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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-24/3/e/solution

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Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 24 / 3 / e / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

For every prime

p \neq = 37

, the elliptic-curve discriminant is a unit modulo

p

, giving good reduction. Hasse's theorem on elliptic curves gives

# E_{p} (F_{p}) - (p + 1) \leq 2 p .

Equivalently, the elliptic-curve point count over a finite field lies between

p + 1 - 2 p

and

p + 1 + 2 p

, including for

p = 2

.

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