Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 337

10.2 Let R be an order in an imaginary quadratic field. Regard R as a subset
of C . Show that if r R , then its complex conjugate r is also in R .

This means that if L is a lattice with complex multiplication by R ,then
there are two ways to embed R into the endomorphisms of L ,namely
via the assumed inclusion of R in C and also via the complex conjugate
embedding (that is, if r ∈ R and ∈ L , define r ∗ = r ). This means
that when we say that R is contained in the endomorphism ring of a
lattice or of an elliptic curve, we should specify which embedding we
are using. For elliptic curves over C ,thisisnotaproblem,sincewe
can implicitly regard R as a subset of C and take the action of R on L
as being the usual multiplication. But for elliptic curves over fields of
positive characteristic, we cannot use this complex embedding.
10.3 Use the fact that Z 1+ √ − 4 2 is a principal ideal domain to show that
e π √ 43 is very close to an integer.

∈

10.4 Let x = a + b i + c j + d k lie in the Hamiltonian quaternions.

(a) Show that

( a + b i + c j + d k )( a − b i − c j − d k )= a 2 + b 2 + c 2 + d 2 .

(b) Show that if x = 0, then there exists a quaternion y such that

xy =1.
(c) Show that if we allow a, b, c, d

Q 2 (= the 2-adics), then a 2 + b 2 +
c 2 + d 2 =0ifandonlyif a = b = c = d =0. ( Hint: Clearing
denominators reduces this to showing that a 2 + b 2 + c 2 + d 2

∈

≡ 0

0(mod8).)
(d) Show that if x, y are nonzero Hamiltonian quaternions with 2-adic
coe cients, then xy

(mod 8) implies that a, b, c, ≡

=0.
(e) Let p be an odd prime. Show that the number of squares a 2 mod p ,
including 0, is ( p +1) / 2 and that the number of elements of F p of
the form 1 b 2 (mod p )isalso( p +1) / 2.

(f) Show that if p is a prime, then a 2 + b 2 +1

−

≡ 0(mod p )hasa

solution a, b .
(g) Use Hensel's lemma (see Appendix A) to show that if p is an odd
prime, then there exist a, b Q p such that a 2 + b 2 +1=0. (The

hypotheses of Hensel's lemma are not satisfied when p =2.)
(h) Let p be an odd prime. Show that there are nonzero Hamiltonian
quaternions x, y with p -adic coe cients such that xy =0.

∈

10.5 Show that a nonzero element in a definite quaternion algebra has a
multiplicative inverse. ( Hint: Use the ideas of parts (1) and (2) of

Exercise 10.4.)