Modular Forms¶

Summary

Modular forms are holomorphic functions with extreme symmetry – they transform predictably under the action of \(\text{SL}_2(\mathbb{Z})\). Their Fourier coefficients contain deep number-theoretic information, and their \(L\)-series are the "twin souls" of the \(L\)-series of elliptic curves.

Prerequisites¶

Helpful but not essential: - Elliptic Curves

1. The Upper Half-Plane¶

The upper half-plane is the set of complex numbers with positive imaginary part:

\[ \mathbb{H} = \{z \in \mathbb{C} \mid \text{Im}(z) > 0\} \]

The group \(\text{SL}_2(\mathbb{Z})\) acts on \(\mathbb{H}\) – the set of \(2 \times 2\) matrices with integer entries and determinant \(1\):

\[ \text{SL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mid a, b, c, d \in \mathbb{Z}, \, ad - bc = 1 \right\} \]

The action is by Möbius transformations:

\[ \gamma \cdot z = \frac{az + b}{cz + d} \qquad \text{for } \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

One can verify: if \(z \in \mathbb{H}\), then \(\gamma \cdot z \in \mathbb{H}\) as well. The upper half-plane is thus closed under this action.

\(\text{SL}_2(\mathbb{Z})\) is generated by two matrices:

\[ T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}: z \mapsto z + 1 \qquad \text{(translation)} \]

\[ S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}: z \mapsto -\frac{1}{z} \qquad \text{(inversion)} \]

2. Definition of a Modular Form¶

A modular form of weight \(k\) for \(\text{SL}_2(\mathbb{Z})\) is a holomorphic function \(f: \mathbb{H} \to \mathbb{C}\) satisfying two conditions:

(M1) Transformation law. For all \(\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}_2(\mathbb{Z})\):

\[ f\left(\frac{az + b}{cz + d}\right) = (cz + d)^k f(z) \]

(M2) Holomorphy at the boundary. \(f\) is also holomorphic "at \(z \to i\infty\)" (i.e., the Fourier expansion has no negative powers).

In particular, from the translation \(T: z \mapsto z + 1\):

\[ f(z + 1) = f(z) \]

So \(f\) is periodic with period \(1\) – and thus has a Fourier expansion.

The weight

The weight \(k\) must be an even natural number (for \(\text{SL}_2(\mathbb{Z})\)). The factor \((cz + d)^k\) is the "price" of symmetry: \(f\) is not invariant under \(\text{SL}_2(\mathbb{Z})\), but transforms with a correction factor.

A modular form is called a cusp form if additionally \(f(z) \to 0\) as \(z \to i\infty\), that is, if the constant term of the Fourier expansion vanishes.

3. Fourier Expansion¶

Since \(f(z + 1) = f(z)\), we can introduce the variable \(q = e^{2\pi i z}\). For \(z \in \mathbb{H}\) we have \(|q| < 1\), and \(f\) has an expansion:

\[ f(z) = \sum_{n=0}^{\infty} a_n q^n = a_0 + a_1 q + a_2 q^2 + a_3 q^3 + \cdots \]

The coefficients \(a_n\) are called the Fourier coefficients of the modular form. For cusp forms, \(a_0 = 0\).

The Fourier coefficients are the heart of the matter: they carry all the arithmetic information of the modular form. That these coefficients have deep number-theoretic significance is one of the great wonders of mathematics.

4. Examples¶

Eisenstein Series¶

For even \(k \geq 4\), the Eisenstein series is defined as:

\[ G_k(z) = \sum_{\substack{(c,d) \in \mathbb{Z}^2 \\ (c,d) \neq (0,0)}} \frac{1}{(cz + d)^k} \]

The normalised version \(E_k(z) = \frac{G_k(z)}{2\zeta(k)}\) has the Fourier expansion:

\[ E_k(z) = 1 - \frac{2k}{B_k} \sum_{n=1}^{\infty} \sigma_{k-1}(n) \, q^n \]

where \(B_k\) is the \(k\)-th Bernoulli number and \(\sigma_{k-1}(n) = \sum_{d \mid n} d^{k-1}\) is the divisor sum function.

Example: \(E_4(z) = 1 + 240(q + 9q^2 + 28q^3 + 73q^4 + \cdots)\)

The Discriminant \(\Delta\)¶

The most famous cusp form is the Ramanujan discriminant:

\[ \Delta(z) = q \prod_{n=1}^{\infty} (1 - q^n)^{24} = \sum_{n=1}^{\infty} \tau(n) q^n \]

It has weight \(12\) and is the (up to scaling) unique cusp form of weight \(12\) for \(\text{SL}_2(\mathbb{Z})\).

The coefficients \(\tau(n)\) are the Ramanujan \(\tau\)-function:

\[ \tau(1) = 1, \quad \tau(2) = -24, \quad \tau(3) = 252, \quad \tau(4) = -1472, \quad \ldots \]

Ramanujan conjectured in 1916 that \(|\tau(p)| \leq 2p^{11/2}\) for all primes \(p\) – this was proved only in 1974 by Deligne (as a consequence of the Weil conjectures).

The \(j\)-Invariant¶

The \(j\)-invariant \(j(z) = E_4(z)^3 / \Delta(z)\) is not a modular form (it has weight \(0\) and a pole at \(i\infty\)), but a modular function. It classifies elliptic curves: two curves are isomorphic (over \(\overline{K}\)) if and only if they have the same \(j\)-invariant.

5. Hecke Operators¶

The spaces of modular forms carry additional symmetries – the Hecke operators \(T_n\). For a modular form \(f(z) = \sum a_m q^m\) and a prime \(p\):

\[ (T_p f)(z) = \sum_{m=0}^{\infty} (a_{mp} + p^{k-1} a_{m/p}) \, q^m \]

(where \(a_{m/p} = 0\) if \(p \nmid m\)).

Eigenforms. A modular form is called a Hecke eigenform if it is an eigenvector of all \(T_n\):

\[ T_n f = \lambda_n f \quad \text{for all } n \]

For normalised Hecke eigenforms (\(a_1 = 1\)), a remarkable result holds: the eigenvalues are the Fourier coefficients: \(\lambda_n = a_n\).

Multiplicativity

The Fourier coefficients of a Hecke eigenform are multiplicative: \(a_{mn} = a_m a_n\) for \(\gcd(m, n) = 1\). Furthermore, \(a_{p^{r+1}} = a_p a_{p^r} - p^{k-1} a_{p^{r-1}}\). Thus the \(a_p\) (for primes \(p\)) determine the entire Fourier expansion.

6. \(L\)-Series of Modular Forms¶

Every modular form \(f(z) = \sum_{n=1}^{\infty} a_n q^n\) (cusp form) defines an \(L\)-series:

\[ L(f, s) = \sum_{n=1}^{\infty} a_n n^{-s} \]

For Hecke eigenforms, this \(L\)-series has an Euler product:

\[ L(f, s) = \prod_p \frac{1}{1 - a_p p^{-s} + p^{k-1-2s}} \]

The \(L\)-series converges for \(\text{Re}(s) > (k+1)/2\) and admits analytic continuation to all of \(\mathbb{C}\). It satisfies a functional equation relating \(L(f, s)\) to \(L(f, k - s)\).

Comparison with Elliptic Curves¶

For a Hecke eigenform \(f\) of weight \(2\) and an elliptic curve \(E\), compare the \(L\)-series:

\[ L(f, s) = \prod_p \frac{1}{1 - a_p(f) p^{-s} + p^{1-2s}} \]

\[ L(E, s) = \prod_p \frac{1}{1 - a_p(E) p^{-s} + p^{1-2s}} \]

The structures are identical! The Taniyama–Shimura conjecture states: for every elliptic curve \(E\) there exists a Hecke eigenform \(f\) of weight \(2\) with \(a_p(E) = a_p(f)\) for all (but finitely many) primes \(p\).

7. Congruence Subgroups¶

For Wiles' proof, modular forms for \(\text{SL}_2(\mathbb{Z})\) are not sufficient – one needs modular forms for congruence subgroups. The most important is:

\[ \Gamma_0(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \text{SL}_2(\mathbb{Z}) \mid N \mid c \right\} \]

The level \(N\) determines how much symmetry the modular form has – less symmetry (larger \(N\)) allows more modular forms.

An elliptic curve \(E\) with conductor \(N_E\) (a measure of the "complexity" of the bad reduction) corresponds to a Hecke eigenform of weight \(2\) and level \(N_E\).

8. The Bridge to Elliptic Curves¶

The connection between modular forms and elliptic curves is one of the deepest links in mathematics:

Elliptic curve \(E\)	Modular form \(f\)
Coefficients \(a, b\) in \(y^2 = x^3 + ax + b\)	Fourier coefficients \(a_n\)
\(a_p(E) = p + 1 - \#E(\mathbb{F}_p)\)	\(a_p(f)\) = Hecke eigenvalue
\(L(E, s)\)	\(L(f, s)\)
Conductor \(N_E\)	Level \(N\)
Galois representation \(\rho_{E,\ell}\)	Galois representation \(\rho_{f,\ell}\)

Theorem (Wiles 1995, Breuil–Conrad–Diamond–Taylor 2001). Every elliptic curve over \(\mathbb{Q}\) is modular: there exists a Hecke eigenform \(f\) of weight \(2\) with \(L(E, s) = L(f, s)\).

Wiles proved the semistable case in 1995 – sufficient for FLT. The full conjecture was proved in 2001 by Breuil, Conrad, Diamond, and Taylor.

The Implication for FLT¶

Why does modularity imply Fermat's Last Theorem? The proof proceeds by contradiction:

Assumption: There exists a solution \(a^p + b^p = c^p\).
Frey: Construct the elliptic curve \(E: y^2 = x(x - a^p)(x + b^p)\).
Ribet: This Frey curve cannot be modular (its conductor would be "too small").
Wiles: But every semistable elliptic curve is modular.
Contradiction: The Frey curve does not exist → there is no solution → FLT is true.