Lecture 16: Structure of the Hecke algebra
In this lecture, we establish basic results on the structure of the Hecke algebra and some of its natural modules. In particular, we show that the Hecke algebra $\bT$ is a finite rank free $\bZ$algebra and that $\bT \otimes \bQ$ is semisimple. We also show that the space of weight 2 cusp forms (at prime level) is a free module of rank 1; this is the multiplicity one theorem.
Review
Recall that for a modular form $f$ for $\Gamma(1)$ and weight $k$, we defined the action of the Hecke operator $T_n$ on $f$ by
$$ (T_nf)(\Lambda) = n^{k1} \sum_{[\Lambda:\Lambda']=n} f(\Lambda'). $$In fact, the same formula can be used to define $T_nf$ when $f$ is a modular form on $\Gamma_0(N)$ of weight $k$ if $n$ is prime to $N$. From now on, we only care about the $k=2$ case.
We proved the following results:

The $T_n$ commute with each other.

If $n$ and $m$ are coprime then $T_{nm}=T_nT_m$.

We have the recurrenceĀ $T_{p^{n+1}}=T_pT_{p^n}pT_{p^{n1}}$.

If $f=\sum_{n \ge 1} a_n q^n$ and $p$ is prime then
$$ T_pf = \sum_{n \ge 1} (a_{pn}+p^{k1} a_{n/p}) q^n $$
In particular, $a_1(T_pf)=a_p(f)$. In fact, $a_1(T_nf)=a_n(f)$ holds for all $n$ prime to $N$.
We define $\wt{\bT}$ to be the infinite polynomial ring $\bZ[T_p]$ with $p$ prime to $N$. We define $\bT$ to be the image of $\wt{\bT}$ in $\End(S_2(N))$. Our goal today is to understand this ring and the structure of $S_2(N)$ as a module over it.
The Petersson inner product
Let $f,g \in S_2(N)$. Then $f(z) dz$ and $g(z) dz$ are 1forms on the upper halfplane invariant under $\Gamma_0(N)$. It follows that $\ol{g(z) dz}$ is also invariant, and so $f(z) dz \wedge \ol{g(z) dz}=2i f(z) \ol{g(z)} dx dy$ is also invariant. We define
$$ \langle f, g \rangle = \int_{\fh/\Gamma_0(N)} f(z) \ol{g(z)} dx dy. $$This is called the Petersson inner product. The integral converges since $f$ and $g$ decay rapidly at the cusps (as they are cusp forms). It is clear that $\langle, \rangle$ is a positive definite Hermitian form on $S_2(N)$. Furthermore, we have the following result:
Proposition. The operators $T_p$ are selfadjoint. That is, $\langle T_pf, g \rangle=\langle f, T_p g \rangle$.
Corollary. The algebra $\bT \otimes \bC$ is semisimple. The space $S_2(\Gamma)$ admits a basis consisting of Hecke eigenforms (i.e., forms $f$ which are eigenvectors of $T_p$ for all $p \nmid N$).
We call a Hecke eigenform $f$ normalized if $a_1(f)=1$. We note that for such forms, $a_n(f)$ is the eigenvalue of $T_n$ acting on $f$, for $n$ prime to $N$. If $f$ is an eigenform then we get a ring homomorphism $\alpha \colon \bT \to \bC$ by mapping $T_p$ to the eigenvalue of $f$ under $T_p$. One calls $\alpha$ a system of eigenvalues. The space of cusp forms decomposes as
$$ S_2(N) = \bigoplus S_2(N)_{\alpha}, $$where $S_2(N)_{\alpha}$ is the space of forms $f$ with $T_pf=\alpha(T_p) f$ for all $p \nmid N$.
Multiplicity one
Theorem. Suppose $N$ is prime and $f,g \in S_2(N)$ are two normalized Hecke eigenforms with the same eigenvalues for all $p \ne N$. Then $f=g$.
Proof. We first introduce a piece of notation. For $\gamma \in \SL_2(\bR)$ and a function $f$ on the upper halfplane, define $(f  [\gamma] )(z) = (cz+d)^{2} f(\gamma)(z)$. Then one can show that this defines an action of $\SL_2(\bR)$. Furthermore, $f$ belongs to $S_2(N)$ if and only if $f \vert [\gamma] = f$ for all $\gamma \in f$, and $f$ satisfies the appropriate holomorphicity conditions.
Now, for $p \ne N$ we have $a_p(f)=a_p(g)$ since the $T_p$ eigenvalues of $f$ and $g$ are assumed equal. It follows that $a_n(f)=a_n(g)$ for all $n$ prime to $N$ by the multiplicativity and recurrence properties of the $T_n$. Let $h=fg$. Then $a_n(h)=0$ unless $N \mid n$. It follows that $h(z+1/N)=h(z)$. Thus $h  [\gamma] = h$ if $\gamma$ belongs to $\Gamma_0(N)$ or if
$$ \gamma = \mat{1}{1/N}{}{1}. $$The group $\Gamma_0(N)$ and the above matrix together generate the group $\sigma^{1} \Gamma(1) \sigma$, where
$$ \sigma = \mat{N}{}{}{1}. $$Thus $h  [\sigma]$ is invariant under $\Gamma(1)$, and therefore belongs to $S_2(1)$. But $S_2(1)=0$, so $h=0$ and $f=g$. ◾
Corollary. For any system of eigenvalues $\alpha$, the space $S_2(N)_{\alpha}$ is onedimensional.
Corollary. There is a bijection between homomorphisms $\bT \to \bC$ and normalized cusp forms.
Corollary. $S_2(N)$ is free of rank 1 as a $\bT \otimes \bC$ module.
Remark. There is also a stronger version of this theorem: if $f$ and $g$ are two normalized Hecke eigenforms that have the same eigenvalues under $T_p$ for all but finitely many $p$ (or even all $p$ in a density 1 set of primes) then $f=g$.
Remark. The above theorem is false for $N$ composite, but for a somewhat silly reason. If $p \mid N$ and $f \in S_2(N/p)$ then both $f(z)$ and $f(pz)$ belong to $S_2(N)$, and they have the same $T_{\ell}$ eigenvalues for $\ell \nmid N$. This is the only problem. More precisely, we define the old subspace to be the subspace of $S_2(N)$ spanned by $f(z)$ and $f(pz)$ where $pN$ and $f \in S_2(N/p)$. We define the new subspace to be the orthogonal complement of the old subspace. Then multiplicity one holds on the new subspace.
Hecke correspondences
In terms of lattices, $(T_pf)(\Lambda)$ is defined by summing the values of $f$ over the index $p$ sublattices of $\Lambda$. If $\Lambda$ corresponds to the elliptic curve $E$, then the sublattices correspond to isogenies $E \to E'$ of degree $p$. The set of such isogenies is exactly the fiber of $X_0(pN) \to X_0(N)$ above $E$. Summing over the $E'$ the corresponds to pushing forward along the map taking $E \to E'$ to $E'$.
Let us be more precise. We previously defined a $\Gamma_0(p)$ structure on $E$ to be a cyclic subgroup $G$ of order $p$. But it is equivalent to define a $\Gamma_0(p)$ structure to be an isogeny $E \to E'$ of degree $p$ with cyclic kernel. For the moment, it will be more convenient to think of $X_0(p)$ as the space of such isogenies. We think of $X_0(Np)$ as parametrizing data $(E \to E', G)$ where $E \to E'$ is an isogeny of degree $p$ and $G \subset E$ is a cyclic subgroup of size $N$.
Let $p$ be prime to $N$. There are two natural maps $X_0(Np) \to X_0(N)$, namely, take $(f:E \to E', G)$ to $(E, G)$ or $(E', f(G))$. Call these maps $p_1$ and $p_2$. The diagram
$$ \xymatrix{ & X_0(pN) \ar[rd]^{p_2} \ar[ld]_{p_1} \\ X_0(N) && X_0(N) } $$is called the Hecke correspondence.
Generalities on correspondences
Let $C$ be a smooth projective curve. A correspondence $C \dashrightarrow C$ is a pair of maps $p_1, p_2 \colon C' \to C$ with $p_1$ and $p_2$ finite maps; we assume $C'$ is a smooth projective curve as well. We can think of a nonconstant function $f \colon C \to C$ as a correspondence by taking $C'=C$, $p_1=\id$, and $p_2=f$. In general, one thinks of correspondences as multivalued functions, where $x \in C$ is mapped to the set $p_2(p_1^{1}(x))$.
Correspondences act on the singular cohomology $\rH^1(C, \bZ)$ by the formula $(p_2)_* p_1^*$. Here $p_1^*$ is the usual pullback operation $\rH^1(C, \bZ) \to \rH^1(C', \bZ)$, and $(p_2)_*$ is the adjoint to $p_2^*$ under the cup product pairing.
Correspondences also act on differential forms, via the same formula. Over the complex numbers, the isomorphism
$$ \rH^1(C, \bZ) \otimes \bC = \rH^0(C, \Omega^1) \oplus \ol{\rH^0(C, \Omega^1)} $$from Hodge theory is compatible with the action of correspondences.
Correspondences also act on divisors, by the same formula. This action preserves principal divisors, and thus induces a map on the Jacobian. If $f$ is the correspondence $(p_1,p_2)$ then the induced endomorphism of $\Jac(C)$ is $(p_2^*)^{\vee} p_1^*$, where $p_i^* \colon \Jac(C) \to \Jac(C')$ is the natural map, and $()^{\vee}$ is the dual isogeny.
Back to Hecke operators
By the above discussion, the big Hecke algebra $\wt{\bT}$ acts on $\rH^1(X_0(N), \bZ)$, and the isomorphism
$$ \rH^1(X_0(N), \bZ) \otimes \bC = S_2(N) \oplus \ol{S_2(N)} $$is compatible with the actions on each side. (Here we have identified $S_2(N)$ with $\rH^0(X_0(N), \Omega^1)$.) If an element of $\wt{\bT}$ acts by zero on $S_2(N)$, then it does so on $\ol{S_2(N)}$ as well, and therefore on $\rH^1(X_0(N), \bZ)$. We therefore see that the image of $\wt{\bT}$ in $\End(\rH^1(X_0(N), \bZ))$ is just $\bT$. Since this endomorphism ring is $M_{2g}(\bZ)$, we see that:
Proposition. The Hecke algebra $\bT$ is a free $\bZ$module of finite rank.
Corollary. The Hecke eigenvalues of a normalized cusp form are algebraic integers.
Corollary. $\bT \otimes \bQ$ is a product of finitely many number fields.
We have shown that $S_2(N)$ is free of rank 1 as a module over $\bT \otimes \bC$. The same is clearly true for $\ol{S_2(N)}$. Thus $\rH^1(X_0(N), \bZ) \otimes \bC$ is free of rank 2 over $\bT \otimes \bC$. Now, $\rH^1(X_0(N), \bQ)$ is a module over the semisimple ring $\bT \otimes \bQ$ which becomes free of rank 2 when tensored up to $\bC$; it is therefore necessarily free of rank 2 itself. We've proved:
Proposition. $\rH^1(X_0(N), \bQ)$ is free of rank 2 as a $\bT \otimes \bQ$ module.
The AtkinLehner involution
The space $X_0(N)$ admits a natural involution $w$ defined as follows. If we think of the points of $X_0(N)$ as cyclic isogenies of degree $N$, then $w$ takesĀ $f \colon E \to E'$ to the dual isogeny $f^{\vee} \colon E' \to E$. If we think of the points of $X_0(N)$ as elliptic curves with a cyclic subgroup of order $N$ then $w$ takes $(E,G)$ to $(E/G, E[N]/G)$.
Thinking of weight 2 cusp forms as 1forms on $X_0(N)$, we get an action of $w$ by pullback. In terms of functions on the upper halfplane, $(wf)(z)=f(1/Nz)$. One verifies that this action of $w$ on $S_2(N)$ commutes with the Hecke operators $T_p$ for $p \nmid N$. Assuming now that $N$ is prime (or $N$ is arbitrary and we use the new subspace). Since $w$ commutes with $\bT$, it preserves the $\bT$ eigenspaces, and these are one dimensional. It follows that if $f$ is an eigenform then $wf=\pm f$.