Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 125 / 3

Forthe chord-and-tangent group law gives, for distinct nonopposite points $P=(x_1,y_1)$ and $Q=(x_2,y_2)$,For doubling, replace the slope byThe point at infinity is the identity and $-(x,y)=(x,-y)$.

Let the $x$-coordinates of $P,Q,P+Q,P-Q$ be $x_1,x_2,x_3,x_4$. Applying the addition formula with $Q$ and $-Q$ givesSubstituting $y_i^2=x_i^3+a{x}_i+b$ and simplifying yieldsMultiplying the two addition formulas and eliminating $y_1{y}_2$ in the same way gives

These identities are the algebraic source of two parallelogram laws. Applied to pullbacks of the pole divisor of $x$, they implyfor isogenies of elliptic curves. Together with $\mathrm{deg}(n\phi )=n^2\mathrm{deg}\phi$, this makes the degree a quadratic form. Applied to the Absolute logarithmic Weil height of the four $x$-coordinates, with bounded terms removed by passage to the limit defining the canonical height of an elliptic curve, they similarly give

For $E:y^2=x^3+1$, let $\alpha (x,y)=(\zeta x,y)$. The three points $P,\alpha P,{\alpha }^{2}P$ lie on the horizontal line through $P$, so their sum is zero. Thusin the endomorphism ring of an elliptic curve. Since $\mathrm{deg}\alpha =1$ and complex conjugation sends $\alpha$ to ${\alpha }^2$, degree on $\mathbb{Z}[\alpha ]$ is the Eisenstein-integer norm:

A separable isogeny is determined by its kernel up to unique isomorphism of its target, and every finite Galois-stable subgroup of an elliptic curve is the kernel of the corresponding quotient isogeny. Put $T=(-1,0)$. The three nonzero points of $E[2]$ are $T,\alpha T,{\alpha }^{2}T$, andHence $\phi (\alpha -{\alpha }^2)$ kills $\ker \phi =\{O,T\}$ and factors uniquely through $\phi$:for an isogeny $\psi :E^{\prime }\to E^{\prime }$. Degrees giveMoreover,Composing the factorization twice and using the surjectivity of $\phi$ gives ${\psi }^2=[-3]$.