1. Modular forms and Hecke operators

  See Lang's book, [4], for the details of this section. Let $ k $ be a field. An elliptic curve over $ k $ may equivalently be defined as

  1. $ E $ is an abelian variety of dimension 1 over $ k $;

  2. $ E $ is a smooth, projective, geometrically connected curve of genus one with a chosen $ k $-rational origin;

the chosen point supplies the group law.

  A complex lattice is a discrete subgroup $ L\subset\mathbb C $ which spans $ \mathbb C $ over $ \mathbb R $. It has the form

\[L=\mathbb Z\omega_1+\mathbb Z\omega_2 =\{a_1\omega_1+a_2\omega_2:a_i\in\mathbb Z\},\]

where $ \omega_1/\omega_2\in\mathcal H $ and $ \mathcal H $ is the upper half-plane. The quotient $ \mathbb C/L $ is an elliptic curve, every complex elliptic curve arises this way, and two lattices give isomorphic elliptic curves precisely when they are homothetic. In higher dimensions an arbitrary complex torus need not be an abelian variety.

  Elements of the modular group $ \gamma \in \text{SL}_2(\mathbb{Z}) $ act on $ \mathcal{H} $ by fractional linear transformations; meaning for $ z \in \mathcal{H} $ and

\[\gamma = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \text{ we put } \gamma(z) = \frac{az + b}{cz + d}.\]

  A holomorphic function $ f\colon\mathcal H\to\mathbb C $ is a modular form of weight $ k $ for $ \mathrm{SL}_2(\mathbb Z) $ if, for every $ \gamma\in\mathrm{SL}_2(\mathbb Z) $,

\[f(\gamma(z)) = (cz+d)^k f(z),\]

and $ f $ is holomorphic at the cusp, equivalently has a Fourier expansion with no negative powers of $ q=e^{2\pi iz} $. One may express the same definition in terms of lattices. If $ L=\mathbb Z\omega_1+\mathbb Z\omega_2 $ with $ \omega_1/\omega_2\in\mathcal H $, put

\[F(L)=\omega_2^{-k}f(\omega_1/\omega_2).\]

The transformation law makes this independent of the chosen oriented basis, and $ F(\lambda L)=\lambda^{-k}F(L) $. Conversely, $ f(z)=F(\mathbb Zz+\mathbb Z) $. Holomorphic dependence on $ L $ and the cusp condition are part of this equivalence. We write $ M_k $ for the resulting space of modular forms.

  Put $ \Gamma=\mathrm{SL}_2(\mathbb Z) $. The Hecke algebra consists of finitely supported functions on the double-coset set $ \Gamma\backslash\mathrm{GL}_2^+(\mathbb Q)/\Gamma $, with multiplication defined by decomposing double cosets into finitely many one-sided cosets and convolving. Equivalently, its elements are finite formal linear combinations of double cosets. It acts on modular forms through the weight-$ k $ slash action, where $ j\left(\left(\begin{smallmatrix}a&b\c&d\end{smallmatrix}\right),z\right)=cz+d $. For the classical operators it is clearest to write the formula directly:

\[(T_nf)(z) =n^{k-1}\sum_{\substack{ad=n\\d>0}}d^{-k} \sum_{b\bmod d}f\left(\frac{az+b}{d}\right).\]

For composite $ n $, this is generally a sum of more than one double coset; it should not be identified with the single double coset of $ \operatorname{diag}(n,1) $.

  In the lattice description, define

\[(T_n^{\mathcal L}F)(L) = n^{k-1} \sum_{[L : L'] = n} F(L');\]

the sum runs over all index-$ n $ sublattices $ L'\subset L $. If $ L=\mathbb Zz+\mathbb Z $, these sublattices have unique representatives

\[L'=\mathbb Z(az+b)+\mathbb Zd, \qquad ad=n,\quad 0\leq b<d.\]

Homogeneity therefore gives

\[(T_n^{\mathcal L}F)(\mathbb Zz+\mathbb Z) =n^{k-1}\sum_{\substack{ad=n\\d>0}}d^{-k} \sum_{b\bmod d}f\left(\frac{az+b}{d}\right) =(T_nf)(z).\]

On weight $ k $ forms, the operators satisfy

\[T_mT_n=T_{mn}\quad\text{if }(m,n)=1,\]

and, for $ r\geq1 $,

\[T_{p^r}T_p=T_{p^{r+1}}+p^{k-1}T_{p^{r-1}}.\]

  We say a modular form $ f $ of weight $ k $ is an eigenform if it is an eigenvector for every $ T_n $. Suppose $ f $ has a $ q $-expansion

\[f(q) = \sum_{n=0}^{\infty} a_n q^n\]

with $ q= e^{2 \pi i z} $. We can then attach a Dirichlet series to $ f $ by

\[D_f(s) = \sum_{n=1}^{\infty} a_n n^{-s}.\]

If $ f $ is a normalized eigenform, meaning $ a_1=1 $, this series factors.

Theorem 1.1.   Let $ f $ be a normalized eigenform of weight $ k $. Then its Dirichlet series factors as

\[D_f(s) = \prod_{p \text{ prime}} (1-a_p p^{-s} + p^{k-1-2s})^{-1}.\]

2. Ramanujan's conjectures

  Consider the space $ S_{12} $. It contains the cusp form

\[\Delta(z) = \sum_{n>0} \tau(n) q^n = q \prod_{n > 0} (1-q^n)^{24},\]

and since $ S_{12} $ is of dimension 1, $ \Delta(z) $ is an eigenform. The coefficient function $ \tau \colon \mathbb{N} \to \mathbb{C} $ is called the Ramanujan tau function.

Ramanujan Conjectures.   The following should hold:

  1. $\tau(m) \tau(n) = \tau(mn)$ for $m$ and $n$ coprime;
  2. $\tau(p^r) \tau(p) = \tau(p^{r+1}) + p^{11} \tau(p^{r-1})$ for $p$ prime; and
  3. $ \lvert\tau(p)\rvert \leq 2 p^{11/2}$.

  Ramanujan conjectured these statements in 1916. Mordell proved the first two in 1917 using the theory of modular functions; Hecke operators were introduced later. In his 1969 Bourbaki exposé, Deligne proved that the Weil conjectures imply the third statement. It became unconditional with Deligne's proof of the Riemann-hypothesis part of the Weil conjectures in 1974. We refer to the third statement as Ramanujan's conjecture below.

3. Deligne's proof

Overview

  Let $ D $ denote the Dirichlet series associated to $ \Delta $. By theorem 1.1,

\[D(s) = \prod_{p \text{ prime}} H_p(p^{-s})^{-1},\]

where

\[H_p(z) = 1 - \tau(p) z + p^{11} z^2.\]

Lemma 3.1.   Ramanujan's conjecture is true if the reciprocal roots of $ H_p $ have absolute value $ p^{11/2} $.

Proof. Factor

\[H_p(X)=(1-\alpha_pX)(1-\beta_pX).\]

Then

\[\alpha_p+\beta_p=\tau(p),\qquad \alpha_p\beta_p=p^{11}.\]

Thus, if $ \lvert\alpha_p\rvert=\lvert\beta_p\rvert=p^{11/2} $, the triangle inequality gives

\[\lvert\tau(p)\rvert=\lvert\alpha_p+\beta_p\rvert\leq2p^{11/2}.\]

The actual zeros of $ H_p(X) $ are $ \alpha_p^{-1} $ and $ \beta_p^{-1} $, of absolute value $ p^{-11/2} $. Q.E.D.

  To calculate the absolute values of these reciprocal roots, we construct a Frobenius action on a \(\mathbb{Q}_\ell\)-vector space \({_1^{10}}W_\ell\). Deligne proves

\[H_p(z) = \det(1-Fz; \, {_1^{10}} W_{\ell}),\]

and hence the reciprocal roots of $ H_p $ are the eigenvalues of this action. A purity theorem then gives their absolute values.

The Shimura isomorphism

  Let $ S $ be a complex analytic space. An elliptic curve over $ S $ is a proper smooth morphism $ f\colon E\to S $ of relative dimension one, with geometrically connected genus-one fibers and a section $ e\colon S\to E $ giving their origins.

  Following Deligne's convention, we say an isomorphism \(\underline{\mathbb{Z}}^2\to R^1f_*\underline{\mathbb{Z}}\) of sheaves on $ S $ is permitted if it reverses the standard orientation relative to the canonical cup-product orientation; equivalently, under the canonical identifications on second exterior powers, the induced map

\[\bigwedge^2\underline{\mathbb{Z}}^2 \longrightarrow \bigwedge^2R^1f_*\underline{\mathbb{Z}}\]

is multiplication by $-1$. Let $ \operatorname{Isom}^-(\mathbb{R}^2,\mathbb{C}) $ be the collection of orientation-reversing $ \mathbb{R} $-linear isomorphisms, and set

\[X=\mathbb{C}^\times\backslash\operatorname{Isom}^-(\mathbb{R}^2,\mathbb{C}).\]

We identify $ X $ with the upper half-plane $ \mathcal H $ by sending the class of $ f $ to $ f(e_1)/f(e_2) $.

Proposition 3.2.   $ X $ represents the functor which sends an analytic space $ S $ to the isomorphism classes of elliptic curves on $ S $, equipped with a permitted isomorphism \(\underline{\mathbb{Z}}^2\cong R^1f_*\underline{\mathbb{Z}}\).

  Unmarked elliptic curves correspond to complex lattices up to homothety, hence to points of $ X/\mathrm{SL}_2(\mathbb Z) $. The space $ X $ itself classifies elliptic curves with a permitted marking, and carries the corresponding universal marked elliptic curve $ f\colon E_X\to X $.

  Let $ \Gamma $ be a discrete subgroup of $ \text{SL}_2(\mathbb{Z}) $ without torsion elements such that $ X/\Gamma $ has finite volume (with respect to the Poincaré metric). We can identify $ X/\Gamma $ to a smooth projective curve $ \overline{X/\Gamma} $ minus a finite number of points.

  One of the main ideas of Deligne's paper is to use a cohomological description of cusp forms. We define a cusp form of weight $ k $ with respect to $ \Gamma $ by

  1. for any $\gamma \in \Gamma$ we have $f(\gamma(z)) = (cz+d)^k f(z)$; and
  2. for any $\gamma \in \text{SL}_2(\mathbb{Z})$ we have $(cz+d)^{-k} f(\gamma(z)) \to 0$ as $\text{im}(z) \to \infty$.

We denote the resulting space $ S_k(\Gamma) $.

  Alternatively, we see for $ \gamma \in \text{SL}_2(\mathbb{Z}) $,

\[\frac{d \gamma(z)}{dz} = (cz+d)^{-2}.\]

Hence, the condition

\[f(\gamma(z)) = (cz+d)^{2k} f(z)\]

is equivalent to

\[f(\gamma(z)) d\gamma(z)^k = f(z) dz^k.\]

Thus a modular form of weight $ 2k $ defines an invariant holomorphic $ k $-differential $ f(z)\,dz^k $. Let $ Y=X/\Gamma $, let $ \overline Y $ be its smooth compactification, and let $ D=\overline Y\setminus Y $ be the cusp divisor. The Hodge bundle \(\omega=f_*\Omega^1_{E_X/X}\) descends to $ Y $ and has a canonical extension to $ \overline Y $. The Kodaira-Spencer isomorphism gives \(\omega^2\cong\Omega^1_{\overline Y}(D)\), and hence

Proposition 3.3.   $ S_{k+2}(\Gamma) $ can be identified with $ H^0(\overline Y,\Omega_{\overline Y}^1\otimes\omega^k) $.

  Put \(\mathcal U_k=\operatorname{Sym}^k(R^1f_*\underline{\mathbb Z})\) on $ Y $. Its parabolic cohomology is

\[\widetilde H^1(Y,\mathcal U_k) =\operatorname{im}\left(H_c^1(Y,\mathcal U_k)\to H^1(Y,\mathcal U_k)\right).\]

We will use the following theorem without constructing the comparison map.

Theorem (Shimura isomorphism).   There exists an isomorphism

\[S_{k+2}(\Gamma)\oplus\overline{S_{k+2}(\Gamma)} \cong\widetilde H^1(Y,\mathcal U_k)\otimes_{\mathbb Z}\mathbb C.\]

Full level $ n $ structures

  An elliptic curve over a scheme $ S $ is a proper smooth morphism $ f\colon E\to S $ of relative dimension one with geometrically connected genus-one fibers, equipped with a section. Let

\[\Gamma(n)=\ker\left(\mathrm{SL}_2(\mathbb Z)\to \mathrm{SL}_2(\mathbb Z/n\mathbb Z)\right).\]

For $ n\geq3 $, write $ M_n $ for the fine moduli scheme over $ \mathbb Z[1/n] $ of elliptic curves with full level-$ n $ structure; it carries a universal elliptic curve $ f_n\colon E_n\to M_n $. After choosing a primitive $ n $-th root of unity, one determinant component is defined over $ \mathbb Z[1/n,\zeta_n] $, and its complex points form $ \mathcal H/\Gamma(n) $. Deligne retains the union of all determinant components, on which $ \mathrm{GL}_2(\mathbb Z/n\mathbb Z) $ acts; our notation $ M_n $ refers to this full union.

  Fix embeddings $ \overline{\mathbb Q}\hookrightarrow\mathbb C $ and $ \overline{\mathbb Q}\hookrightarrow\overline{\mathbb Q}_\ell $. Define the parabolic Betti and étale cohomology spaces

\[\begin{aligned} {}_n^kW &=\widetilde H^1\left(M_n^{\mathrm{an}}, \operatorname{Sym}^k(R^1f_{n*}\underline{\mathbb Q})\right),\\ {}_n^kW_\ell &=\widetilde H_{\mathrm{\acute et}}^1\left( M_n\otimes\overline{\mathbb Q}, \operatorname{Sym}^k(R^1f_{n*}\underline{\mathbb Q}_\ell)\right). \end{aligned}\]

Comparison identifies these after extending scalars. Taking invariants under the finite level-change group gives the level-one summand \({}_1^kW\) and its $ \ell $-adic realization. Set \({}_1^kW_\infty={}_1^kW\otimes\mathbb C\). The Shimura isomorphism yields

\[{}_1^kW\otimes\mathbb C \cong S_{k+2}\oplus\overline{S_{k+2}}.\]

Lemma 3.5.   We have $ _1^k W_{\infty} \cong S_{k+2} \oplus \overline{S_{k+2}} $.

Eichler–Shimura congruence relation

  We now compare $ T_p $ with Frobenius under the Shimura isomorphism. Fix primes $ p\nmid n $ and $ \ell\neq p $. The special fiber of the $ p $-isogeny correspondence decomposes into a Frobenius component and its transpose, the Verschiebung component. Over the ordinary locus this says that a degree-$ p $ isogeny is locally Frobenius or dual to Frobenius; the extension across the supersingular points is part of the Eichler-Shimura congruence theorem.

  Fix a pair $ (E,\alpha) $ with full level-$ n $ structure, and let $ F\colon E\to E^{(p)} $ denote relative Frobenius. Since $ p\nmid n $, Frobenius is an isomorphism on $ n $-torsion and transports $ \alpha $ to a level structure $ \alpha^{(p)} $:

\[\xymatrix{ & (\mathbb{Z}/n\mathbb{Z})^2 & \\ E[n] \ar[ru]^{\alpha} \ar[rr] & & E^{(p)}[n] \ar[lu]_{\alpha^{(p)}} \\ E \ar[u] \ar[rr]^{F} & & E^{(p)} \ar[u] }\]

Together with pullback and trace, this correspondence defines an endomorphism of

\[\widetilde{H}^1(M_n \otimes \overline{\mathbb{F}}_p, \text{Sym}^k(R^1 f_{n*} \underline{\mathbb{Z}}_{\ell}))\]

compatible with the coefficient local system.

  We likewise have a diagram

\[\xymatrix{ & (\mathbb{Z}/n\mathbb{Z})^2 & \\ E^{(p)}[n] \ar[ru]^{p \alpha^{(p)}} \ar[rr] & & E[n] \ar[lu]_{\alpha} \\ E^{(p)} \ar[u] \ar[rr]^{V} & & E \ar[u] }\]

which gives Verschiebung, the transpose of Frobenius with respect to the pairing inducing the Petersson inner product.

  Finally, we define the Hecke operator $ T_p $ acting on \({_1^k} W_{\infty}\) by the coset of

\[\begin{bmatrix} 1 & 0 \\ 0 & p^{-1} \end{bmatrix}\]

in $ \text{GL}_2(\mathbb{Z}_p) \backslash \text{GL}_2(\mathbb{Q}_p)/\text{GL}_2(\mathbb{Z}_p) $. Likewise define $ R_p $ to be the coset of

\[\begin{bmatrix} p^{-1} & 0 \\ 0 & p^{-1} \end{bmatrix}.\]

Since $ p $ is invertible modulo $ n $, let $ I_p $ denote the automorphism induced by $ (E,\alpha)\mapsto(E,\alpha/p) $. Deligne proved:

Proposition 3.6.   $ F $ is geometric Frobenius, the inverse of arithmetic Frobenius in $ \operatorname{Gal}(\overline{\mathbb F}_p/\mathbb F_p) $, on

\[\widetilde{H}^1(M_n \otimes \overline{\mathbb{F}}_p, \text{Sym}^k(R^1 f_{n*} \underline{\mathbb{Z}}_{\ell})).\]

Over $ \mathbb{F}_p $, we have \(T_p = F + I_p^* V\) and \(FV = VF = p^{k+1}\).

The scalar double coset satisfies $ R_p=p^kI_p^* $ on this cohomology.

  From this, we get

Theorem 3.7.   Let $ K_{n,\ell}\subset\overline{\mathbb Q} $ be the largest subextension unramified outside the primes dividing $ n\ell $. Let $ p\nmid n\ell $, let $ \varphi_p $ be an arithmetic Frobenius element at $ p $, defined up to conjugacy in $ \operatorname{Gal}(K_{n,\ell}/\mathbb Q) $, let $ F=\varphi_p^{-1} $ act on $ {}_n^kW_\ell $, and let $ V $ be its transpose. Then

\[T_p=F+I_p^*V,\qquad FV=p^{k+1},\qquad R_p=p^kI_p^*,\] \[1-T_pX+pR_pX^2=(1-FX)(1-I_p^*VX).\]

Applying Weil II

  Applying the results of [3], we arrive at

Theorem 3.8.   For $ p\nmid n\ell $, the eigenvalues of geometric Frobenius $ F $ on $ {}_n^kW_\ell $ are algebraic integers, and every complex conjugate has absolute value $ p^{(k+1)/2} $.

Proof. The local system

\[\operatorname{Sym}^k(R^1f_{n*}\underline{\mathbb Q}_\ell)\]

is punctually pure of weight $ k $: $ R^1f_{n*}\underline{\mathbb Q}_\ell $ has weight $ 1 $, and symmetric powers add weights. Weil II implies that the parabolic cohomology

\[\operatorname{im}\left( H_c^1(M_n\otimes\overline{\mathbb F}_p,\mathcal U_k) \longrightarrow H^1(M_n\otimes\overline{\mathbb F}_p,\mathcal U_k) \right)\]

is pure of weight $ k+1 $. Deligne's integrality theorem for étale cohomology (equivalently, the smooth-compactification argument in this case) shows that these Frobenius eigenvalues are algebraic integers. Q.E.D.

  Since $ I_p^* $ induces the identity on \({_1^k} W_{\ell}\), we have

\[1 - T_p X + p^{k+1} X^2 = (1-FX)(1-VX);\]

and because $ F $ and $ V $ are transposes,

\[\det(1-FX; \, {_1^k W_{\ell}}) = \det(1-VX; \, {_1^k W_{\ell}}).\]

Thus,

\[\det(1-T_pX+p^{k+1}X^2; \, {_1^k W_{\ell}}) = \det(1-FX; \, {_1^k W_{\ell}})^2.\]

  The action of $ T_p $ on \({}_1^kW_\ell\) is compatible with its action on \({}_1^kW\otimes\mathbb C\). Under lemma 3.5 it preserves the two conjugate summands, and its Petersson self-adjointness makes their characteristic polynomials equal. Hence

\[\det(1-T_pX +p^{k+1}X^2; \, {_1^k W_{\ell}}) = \det(1-T_pX+p^{k+1}X^2; \, S_{k+2})^2,\]

Both sides are squares of polynomials with constant term $ 1 $, and hence

\[\det(1-T_pX+p^{k+1}X^2; \, S_{k+2}) = \det(1-FX; \, {_1^k W_{\ell}}).\]

  In the case $ k = 10 $, because $ S_{12} $ is of dimension 1, this tells us

\[H_p(X) = \det(1-FX; \, {_1^{10} W_{\ell}}).\]

Applying theorem 3.8 completes Deligne's proof of the Ramanujan conjecture. Q.E.D.

4. References

  1. Pierre Deligne. "Formes modulaires et représentations $ \ell $-adiques". Séminaire Bourbaki, exp. no. 355 (1968–1969), published 1971, pp. 139–172.
  2. Pierre Deligne. "La conjecture de Weil: I". Publications Mathématiques de l'IHÉS, Volume 43 (1974), pp. 273-307. Link
  3. Deligne, Pierre. "La conjecture de Weil: II". Publications Mathématiques de l'IHÉS, Volume 52 (1980), pp. 137-252. Link
  4. Serge Lang, Introduction to modular forms, Springer-Verlag, 1976.
  5. Jean-Pierre Serre, A course in arithmetic, Springer-Verlag, 1973.