Modular forms are holomorphic functions constrained by a large symmetry group. Their Fourier coefficients encode arithmetic, and Mellin transforms turn them into \(L\)-functions with analytic continuation and functional equations.
The modular action
The group \(\operatorname{SL}_2(\mathbb Z)\) acts on the upper half-plane \(\mathbb H\) by \[ \gamma z=\frac{az+b}{cz+d},\qquad \gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}. \]
Definition. A modular form of weight \(k\) for \(\operatorname{SL}_2(\mathbb Z)\) is a holomorphic \(f:\mathbb H\to\mathbb C\) satisfying \[ f(\gamma z)=(cz+d)^kf(z) \] and holomorphic at the cusp. Since \(f(z+1)=f(z)\), it has a Fourier expansion \[ f(z)=\sum_{n=0}^{\infty}a_nq^n,\qquad q=e^{2\pi iz}. \] It is a cusp form if \(a_0=0\).
Eisenstein series and the discriminant
Theorem. For even \(k\ge4\), \[ G_k(z)=\sum_{(m,n)\ne(0,0)}(mz+n)^{-k} \] converges absolutely and is a modular form of weight \(k\).
Proof. Absolute convergence is locally uniform for \(k>2\). Reindexing the lattice sum by the bijection \((m,n)\mapsto(ma+nc,mb+nd)\) gives \[ G_k(\gamma z)=(cz+d)^kG_k(z). \] Uniform convergence permits termwise holomorphy, and its Fourier expansion shows regularity at the cusp.
The normalized Eisenstein series satisfy \[ E_4=1+240\sum_{n\ge1}\sigma_3(n)q^n,\qquad E_6=1-504\sum_{n\ge1}\sigma_5(n)q^n. \] The discriminant cusp form \[ \Delta(z)=q\prod_{n\ge1}(1-q^n)^{24} =\frac{E_4^3-E_6^2}{1728} \] has weight \(12\).
Hecke operators and Euler products
Theorem. If a normalized cusp form \(f=\sum_{n\ge1}a_nq^n\) of weight \(k\) is a simultaneous Hecke eigenform, then \[ a_{mn}=a_ma_n\quad(\gcd(m,n)=1),\qquad a_{p^{r+1}}=a_pa_{p^r}-p^{k-1}a_{p^{r-1}}. \] Consequently \[ L(f,s)=\sum_{n\ge1}\frac{a_n}{n^s} =\prod_p(1-a_pp^{-s}+p^{k-1-2s})^{-1}. \]
The Hecke algebra turns geometric symmetries of modular curves into multiplicativity of Fourier coefficients.
Mellin transform and functional equation
Theorem. For a cusp form \(f\) of weight \(k\), \[ \Lambda(f,s)=(2\pi)^{-s}\Gamma(s)L(f,s) =\int_0^\infty f(iy)y^{s-1}\,\mathrm{d}y \] extends to an entire function and satisfies \[ \Lambda(f,s)=i^k\Lambda(f,k-s). \]
Proof. Insert the Fourier expansion and integrate termwise to obtain the Mellin identity. Split the integral at \(1\), substitute \(y\mapsto1/y\) in \((0,1)\), and use \(f(-1/z)=z^kf(z)\). Exponential decay at both ends gives entire continuation and the functional equation.
Arithmetic correspondences
Weight-two cusp forms for congruence subgroups correspond to holomorphic differentials on modular curves. Newforms with rational coefficients correspond to isogeny classes of elliptic curves over \(\mathbb Q\), matching \[ a_p=p+1-\#E(\mathbb F_p). \] More broadly, automorphic \(L\)-functions package local representation data into Euler products. Reciprocity conjectures seek to identify Galois representations and automorphic representations through equality of these local factors.
Critical values
Special values of \(L\)-functions often encode algebraic invariants: class numbers and regulators for Dedekind zeta functions, generalized Bernoulli numbers for Dirichlet \(L\)-functions, and ranks or periods for elliptic curves. This is not incidental; functional equations and cohomological interpretations make special values a central meeting point of analysis, geometry, and arithmetic.