MATH 845 Homework 5: Primes, Fields, and Elliptic Curves, Assignments of Mathematics

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MATH 845: HOMEWORK 5, DUE MAY 5.

√^ 1. (a) Let^ p^ be prime. Show there exists a field^ K^ of degree 4 over^ Q, containing 5, with absolute discriminant 25p if and only if p 6 ≡ 2 , 3 (mod 5). Show that such a field is determined by p - call it Kp. Show that the only Galois Kp/Q is for p = 5. (b) Show that Kp is neither totally real nor totally complex if and only if p ≡ 3 (mod 4). If p ≡ 1 (mod 8), show that Kp is totally real if and only if p divides the (p − 1)/4th Fibonacci number and is totally complex otherwise. If p ≡ 5 (mod 8), show that Kp is totally complex if p divides the (p − 1)/4th Fibonacci number and totally real otherwise.

2. An elliptic curve E over Q of conductor N gives rise, for each prime p not dividing N , to a Galois extension (its p-division field) K/Q that is unramified at all primes dividing pN. Moreover Gal(K/Q) embeds in GL 2 (Fp) such that the image of any Frobenius element at q (a prime not dividing pN ) has trace q + 1 − |E(Fq | and determinant q (both taken modulo p). (a) Let E be an elliptic curve over Q of conductor 11. We know that if p = 2, then K/Q has ramification indices e 2 = 3, e 11 = 2. Find K. (You cannot quote any classification of elliptic curves of conductor 11, unless you write out e.g. Wiles’s proof in full). Characterize the parity of |E(Fq )| as q varies. (b) If E and E′^ are two non-isogenous elliptic curves over Q of conductor N , then Faltings’s work yields an extension L/Q containing K, whose Galois group embeds in the subgroup H of GL 2 (Fp[T ]/(T 2 )) consisting of matrices whose determinant is in Fp. Show that in the case p = 2, H ∼= S 4 × Z/2. L/Q is also ramified only at the primes dividing pN. Suppose N = 11 and p = 2. Find all possible L (the result in fact implies there is only one isogeny class of elliptic curves over Q of conductor 11).