Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 461

for Γ 0 (17). The first few values of b are as follows:

2357 1 3 7 9 3

b − 10 − 24 0 − 2 −− 44

It can be shown that a ≡ b (mod 5) for all =17 , 37 (we ignore these

bad primes), so

g E ≡ g 0 (mod 5) .

Can we prove that g E is a modular form?
Let A be the set of all potential modular forms g with g ≡ g 0 (mod 5) and
where the level N for g is allowed to contain only the primes 5 , 17 , 37 in its
factorization. There is also a technical condition, which we omit, on the ring
generated by the coe cients of g . The subspace M of true modular forms
contains g 0 . Here are pictures of A and M :

A : • •

g 0 g E

M : • or • •

g 0 g 0 g E

Therefore, our intuitive picture given in Figure 15.2 is not quite accurate.
In particular, the sets A and M are finite. However, by reinterpreting the
geometric picture algebraically, we can still discuss tangent spaces.

Since the sets A and M are finite, why not count the elements in both sets
and compare? First of all, this seems to be hard to do. Secondly, the tangent
spaces yield enough information. Consider the following situation. Suppose
you arrive at a train station in a small town. There are no signs telling you
which town it is, but you know it must be either I or II. You have the maps
given in Figure 15.3, where the large dot in the center indicates the station.

I II

Figure 15.3
Two Small Towns