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Math 346/646 - Homework 8 Assigned: 10/27? Due: 11/3 at the start of class. Notation: Exercise a.b stands for exercise b in Chapter a of Silverman and Tate. Note that the exercises are all together at the end of the chapter. A problem marked with a ∗ is a challenge problem. Problems: 1. Prove the upper bound in Lemma 3’. (In class we only proved the lower bound. This problem is also problem 3.4 from the text.) 2. Let E : y 2 = x3 − 2x. Find an explicit constant κ for which h(2P ) ≥ 4h(P ) − κ for all points P ∈ E(Q). (This exercise is asking you to make effective Lemma 3 for this particular curve.) 3. ∗ (a) Let H : x2 + y 2 = 1. Define a binary operation on H by (x1 , y1 ) ∗ (x2 , y2 ) = (x1 x2 − y1 y2 , x1 y2 + x2 y1 ). Show that H is a group with respect to this operation. (b) Prove that H(Q) = {(x, y) ∈ Q : x2 + y 2 = 1} is not finitely generated. 1