Add dual isogeny

docs/ristretto-notes.md

@@ -118,8 +118,14 @@ $$
 $$
 \theta\_{a,d} : (s,t) \mapsto \left( \frac{1}{\sqrt{ad-1}} \cdot \frac{2s}{t},\quad \frac{1+as\^2}{1-as\^2} \right).
 $$
-
-XXX Its dual is ... ?
+Its dual is 
+$$
+\hat{\theta}\_{a,d} : \mathcal E\_{a,d} \longrightarrow \mathcal J\_{a\^2, -a(a+d)/(a-d)},
+$$
+defined by
+$$
+\hat{\theta}\_{a,d} : (x,y) \mapsto \left( \sqrt{ad-1} \cdot \frac{xy}{1-ax\^2}, \frac{y^2 + ax^2}{1-ax^2} \right)
+$$
 
 The kernel of the isogeny is \\( \{(0, \pm 1)\} \\).
 The image of the isogeny is \\(\[2\](\mathcal E)\\).  To see this,
@@ -134,10 +140,12 @@ $$
 $$
 
 To determine the image \\(\theta(\mathcal J[2])\\) of the
-\\(2\\)-torsion, we consider the image of the coset \\(\theta((s,t)
-+ \mathcal J[2])\\).  Let \\((x,y) = \theta(s,t)\\); then
-\\(\theta(-s,-t) = (x,y)\\) and \\(\theta(1/as, -t/as\^2) = (-x,
--y)\\), so that \\(\theta(\mathcal J[2]) = \mathcal E[2]\\).
+\\(2\\)-torsion, we consider the image of the coset
+\\(\theta((s,t) + \mathcal J[2])\\).
+Let \\((x,y) = \theta(s,t)\\); then
+\\(\theta(-s,-t) = (x,y)\\) and
+\\(\theta(1/as, -t/as\^2) = (-x, -y)\\),
+so that \\(\theta(\mathcal J[2]) = \mathcal E[2]\\).
 
 The Decaf paper recalls that, for a group \\( G \\) with normal
 subgroup \\(G' \leq G\\), a group homomorphism \\( \phi : G