Milne. .Elliptic.Curves.And.Algebraic.Geometry.(1996).[sharethefiles.com]

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" this means that
f(z) = c(n)qn, q = e2�iz/h.
ne"1
a b
Remark 25.2. Note that, for � = " SL2(Z),
c d
az + b a(cz + d) - c(az + b)
d�z = d = dz = (cz + d)-2dz.
cz + d (cz + d)2
Thus condition (25.1b) says that f(z)(dz)k is invariant under the action of �.
Write M2k(�) for the vector space of modular forms of weight 2k, and S2k(�) for the
subspace28 of cusp forms. A modular form of weight 0 is a holomorphic modular function (i.e.,
a holomorphic function on the compact Riemann surface X(�)), and is therefore constant:
M0(�) = C. The product of modular forms of weight 2k and 2k is a modular form of weight
2(k +k ), which is a cusp form if one of the two forms is a cusp form. Therefore �"ke"0M2k(�)
is a graded C-algebra.
Proposition 25.3. Let � be the quotient map H" �! �0(N)\H", and for any holomorphic
differential � on �0(N)\H", set �"� = fdz. Then � �! f is an isomorphism from the space
of holomorphic differentials on �0(N)\H" to S2(�0(N)).
27
k and -k are also used.
28
The S is for Spitzenform , the German name for cusp form. The French call them forme parabolique .
ELLIPTIC CURVES 127
Proof. The only surprise is that f is necessarily a cusp form rather than just a modular
form. I explain what happens at ". Recall (p122) that there is a neighbourhood U of " in
�0(N)\H" and an isomorphism q : U �! D (some disk) such that q �% � = e2�iz. Consider the
differential g(q)dq on U. Its inverse image on H is
g(e2�iz)d(e2�iz) = 2�i � g(e2�iz) � e2�izdz = 2�ifdz
where f(z) = g(e2�iz) � e2�iz. If g is holomorphic at 0, then g(q) = c(n)qn, and so the
ne"0
q-expansion of f is q c(n)qn, which is zero at ".
ne"0
Corollary 25.4. The C-vector space S2(�0(N)) has dimension equal to the genus of X0(N).
Proof. It is part of the theory surrounding the Riemann-Roch theorem that the holomorphic
differential forms on a compact Riemann surface form a vector space equal to the genus of
the surface.
Hence, there are explicit formulas for the dimension of S2(�0(N)) see p123. For example,
it is zero for N d" 10, and has dimension 1 for N = 11. In fact, the Riemann-Roch theorem
gives formulas for the dimension of S2k(�0(N)) for all N.
The modular forms for �0(1). In this section, we find the C-algebra �"ke"0M2k(�0(1)).
We first explain a method of constructing functions satisfying (25.1b). As before, let L be
the set of lattices in C, and let F : L �! C be a function such that
F (��) = �-2kF (�), � " C, � " L.
Then
2k
�2 F (�(�1, �2))
depends only on the ratio �1 : �2, and so there is a function f(z) defined on H such that
2k
�2 F (�(�1, �2)) = f(�1/�2), whenever (�1/�2) > 0.
a b
For � = " SL2(Z), �(a�1 + b�2, c�1 + d�2) = �(�1, �2) and so
c d
az + b
f( ) = (cz + d)-2kF (�(z, 1)) = (cz + d)-2kf(z).
cz + d
When we apply this remark to the Eisenstein series
1
G2k(�) = ,
�2k
�"�,� =0
we find that the function G2k(z) =df G2k(�(z, 1)) satisfies (25.1b). In fact:
Proposition 25.5. For all k > 1, G2k(z) is a modular form of weight 2k for �0(1), and "
is a cusp form of weight 12.
Proof. We know that G2k(z) is holomorphic on H, and the formula on p125 shows that it is
holomorphic at ", which is the only cusp for �0(1) (up to �0(1)-equivalence). The statement
for " is obvious from its definition " = g4(z)3 - 27g4(z)2, and its q-expansion (p125).
128 J.S. MILNE
Theorem 25.6. The C-algebra �"ke"0M2k(�0(1)) is generated by G4 and G6, and G4 and G6
are algebraically independent over C. Therefore
H"
C[G4, G6] - �"ke"0M2k(�0(1)), C[G4, G6] H" C[X, Y ]
�!
(isomorphisms of graded C-algebras if X and Y are given weights 4 and 6 respectively).
Moreover,
f �! f � " : M2k-12(�0(1)) �! S2k(�0(1))
is a bijection.
Proof. Straightforward see Serre, Cours..., VII.3.2.
Therefore, for k e" 0,
[k/6] if k a" 1 mod 6
dim M2k(�0(N)) =
[k/6] + 1 otherwise.
Here [x] is the largest integer d" x.
Theorem 25.7 (Jacobi). There is the following formula:
"
" = (2�)12q (1 - qn)24, q = e2�iz.
n=1
Proof. Let
"
F (z) = q (1 - qn)24.
n=1
From the theorem, we know that the space of cusp forms of weight 12 has dimension 1, and
therefore if we can show that F (z) is such a form, then we ll know it is a multiple of ", and
it will be follow from the formula on p125 that the multiple is (2�)12.
1 1 0 -1
Because SL2(Z)/{�I} is generated by T = and S = , to verify the
0 1 1 0
conditions in (25.1), it suffices to verify that F transforms correctly under T and S. For T
this is obvious from the way we have defined F , and for S it amounts to checking that
1
F (- ) = z12F (z).
z
This is trickier than it looks, but there are short (2 page) elementary proofs see for example,
Serre, ibid., VII.4.4.
26. Modular Forms and the L-series of Elliptic Curves
In this section, I ll discuss how the L-series classify the elliptic curves over Q up to isogeny,
and then I ll explain how the work of Hecke, Petersson, and Atkin-Lehner leads to a list of
candidates for the L-series of such curves, and hence suggests a classification of the isogeny
classes.
ELLIPTIC CURVES 129
Dirichlet Series. A Dirichlet series is a series of the form
f(s) = a(n)n-s, a(n) " C, s " C.
ne"1
The simplest example of such a series is, of course, the Riemann zeta function n-s. If
ne"1
there exist positive constants A and b such that | a(n)| d" Axb for all large x, then the
nd"x
series for f(s) converges to an analytic function on the half-plane (s) > b.
It is important to note that the function f(s) determines the a(n) s, i.e., if a(n)n-s and
b(n)n-s are equal as functions of s on some half-plane, then a(n) = b(n) for all n. In
fact, by means of the Mellin transform and its inverse (see 26.4 below), f determines, and is
determined by, a function g(q) convergent on some disk about 0, and g(q) = a(n)qn.
We shall be especially interested in Dirichlet series that are equal to Euler products, i.e.,
those that can be expressed as
1
f(s) =
1 - Pp(p-s)
p
where each Pp is a polynomial.
Dirichlet series arise in two essentially different ways: from analysis and from geometry
and number theory. One of big problems mathematics is to show that the second set of
Dirichlet series is a subset of the first, and to identify the subset. This is a major theme
in Langlands s philosophy, and the rest of the course will be concerned with explaining how
Wiles was able to identify the L-series of (almost all) elliptic curves over Q with certain
L-series attached to modular forms.
The L-series of an elliptic curve. Recall that for an elliptic curve E over Q, we define
1 1
L(E, s) = �
1 - app-s + p1-s p bad 1 - app-s
p good
where
��
�� p + 1 - Np p good;
��
��
��
1 p split nodal;
ap =
�� -1 p nonsplit nodal;
��
��
ó�
0 p cuspidal.
p
Recall also that the conductor N = NE/ of Q is pf where fp = 0 if E has good reduction
p
at p, fp = 1 if E has nodal reduction at p, and fp e" 2 otherwise (and = 2 unless p = 2, 3).
On expanding out the product (cf. below), we obtain a Dirichlet series
L(E, s) = ann-s.
This series has, among others, the following properties:
(a) (Rationality) Its coefficients an lie in Q.
(b) (Euler product) It can be expressed as an Euler product ; in fact, that s how it is
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