 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2026

Paper 125

For an endomorphism $\phi$ of an <elliptic curve>, define its <trace of an elliptic-curve endomorphism> by
$$
\operatorname{tr}(\phi)=1+\deg\phi-\deg(1-\phi).
$$
Polarizing the quadratic form $\deg$ shows that this is the integer for which
$$
\phi+\widehat\phi=[\operatorname{tr}(\phi)],
\qquad
\widehat\phi\phi=[\deg\phi],
$$
where $\widehat\phi$ is the <dual isogeny>. Consequently
$$
\phi^2-[\operatorname{tr}(\phi)]\phi+[\deg\phi]
=\phi^2-(\phi+\widehat\phi)\phi+\widehat\phi\phi=0.
$$
Also
$$
\phi^2+\widehat\phi^{,2}
=(\phi+\widehat\phi)^2-2\phi\widehat\phi
=[\operatorname{tr}(\phi)^2-2\deg\phi].
$$
The left side is $[\operatorname{tr}(\phi^2)]$, so
$$
\operatorname{tr}(\phi^2)=\operatorname{tr}(\phi)^2-2\deg\phi.
$$

Solved by gpt-5.6-sol high.

Solution

Paper 125 / 2 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 125 / 2 / a / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 125 / 2 / a / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)