Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 455

every pair ( E,C ) of an elliptic curve over C and a cyclic subgroup C of order
N is isomorphic to a pair ( E τ ,C τ )forsome τ . Therefore, the set of

isomorphism classes of these pairs is in one-to-one correspondence with the
points of

∈H

H mod the action of Γ 0 ( N ). These are the noncuspidal points of

X 0 ( N ).
Of course, over arbitrary fields, we cannot work with the upper half plane
H , and it is much more dicult to show that the pairs ( E,C ) can be collected
together as the points on a curve X 0 ( N ). However, when this is done, it yields
a convenient way to work with the modular curve X 0 ( N ) and its reductions
mod primes.

For a nonsingular algebraic curve C over a field K ,let J ( C ) be the divisors
(over K ) of degree 0 modulo divisors of functions. It is possible to represent
J ( C ) as an algebraic variety, called the Jacobian of C .When C is an elliptic
curve E , we showed (Coroll ar y 11.4; see also the sequence (9.3)) that J ( E )
is a group isomorphic to E ( K ). When K = C ,wethusobtainedatorus. In
general, if K = C and C is a curve of genus g ,then J ( C ) is isomorphic to a
higher dimensional torus, namely, C g mod a lattice of rank 2 g . The Jacobian
of X 0 ( N ) is denoted J 0 ( N ).
The Jacobian J 0 ( N ) satisfies various functorial properties. In particular, a
nonconstant map φ : X 0 ( N ) → E induces a map φ ∗ : E → J 0 ( N ) obtained
by mapping a point P of E to the divisor on X 0 ( N ) formed by the sum of
the inverse images of P minustheinverseimagesof

∞∈ E :

φ ∗ : P −→ [ Q ] − [ R ] .

φ ( Q )= P φ ( R )= ∞

Therefore, we can map E to a subgroup of J 0 ( N ) (this map might have a
nontrivial, but finite, kernel).
An equivalent formulation of the modularity of E is to say that there is a
nonconstant map from X 0 ( N )to E and therefore that E is isogenous to an
elliptic curve contained in some J 0 ( N ).
If p is a prime dividing N , there are two natural maps X 0 ( N ) → X 0 ( N/p ).
If ( E,C ) is a pair corresponding to a point in X 0 ( N ), then there is a unique
subgroup C ⊂ C of order N/p . Sowehaveamap

( E,C ) . α :( E,C ) −→ (15.10)

However, there is also a unique subgroup P ⊂ C of order p .Itcanbeshown
that E/P is an elliptic curve and therefore ( E/P,C/P ) is a pair corresponding
to a point on X 0 ( N/p ). This gives a map

β :( E,C ) −→ ( E/P,C/P ) . (15.11)

These two maps can be interpreted in terms of the complex model of X 0 ( N ).
Since Γ 0 ( N ) ⊂ Γ 0 ( N/p ), we can map H mod Γ 0 ( N )to H mod Γ 0 ( N/p )by