The tangent slope is $3x^2/(2y)=0$, so the elliptic-curve doubling formulas give $2P=(0,-1)=-P$ and hence $3P=O$. This displays a nonzero rational 3-torsion point rather than only asserting its possible existence.

In characteristic three, the geometric-point count need not be nine. The scheme-theoretic multiplication kernel still carries degree information not captured by counting its ordinary points. Thus the characteristic restriction is essential when identifying geometric 3-torsion with two cyclic groups.

An abelian variety over a field is a complete, geometrically connected group variety; equivalently it is a connected projective algebraic group in the usual smooth variety setting. Its group operation is necessarily commutative, a consequence of rigidity rather than an extra condition hidden in the name. An elliptic curve with a chosen origin is a one-dimensional abelian variety; products of elliptic curves give higher-dimensional examples. The additive and multiplicative algebraic groups are not complete and hence are not abelian varieties. Their geometry links line bundles, torsion elements and the Jacobian variety of a curve to questions in number theory.^[ref]

Algebraic geometry studies geometric objects defined by polynomial equations and the algebra of functions on them. For example, $y=x^2$ defines a parabola, while $xy=0$ defines the union of two coordinate lines. The coordinate ring $k[x,y]/(xy)$ records the second example algebraically: the images of $x$ and $y$ are nonzero but their product is zero, reflecting two components. Modern schemes extend this dictionary beyond ordinary point sets and permit geometric reasoning over general commutative rings. Applications range from elliptic curves in number theory to families of solutions depending on parameters.^[ref]

On a quasi-compact scheme $X$, a line bundle $L$ is ample if each point has an affine neighborhood of the formEquivalently, the affine nonvanishing loci of global sections of positive tensor powers form a basis of a topology. This makes ampleness a condition that global sections recover enough affine local geometry.^[ref]

For a proper scheme over a field, this is equivalent to some positive tensor power being very ample: $L^{\otimes m}$ is the pullback of $\mathcal{O}(1)$ by a closed immersion into projective space. A positive tensor power may be needed. On an elliptic curve, a degree-one line bundle is ample but has only one independent global section, so it cannot itself embed the curve. Its third power has degree three and gives a plane cubic embedding. On an affine scheme, every line bundle is ample under the scheme definition, even when the scheme is not proper.

Andrew V. Sutherland works in computational number theory and arithmetic geometry. His research includes algorithms for elliptic curves, their torsion subgroups, and explicit computations with modular curves. These subjects turn structural questions about equations over number fields into computations that can test and sharpen conjectures in number theory.^[ref]

Homepage: math.mit.edu/~drew/

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, fix the absolute logarithmic $x$-height $h_x(P)=h([x(P):1])$, with $h_x(O)=0$. The normalization used here isThe factor $1/2$ makes this the Néron–Tate height associated with the degree-one divisor $(O)$. Some sources omit $1/2$ and obtain twice this function; formulas and the regulator of an elliptic curve must use a consistent convention.^[ref]

The estimate $h_x(2P)=4h_x(P)+O(1)$ makes successive terms differ by a summable geometric series, proving convergence and $\hat{h}(P)=h_x(P)/2+O(1)$. The resulting quadratic form satisfies $\hat{h}([m]P)=m^2\hat{h}(P)$ and the height parallelogram identity. It is nonnegative and vanishes exactly on torsion points of an elliptic curve; bounded-height rational points over a fixed number field form a finite set. For an isogeny $\varphi :E\to E^{\prime }$, the corresponding normalized heights satisfy ${\hat{h}}_{E^{\prime }}(\varphi P)=\mathrm{deg}(\varphi ){\hat{h}}_E(P)$.

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.

Arithmetic operations on complex numbers. Writing two inputs as $a+bi$ and $c+di$ turns multiplication into $(ac-bd)+(ad+bc)i$ and addition into $(a+c)+(b+d)i$. This is distinct from the specialized theory of complex multiplication of elliptic curves.

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.

For an elliptic curve $E:y^2=x^3+Ax+B$ in characteristic different from two and three, division polynomials are elements of its coordinate ring encoding multiplication and torsion points of an elliptic curve. The initial terms includeThe recurrences for higher $n$ yieldFor odd $n$, ${\psi }_n$ is a polynomial in $x$; for even $n$, it has a factor $2y$, and ${\psi }_n/(2y)$ is a polynomial in $x$. Thus the phrase “roots of the division polynomial” must specify whether it means points on the curve or roots in the $x$-line. When $n$ is prime to the characteristic, its zero points are exactly the nonidentity $n$-torsion points of an elliptic curve; for odd $n$, the two opposite points share an $x$-root.^[ref]

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.