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For a singular projective Weierstrass cubic, its smooth locus of a variety contains the point at infinity and carries the chord-and-tangent group law. The singular point is excluded. Over an algebraic closure, a node gives the multiplicative algebraic group and a cusp gives the additive group. A node whose branches are not defined over the base field gives a nonsplit one-dimensional algebraic torus. This group is distinct from an elliptic curve, whose projective model is smooth.

For the square period lattice, multiplying the defining series by the lattice symmetry $z\mapsto iz$ gives $\wp (iz)=-\wp (z)$. At $h=(1+i)/2$, one has $ih=h-1$, so periodicity gives $\wp (h)=-\wp (h)$ and hence the displayed zero. This is one of the three finite branch values of the degree-two Weierstrass elliptic function on its torus. The three values are always distinct but need not all be nonzero. Distinctness is enough for the nonsingularity of the associated elliptic curve $Y^{2}Z=4{\prod }_i(X-e_{i}Z)$.

If an elliptic curve has good reduction over a local field $K$, its integral projective model is smooth. Every point over the residue field lifts by the Hensel lemma, applied in a chart with a unit partial derivative. Thus reduction gives an exact sequence $0\to E_1(K)\to E(K)\to \tilde{E}(k)\to 0$. The kernel of reduction of an elliptic curve is the maximal ideal of the valuation ring, with the formal group of an elliptic curve as its group operation.

For an abelian variety $A$ and a prime number $\ell$ different from the characteristic, the inverse limit of its geometric ${\ell }^n$-torsion, with transition multiplication by $\ell$, is its Tate module. For an elliptic curve, it is free of rank two over the p-adic integers and satisfies $T_{\ell }(E)/{\ell }^n{T}_{\ell }(E)\simeq E[{\ell }^n]$. The absolute Galois group acts continuously and commutes with all base-defined endomorphisms.

For a line bundle $L$ on an abelian variety $A$ and translations $T_x(a)=a+x$, the Theorem of the square gives an isomorphism of line bundlesConsequently $x\mapsto T_x^{\ast }L\otimes L^{-1}$ is a group homomorphism from $A$ to its dual abelian variety. These are isomorphism classes: no canonical trivialization is supplied without normalization data. On an elliptic curve, translating a degree-one divisor changes its degree-zero difference class additively, illustrating the identity. The Theorem of the Cube controls the additional compatibility when the translation points themselves vary.^[ref]

Suppose $15\nmid n$ and both $n,-n$ are nonsquares. Each elliptic curve has just one nonidentity rational 2-torsion point, $(0,0)$. The duplication formula shows that halving it requires the corresponding coefficient $\pm n$ to be a square, so neither has rational order-four points. If $3\nmid n$, both good reductions at three have order four, forcing torsion point of an elliptic curve to be exactly order two. Otherwise $5\nmid n$; the reduction orders for coefficients $1,2,3,4$ modulo five are $4,2,10,8$. Among opposite coefficients at least one has order two or four, giving the same conclusion for at least one curve by reduction of torsion points on an elliptic curve.

For an elliptic curve over $\mathbb{Q}$, reduction at a good prime $p$ is injective on the $\ell$-primary torsion subgroup when $\ell \ne p$: its kernel is a formal group, where multiplication by $\ell$ is bijective. Hence a rational point of order $\ell$ forces $\ell \mid \#\tilde{E}({\mathbb{F}}_p)$. If for every possible prime $\ell$ some good $p\ne \ell$ has a point count not divisible by $\ell$, rational torsion is trivial. In particular two distinct good primes $p,r$ with coprime counts, $p\nmid \#\tilde{E}({\mathbb{F}}_r)$ and $r\nmid \#\tilde{E}({\mathbb{F}}_p)$ suffice: the coprime counts exclude other primes, and the two cross conditions exclude $p,r$. This argument does not incorrectly apply prime-to-$p$ injectivity to $p$-power torsion.

For a fixed point $Q$ on an elliptic curve, translation is the automorphism $P\mapsto P+Q$. Its inverse is translation by $-Q$. Translations let local computations of an invariant differential on an elliptic curve control every point of the curve.

For an elliptic curve over a finite field of characteristic $p$, its rational-point group is finite and abelian. Prime-to-characteristic geometric torsion bounds every other prime-torsion dimension by two. Multiplication by $p$ has degree $p^2$ and inseparable degree at least $p$, so the $p$-torsion dimension is at most one. The structure theorem for finite abelian groups therefore gives two cyclic factors with $m\mid n$ and $p\nmid m$.

Assume $\mathrm{char}K\ne 2$, $b\ne 0$ and $b^{\prime }=a^2-4b\ne 0$. Thenare elliptic curves, andis a degree-two separable isogeny with kernel $\{O,(0,0)\}$. Substitution verifies the target equation. The poles at $O$ and $(0,0)$ extend to the target point at infinity, giving a morphism of the full projective curves. This is a special case of Vélu formulas.

Applying the same construction to $E^{\prime }$ gives a curve with coefficients $4a,16b$. Rescaling that curve to $E$ supplies the dual isogenyThe factors four and eight are essential to this dual normalization. The characteristic restriction permits these divisions, while $b{b}^{\prime }\ne 0$ excludes singular cubic models.

Let an elliptic curve over $\mathbb{Q}$ have good reduction at distinct primes $p,r$, with point counts $N_p,N_r$. For a torsion point, its $\ell$-primary part injects into every reduction whose residue characteristic differs from $\ell$. Consequently its $\ell$-power order has exponent at most $\min (v_{\ell }(N_p),v_{\ell }(N_r))$ when $\ell \ne p,r$, at most $v_p(N_r)$ when $\ell =p$, and at most $v_r(N_p)$ when $\ell =r$. This precise form avoids assuming injectivity on residue-characteristic torsion. Counts five at two and seven at three force trivial rational torsion.

For an elliptic curve $E$ and a finite subgroup $G$ of its geometric points, Jacques Vélu's formulas construct the separable isogeny with kernel $G$. On a Short Weierstrass form away from characteristics two and three, its normalized coordinates areThese rational functions extend to a morphism of the smooth projective curves, sending $G$ to the new identity. Its degree is $\#G$. If $G$ is stable under the absolute Galois group, the summed expressions and quotient curve are defined over the base field, even when the individual points of $G$ are not rational.^[ref]

For $E:y^2=x^3-x$ and $G=\{O,(0,0)\}$, they give $x^{\prime }=x-1/x$ and $y^{\prime }=y(1+1/x^2)$, with quotient $y^{\prime 2}=x^{\prime 3}+4x^{\prime }$. The apparent pole at $(0,0)$ represents its image at infinity, not a missing value of the projective isogeny.

For an abelian variety $A$ over a field $K$, its Weil–Châtelet group is the Galois cohomology group $H^1(K,A(K^{\mathrm{sep}}))$, classifying torsors under $A$. Such an algebraic-group torsor becomes $A$ after passing to a separable closure but need not have a $K$-point. A chosen geometric point identifies the torsor with $A$ and its Galois descent with a cocycle; changing that point adds a coboundary. The zero class is precisely the torsor with a rational point. For an elliptic curve, these torsors are genus-one curves with a specified action and Jacobian. Over a number field the kernel of restriction to every completion is the Tate–Shafarevich group.

Let $E/K$ be an elliptic curve and $n>0$ an integer prime to the characteristic. The Weil pairing is the canonical perfect pairingwhere the arguments are geometric $n$-torsion points of an elliptic curve and ${\mu }_n$ is the group of $n$th roots of unity. It is bilinear and alternating, so $e_n(P,P)=1$ and $e_n(P,Q)=e_n(Q,P{)}^{-1}$. Perfection means that every nonzero first argument pairs nontrivially with some second argument.^[ref]

It is compatible with the absolute Galois group: $e_n(\sigma P,\sigma Q)=\sigma (e_n(P,Q))$. For an isogeny $\varphi :E\to E^{\prime }$ and its dual isogeny, $e_n^{E^{\prime }}(\varphi P,Q)=e_n^E(P,\hat{\varphi }Q)$. In a torsion basis, alternation makes the action on the pairing equal to the determinant; hence the determinant of the torsion Galois representation is the cyclotomic character. In characteristic $p$, the pairing on geometric $p$-torsion points cannot have this same prime-to-characteristic interpretation: ${\mu }_p$ has only one geometric point.