These notes study the passage from classical modular forms to Galois representations. The guiding question is how analytic data, such as the Hecke eigenvalues of a modular eigenform, can encode arithmetic data over $\mathbb Q$. The course develops this question by moving through modular curves, Hecke correspondences, Jacobians, cohomology, elliptic curves, and finally two-dimensional representations of absolute Galois groups.

The chapters are arranged as a sequence of translations. Congruence conditions first become level structures on modular curves. Hecke eigenvalues then become geometric correspondences, and those correspondences act on cohomology and Tate modules. In weight $2$ this construction is visible through Jacobians of modular curves; in higher weight it is completed by Deligne's construction. Later chapters use the resulting compatible systems to discuss residual representations, congruences, elliptic curves, modular parametrizations, the modularity theorem, and the route from modularity to [Fermat's Last Theorem](/theorems/4789).

The aim of this introductory chapter is not to prove the main theorems. It fixes the objects and compatibility conditions that the rest of the course will make precise: which coefficients of a modular form should become Frobenius traces, which geometric spaces carry the relevant operators, and why elliptic curves and modular eigenforms can be compared by their local data.

# Introduction

## What Does It Mean for a Modular Form to Be Arithmetic?

A modular form begins life as a [holomorphic function](/page/Holomorphic%20Function) on the upper half-plane with transformation laws and growth conditions. The first problem of the course is that this analytic definition does not yet explain why its Fourier coefficients should have Galois-theoretic meaning. A $q$-expansion alone gives a sequence of complex numbers, but it does not explain why the prime-index coefficients should satisfy local Euler factor identities, be compatible as $p$ varies, or match point counts over finite fields. Hecke operators provide the bridge: their eigenvalues are simultaneously analytic spectral data and arithmetic counting data on modular curves.

For the first definition, the notation is as follows. The congruence subgroup

\begin{align*} \Gamma_1(N)=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in SL_2(\mathbb Z): c\equiv 0\pmod N,\ a\equiv d\equiv 1\pmod N\right\} \end{align*}

acts on the upper half-plane by fractional linear transformations. The space $S_k(\Gamma_1(N))$ is the complex [vector space](/page/Vector%20Space) of cusp forms of weight $k$ for this subgroup, meaning modular forms whose expansions at all cusps have zero constant term. For each $n\ge 1$, the Hecke operator $T_n$ is the standard double-coset operator acting linearly on $S_k(\Gamma_1(N))$; later chapters rebuild this operator geometrically as a correspondence on modular curves. A simultaneous eigenvector for all $T_n$ is the analytic input whose eigenvalues will later be compared with Frobenius traces. When a character $\varepsilon$ appears later, it is the nebentypus character, the Dirichlet character recording the diamond-operator transformation factor of the form.

[definition: Normalised Hecke Eigenform] Let $N \in \mathbb N$ and $k \ge 2$. A normalised Hecke eigenform of level $N$ and weight $k$ is a cusp form $f \in S_k(\Gamma_1(N))$ such that \begin{align*} f(q) = \sum_{n=1}^{\infty} a_n(f)q^n, \qquad a_1(f)=1, \end{align*} and $f$ is an eigenvector for all Hecke operators $T_n$ acting on $S_k(\Gamma_1(N))$. [/definition]

The normalisation fixes the scale, so the eigenvalues can be read off as the coefficients $a_n(f)$ under the standard Hecke action. In this course, the coefficients $a_p(f)$ for primes $p \nmid N$ will become traces of Frobenius elements in Galois representations.

[example: Ramanujan Delta As A Prototype] The discriminant form is \begin{align*} \Delta(q)=q\prod_{n=1}^{\infty}(1-q^n)^{24}=\sum_{n=1}^{\infty}\tau(n)q^n . \end{align*} The product has constant term $1$, so multiplying by $q$ gives first coefficient $\tau(1)=1$ and no constant term. Thus the displayed $q$-expansion is normalised, and the vanishing constant term is the cusp-form condition at the cusp $\infty$. The classical discriminant form is a level $1$ Hecke eigenform of weight $12$. For a prime $p$, the prime-index Hecke eigenvalue is the coefficient $\tau(p)$. Since the level is $1$, the nebentypus is trivial, so the determinant term predicted for weight $12$ is \begin{align*} \varepsilon(p)p^{k-1}=1\cdot p^{12-1}=p^{11}. \end{align*} Hence the local Euler factor has the form \begin{align*} 1-\tau(p)X+p^{11}X^2. \end{align*} This is exactly the shape of a two-dimensional Frobenius characteristic polynomial written in reciprocal form: if a $2 \times 2$ matrix has trace $t$ and determinant $d$, with eigenvalues $\alpha$ and $\beta$, then \begin{align*} \det(1-XA)&=(1-\alpha X)(1-\beta X)\\ &=1-(\alpha+\beta)X+\alpha\beta X^2\\ &=1-\operatorname{tr}(A)X+\det(A)X^2. \end{align*} Thus, for $\Delta$, in the arithmetic-Frobenius convention used in these notes, the comparison data at $p$ would be $\operatorname{tr}(\operatorname{Frob}_p)=\tau(p)$ and $\det(\operatorname{Frob}_p)=p^{11}$. The reusable phenomenon is that prime-index Hecke eigenvalues are the trace entries one compares with local Galois data once the Frobenius convention has been fixed. [/example]

## Why Modular Curves Enter the Story

If Hecke eigenvalues are to have arithmetic meaning, they need a geometric home. Modular curves supply that home by reinterpreting modular forms and Hecke operators in terms of elliptic curves with level structure. This changes the subject from functions on the upper half-plane to algebraic curves defined over number fields.

[definition: Modular Curve Of Type Gamma Zero] Let $N \in \mathbb N$. The modular curve $Y_0(N)$ is the coarse moduli space whose complex points classify pairs $(E,C)$, where $E$ is an elliptic curve over $\mathbb C$ and $C \subset E[N]$ is a cyclic subgroup of order $N$. [/definition]

The compactification $X_0(N)$ adds cusps, which correspond to degenerating elliptic curves. The analytic description as a quotient of the upper half-plane and the moduli description as elliptic curves with cyclic subgroups are two views of the same object.

The informal phrase "classifies elliptic curves with level structure" is not enough for later arithmetic use: one has to know exactly which data are being identified and which degenerations are represented after compactification. Without that precision, a Hecke correspondence would have no well-defined moduli interpretation at the cusps or on points with automorphisms. The following classification fixes the complex points of $Y_0(N)$ and explains how the cusp points enter $X_0(N)$.

[quotetheorem:4729]

[citeproof:4729]

The hypotheses indicate exactly what the moduli problem remembers. The cyclic subgroup of order $N$ is the geometric datum corresponding to the congruence condition defining $\Gamma_0(N)$, whereas full level structure would remember a basis of $E[N]$. For now this is a classification over $\mathbb C$: it tells us what the complex points and cusps mean. The arithmetic refinements, such as integral models, bad reduction, and cohomological actions, enter only after the basic moduli picture is stable.