 Solution

ID: past-exam-of-the-analytic-number-theory-and-elliptic-curves-course-of-the-university-of-oxford/2007/b/7/i/solution

In the typical case, the line through

P, Q

, or their tangent when they coincide, is

v = m u + c

. Its intersections with

D

have first coordinates given by

u^{3} + (- 2 a - m^{2}) u^{2} + (b_{1} - 2 m c) u - c^{2} = 0.

Their product is

c^{2}

. The third intersection is

- (P + Q)

, which has the same first coordinate as

P + Q

. Consequently

u (P) u (Q) u (P + Q) = c^{2} .

Passing to the quotient by rational squares gives

q (P) q (Q) q (P + Q) = 1

. Every square class is its own inverse, so

q (P + Q) = q (P) q (Q) .

The excluded cases have their specified special values; the source asks only for this chord-and-tangent argument away from them.

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