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Over an algebraically closed field of characteristic different from two, a degree-two map from a smooth genus one curve to the projective line has a deck involution. After choosing an origin, all its fiber divisors on an algebraic curve are linearly equivalent and have the same group sum $s$, by the Abel-Jacobi map of a genus-one curve. The involution is consequently $p\mapsto s-p$, including its fixed ramification points. Composing two such involutions gives a translation on an elliptic curve.

The printed construction is not well defined: the next tangent must pass through $x_1$, rather than $x_0$. If $x_0\ne x_1$ and ${\ell }_1\ne {\ell }_0$, a second line through $x_0$ cannot also pass through $x_1$, because the unique line through both points is ${\ell }_0$. Here is a concrete counterexample satisfying all the conic hypotheses. In the affine chart takewith ${\ell }_0:x=1$, $x_0=(1,3\sqrt{3}/2)$ and $x_1=(1,-3\sqrt{3}/2)$. The second tangent through $x_0$ is $-23x+12\sqrt{3}y=31$. It does not contain $x_1$: its left side there is $-77$. The conics meet transversely at the four complex points with $x^2=32/5$, $y^2=-27/5$. Thus this is a defect in the original PDF, not just in its conversion.

For the corrected construction, let ${\iota }_{\ell }$ exchange the two points of $C_2$ on a fixed tangent line, and let ${\iota }_x$ exchange the two tangents to $C_1$ through a fixed point of $C_2$. Both projections from $E$ are degree-two morphisms to a smooth plane conic: for the line projection this follows from intersecting a line with $C_2$, which has no line component. Since $E$ is smooth and the ground field has characteristic zero, their quadratic function-field extensions define regular involutions on the whole curve. At a ramification point “the second” point is the same point, counted with multiplicity. Each switch is an involution of a degree-two map from a genus one curve. Hence the corrected step is the everywhere-defined automorphism $F={\iota }_x\circ {\iota }_{\ell }$.

Choose an origin $O$ on the genus one curve. The Abel-Jacobi map of a genus-one curve identifies $E$ with ${\mathrm{Pic}}^0(E)$ by $p\mapsto {\mathcal{O}}_E(p-O)$. Fibers of each degree-two projection are linearly equivalent Weil divisors, since they are pullbacks of points of ${\mathbb{P}}^1$. Thus their group sums are constant: for suitable $a,b\in E$,This also holds at the ramification points, where $2p=a$ or $2p=b$. Therefore the corrected step is a translation on an elliptic curve. If $F^n(p_0)=p_0$ for one point, then $n(b-a)=0$. It follows that $F^n(p)=p$ for every $p\in E$. The corrected construction has the Poncelet porism: one periodic orbit implies all orbits are periodic, with the same least period. The literal printed construction fails before this conclusion; the proof establishes the intended, explicitly corrected assertion.

Write the cubic in the Weierstrass equation of an elliptic curve as $f(x)=x^3+a_2{x}^2+a_{4}x+a_6$. A suitable invariant differential on an elliptic curve isIt is regular and nonzero wherever $y\ne 0$. At a point with $y=0$, nonsingularity gives $f^{\prime }(x)\ne 0$, and differentiating the equation shows $\omega =dy/f^{\prime }(x)$, again regular and nonzero. At the identity $O$, use the parameter $t=-x/y$: the expansions start $x=t^{-2}+\cdots$, $y=-t^{-3}+\cdots$, so $\omega =(1+\cdots )dt$. Thus it is a nowhere-vanishing regular differential on the whole smooth projective curve.

Here is a direct proof of translation invariance of a Weierstrass differential. Fix $Q=(u,v)$, let $P=(x,y)$ vary, and write $P+Q=(X,Y)$. On the open set where the chord-and-tangent group law uses an ordinary chord, put $h=(y-v)/(x-u)$. The addition formulas are $X=h^2-a_2-x-u$ and $Y=h(x-X)-y$. The line-intersection identity isDifferentiate this polynomial in $T$ at $T=x$ to obtain $f^{\prime }(x)-2yh=(x-u)(x-X)$. Differentiating $h$ along the curve now gives $dh/dx=(x-X)/(2y)$, and henceThis proves translation invariance on a dense open set. Since the translation on an elliptic curve is an automorphism and both sides are regular differentials, it proves invariance everywhere, including the exceptional addition cases. Translation by $O$ is the identity.

Translation invariance also gives ${\mu }^{\ast }\omega ={\mathrm{pr}}_1^{\ast }\omega +{\mathrm{pr}}_2^{\ast }\omega$ for the addition map: its differential on a tangent pair is the sum of the two translated tangent vectors. Therefore $[j{]}^{\ast }\omega =j\omega$ for every integer $j$.

Let $d_j=\mathrm{deg}[j]$, with the zero endomorphism assigned degree zero. The assumed degree identity, applied to $[j]$ and $[1]$, givesInduction gives $\mathrm{deg}[j]=j^2$ for positive $j$; negation is an automorphism, so the result holds for negative $j$ as well. This is quadratic degree recursion for elliptic multiplication.

For a prime $\ell \ne p$, the invariant differential on an elliptic curve has nonzero pullback under $[\ell ]$, so that map is a separable isogeny. Translation identifies all its fibres and their local multiplicities, so its geometric kernel has exactly its degree ${\ell }^2$ distinct points. It is killed by $\ell$, and thereforeThe algebraic closure is essential here; the assertion is not generally true for the rational-point subgroup over ${\mathbb{F}}_p$ itself. This is prime-to-characteristic geometric torsion.

For $[p]$, the pullback differential vanishes. Its total degree is $p^2$, and its inseparable degree is at least $p$, so its kernel has at most $p$ geometric points. Thus $E({\mathbb{F}}_p)[p]$ has dimension at most one over ${\mathbb{F}}_p$, while every other prime-torsion subgroup has dimension at most two. The rational-point group is finite and abelian. The structure theorem for finite abelian groups says that the number of cyclic factors of an $\ell$-primary component equals the dimension of its subgroup killed by $\ell$. Each component therefore has at most two factors. Combining the smaller primary factors into one cyclic group and the larger ones into another yieldsOne may also choose $p\nmid m$, since the $p$-primary component is cyclic. This explains two generators for elliptic curves over finite fields.

For two smooth plane conics meeting transversely, the operation of crossing a chord of the second conic tangent to the first and choosing the other tangent through the new endpoint acts as a translation on an elliptic curve on their incidence curve. Each switch is an involution of a degree-two map from a genus one curve. One periodic orbit means that the translating point is torsion, so every orbit is periodic with the same least period. The construction extends through coincident choices at ramification points using the regular involutions.

On a nonsingular equation $y^2=x^3+a_2{x}^2+a_{4}x+a_6$ in characteristic different from two, $\omega =dx/(2y)$ is regular and nowhere zero. For a chord of slope $h$ giving $P+Q=(X,Y)$, differentiating the line-intersection identity yields $dX/dx=Y/y$, so $dX/(2Y)=dx/(2y)$. Extension across the exceptional cases proves invariance under every translation on an elliptic curve. The addition map consequently satisfies ${\mu }^{\ast }\omega ={\mathrm{pr}}_1^{\ast }\omega +{\mathrm{pr}}_2^{\ast }\omega$, and $[n{]}^{\ast }\omega =n\omega$.