MATH 845: HOMEWORK 5, DUE MAY 5.
1. (a) Let pbe prime. Show there exists a field Kof degree 4 over Q, containing
√5, with absolute discriminant 25pif and only if p6≡ 2,3 (mod 5). Show that such
a field is determined by p- call it Kp. Show that the only Galois Kp/Qis for p= 5.
(b) Show that Kpis neither totally real nor totally complex if and only if p≡3
(mod 4). If p≡1 (mod 8), show that Kpis totally real if and only if pdivides the
(p−1)/4th Fibonacci number and is totally complex otherwise. If p≡5 (mod 8),
show that Kpis totally complex if pdivides the (p−1)/4th Fibonacci number and
totally real otherwise.
2. An elliptic curve Eover Qof conductor Ngives rise, for each prime pnot
dividing N, to a Galois extension (its p-division field) K/Qthat is unramified at all
primes dividing pN. Moreover Gal(K/Q) embeds in GL2(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 Ebe an elliptic curve over Qof conductor 11. We know that if p= 2,
then K/Qhas ramification indices e2= 3, e11 = 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 qvaries.
(b) If Eand E0are two non-isogenous elliptic curves over Qof conductor N, then
Faltings’s work yields an extension L/Qcontaining K, whose Galois group embeds
in the subgroup Hof GL2(Fp[T]/(T2)) consisting of matrices whose determinant
is in Fp. Show that in the case p= 2, H∼
=S4×Z/2.
L/Qis 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 Qof conductor 11).
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