Elliptic Curves: Homework 5, Assignments of Mathematics

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MATH 845: HOMEWORK 5, DUE APR 15.

9. Let E be an elliptic curve over Fq. (a) Show that E(Fq ) ∼= Z/m×Z/mn for some integers m, n ≥ 1 with gcd(m, q) =

(b) With the notation of (a), show that q ≡ 1 (mod m). (c) Suppose that q is a prime ≥ 5 and that E is supersingular. Show that m = 1 or 2. If q ≡ 1 (mod 4), prove that m = 1.

10. Let E be the elliptic curve y^2 = x^3 + 16. (a) Show that this equation defines an elliptic curve over Fp(p prime) if and only if p ≥ 5. (b) Calculate a 5 and a 7 where ap = p + 1 − |E(Fp)|. (c) Which kind of singularity does the equation have over F 3? What should we take a 3 to be (is the given equation minimal at 3)? (d) For p = 2 we have to proceed differently since the equation is not in “minimal form” for p = 2. Give a Q-linear change of variables taking the equation of E to y^2 + y = x^3. Does this curve have good reduction at p = 2? What is a 2? (e) Calculate L(E, s) =

∑∞

n=1 cn/n

s (^) up to and including the n = 10 term. (f) Let Z[ω] (ω = e^2 πi/^3 ) denote the ring {a + bω : a, b ∈ Z}. Define the norm N (a + bω) = a^2 − ab + b^2. If gcd(N x, 3) = 1, let χ(x) = (−ω)j^ be the unique 6th root of 1 such that∑ xχ(x) ≡ 1 (mod 3). Let χ(x) = 0 for all other x. Compute

x∈Z[ω], 6 =0 χ(x)x/(N x)

s (^) as a Dirichlet series up to N x ≤ 10 and hence conjecture

an expression for L(E, s).