For the <filtration of elliptic-curve points over a local field>, define
$$
E_0(\mathbb Q_p)=\{P:\operatorname{red}(P)\in\bar E_{\mathrm{ns}}(\mathbb F_p)\},
$$
and
$$
E_1(\mathbb Q_p)=\ker\left(E_0(\mathbb Q_p)\to\bar E_{\mathrm{ns}}(\mathbb F_p)\right).
$$
For $r\geq1$, use the <formal group of an elliptic curve> with parameter $z=-x/y$ and put
$$
E_r(\mathbb Q_p)=\{P\in E_1(\mathbb Q_p):v_p(z(P))\geq r\}.
$$
Reduction induces
$$
E_0(\mathbb Q_p)/E_1(\mathbb Q_p)
\cong\bar E_{\mathrm{ns}}(\mathbb F_p),
$$
while the coefficient of $p^r$ in the formal parameter gives
$$
E_r(\mathbb Q_p)/E_{r+1}(\mathbb Q_p)
\cong(\mathbb F_p,+)qquad(r\geq1).
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 125 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-125/1/c/solution