This course develops the classical theory of modular forms from the ground up, beginning with the action of the modular group on the upper half-plane and ending with modular forms as explicit, computable arithmetic objects. The central goal is to understand holomorphic functions with strong symmetry and growth conditions, and to see how those conditions force rich algebraic, analytic, and arithmetic structure. Along the way, the course introduces the basic examples, the Fourier or q-expansion viewpoint, and the special role played by Eisenstein series and cusp forms.

The chapters are arranged to build a usable toolkit step by step. After the geometric and analytic foundations, the course turns to q-expansions and then uses the valence formula to control zeros, poles, and spaces of forms, leading to dimension formulas and concrete bases. Hecke operators and the Hecke algebra then provide the main arithmetic structure, with eigenforms serving as distinguished simultaneous eigenvectors. The final chapters connect this theory to orthogonality via the Petersson inner product, to [analytic continuation](/page/Analytic%20Continuation) and arithmetic information through L-functions, and to the synthesis of modular forms as objects that can be computed, decomposed, and applied in number theory.

# 1. The Modular Group and the Upper Half-Plane

This chapter introduces the geometric stage on which classical modular forms live. The main object is the complex upper half-plane, acted on by integer matrices through fractional linear transformations. We first study the action itself, then the special points and cusps that survive in the quotient, and finally the standard fundamental domain that turns the quotient into a concrete geometric object. The central obstruction is that a quotient by an infinite group is hard to see directly, so the chapter builds explicit representatives, boundary identifications, and stabilizer computations.

## Fractional Linear Transformations on the Upper Half-Plane

How can an arithmetic group act on a complex domain in a way that preserves enough geometry for complex analysis? The answer is that matrices in $SL_2(\mathbb Z)$ act on the upper half-plane by Möbius transformations. This action is the source of the transformation law for modular forms in the next chapter.

[definition: Upper Half-Plane] The complex upper half-plane is \begin{align*} \mathbb H := \{z \in \mathbb C : \operatorname{Im}(z) > 0\}. \end{align*} [/definition]

The domain $\mathbb H$ is stable under the transformations we need because the imaginary part transforms by a positive factor. This is not automatic for arbitrary complex Möbius transformations: a real determinant condition is what prevents points from crossing the real axis. The group responsible for the arithmetic part of the theory is the integral determinant-one subgroup.

[definition: Modular Group] The modular group is \begin{align*} SL_2(\mathbb Z) := \left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} : a,b,c,d \in \mathbb Z,\ ad-bc=1\right\}. \end{align*} [/definition]

Since $-I$ acts in the same way as $I$ on $\mathbb H$, the effective group of transformations is often

\begin{align*} PSL_2(\mathbb Z):=SL_2(\mathbb Z)/\{\pm I\}. \end{align*}

We nevertheless keep $SL_2(\mathbb Z)$ in the notation because modular forms remember signs through their weights. Without this distinction, the geometry of the quotient would be correct but the later transformation law for odd weights would lose information. The next definition records the actual map whose kernel contains exactly this central sign ambiguity.

[definition: Fractional Linear Action] The fractional linear action is the map \begin{align*} SL_2(\mathbb Z)\times \mathbb H &\longrightarrow \mathbb H,\\ (\gamma,z)&\longmapsto \gamma z, \end{align*} where, for $\gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in SL_2(\mathbb Z)$, \begin{align*} \gamma z := \frac{az+b}{cz+d}. \end{align*} [/definition]

The denominator never vanishes on $\mathbb H$: if $cz+d=0$ with $c,d\in\mathbb R$, then $z=-d/c$ is real when $c\ne0$, and if $c=0$ then $d\ne0$.

For this formula to define an action on the upper half-plane, it must do more than avoid division by zero. We also need to know that applying such a matrix keeps the point in $\mathbb H$, and that composition of matrices agrees with composition of the resulting transformations. The next result supplies exactly these two checks, turning the formula into a genuine geometric action rather than just a formal expression.

[quotetheorem:4213]

[citeproof:4213]

The formula also explains why the upper half-plane is natural: these transformations preserve orientation and hyperbolic height up to the factor forced by the denominator. The hypothesis $ad-bc=1$ is essential here; with negative determinant the imaginary part changes sign, and with non-real coefficients the real boundary need not be preserved. This is the first point where arithmetic, complex analysis, and geometry meet in the course: integral matrices give a discrete group, while the imaginary-part formula keeps the action inside a complex domain.

[example: Translation and Inversion] Let \begin{align*} T=\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad S=\begin{pmatrix}0&-1\\1&0\end{pmatrix}. \end{align*} For $T$, the entries are $a=1$, $b=1$, $c=0$, and $d=1$, so the fractional linear action gives \begin{align*} Tz=\frac{1\cdot z+1}{0\cdot z+1} =\frac{z+1}{1} =z+1. \end{align*} Thus $T$ adds $1$ to the real part and leaves the imaginary part unchanged: \begin{align*} T(x+iy)=x+1+iy. \end{align*} So vertical strips are shifted horizontally by one unit. For $S$, the entries are $a=0$, $b=-1$, $c=1$, and $d=0$, hence \begin{align*} Sz=\frac{0\cdot z-1}{1\cdot z+0} =-\frac{1}{z}. \end{align*} If $z=re^{i\theta}$ with $r>0$ and $0<\theta<\pi$, then \begin{align*} Sz=-\frac{1}{re^{i\theta}} =-r^{-1}e^{-i\theta} =r^{-1}e^{i(\pi-\theta)}. \end{align*} Therefore $|Sz|=r^{-1}=1/|z|$, so points with $|z|<1$ are sent to points with $|Sz|>1$, and points with $|z|>1$ are sent to points with $|Sz|<1$. Also the argument changes from $\theta$ to $\pi-\theta$, which is reflection across the imaginary axis. These two transformations preview the horizontal translation and circle-inversion moves that build the quotient geometry. [/example]

The preceding example also shows a problem that will reappear for functions: the denominator $cz+d$ is not a harmless by-product of the action. It controls both the change in height and the derivative of the transformation. To state modular transformation laws without rewriting this denominator every time, we isolate it as a separate function.