Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 456

mapping the equivalence class of τ mod Γ 0 ( N ) to the equivalence class of τ
mod Γ 0 ( N/p ). This corresponds to the map α .Themap β can be shown to
correspond to the map τ pτ . Note that these two maps represent the two

methods of using modular forms for Γ 0 ( N/p ) to produce oldforms for Γ 0 ( N ).
The Hecke algebra T acts on J 0 ( N ). Let P be a point on X 0 ( N ). Recall
that P corresponds to a pair ( E,C ), where E is an elliptic curve and C is
a cyclic subgroup of order N .Let p be a prime. For each subgroup D of E
of order p with D ⊆ C , we can form the pair ( E/D, ( C + D ) /D ). It can be
shown that E/D is an elliptic curve and ( C + D ) /D is a cyclic subgroup of
order N . Therefore, this pair represents a point on X 0 ( N ). Define
T p ([( E,C )]) =
D

→

[( E/D, ( C + D ) /D )] ∈ Div( X 0 ( N )) ,

where the sum is over those D of order p with D C and where Div( X 0 ( N ))

denotes the divisors of X 0 ( N ) (see Chapter 11). It is not hard to show that
this corresponds to the formulas for T p given in (15.6). Clearly T p maps
divisors of degree 0 to divisors of degree 0, and it can be shown that it maps
principal divisors to principal divisors. Therefore, T p gives a map from J 0 ( N )
to itself. This yields an action of T on J 0 ( N ), and these endomorphisms are
defined over Q .
Let α

⊆

T and let J 0 ( N )[ α ] denote the kernel of α on J 0 ( N ). More generally,
let I be an ideal of T . Define
J 0 ( N )[ I ]=
α

∈

J 0 ( N )[ α ] .

∈ I

For example, when I = n T for an integer n ,then J 0 ( N )[ I ] is just J 0 ( N )[ n ],
the n -torsion on J 0 ( N ).
Now let's consider the representation ρ of Theorem 15.6. Since ρ is assumed
to be modular, it corresponds to a maximal ideal M of T .Let F = T / M ,

which is a finite field. Then W = J 0 ( N )[ ] has an action of F ,whichmeans

that it is a vector space over F .Let be the characteristic of F .Since =0
in F , it follows that

M

W ⊆ J 0 ( N )[ ] ,

the -torsion of J 0 ( N ). Since G acts on W , we see that W yields a represen-
tation ρ of G over F .Itcanbeshownthat ρ is equivalent to ρ ,sowecan
regard the representation space for ρ as living inside the -torsion of J 0 ( N ).
This has great advantages. For example, if M N then there are natural maps

X 0 ( N ) → X 0 ( M ). These yield (just as for the map X 0 ( N ) → E above) maps
J 0 ( M ) → J 0 ( N ). Showing that the level can be reduced from N to M is
equivalent to showing that this representation space lives in these images of
J 0 ( M ). Also, we are now working with a representation that lives inside a
fairly concrete object, namely the -torsion of an abelian variety, rather than
a more abstract situation, so we have more control over ρ .

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