63.31 Automorphic forms and sheaves

References: See especially the amazing papers [D1], [D2] and [D0] by Drinfeld.

Unramified cusp forms. Let $k$ be a finite field of characteristic $p$. Let $X$ geometrically irreducible projective smooth curve over $k$. Set $K = k(X)$ equal to the function field of $X$. Let $v$ be a place of $K$ which is the same thing as a closed point $x\in X$. Let $K_ v$ be the completion of $K$ at $v$, which is the same thing as the fraction field of the completion of the local ring of $X$ at $x$. Denote $O_ v\subset K_ v$ the ring of integers. We further set

$O = \prod \nolimits _ v O_ v \subset \mathbf{A} = \prod _ v' K_ v$

and we let $\Lambda$ be any ring with $p$ invertible in $\Lambda$.

Definition 63.31.1. An unramified cusp form on $\text{GL}_2(\mathbf{A})$ with values in $\Lambda$1 is a function

$f : \text{GL}_2(\mathbf{A}) \to \Lambda$

such that

1. $f(x\gamma ) = f(x)$ for all $x\in \text{GL}_2(\mathbf{A})$ and all $\gamma \in \text{GL}_2(K)$

2. $f(ux) = f(x)$ for all $x\in \text{GL}_2(\mathbf{A})$ and all $u\in \text{GL}_2(O)$

3. for all $x\in \text{GL}_2(\mathbf{A})$,

$\int _{\mathbf{A} \mod K} f \left(x \left( \begin{matrix} 1 & z \\ 0 & 1 \end{matrix} \right) \right) dz = 0$

see [Section 4.1, dJ-conjecture] for an explanation of how to make sense out of this for a general ring $\Lambda$ in which $p$ is invertible.

Hecke Operators. For $v$ a place of $K$ and $f$ an unramified cusp form we set

$T_ v(f)(x) = \int _{g\in M_ v}f(g^{-1}x)dg,$

and

$U_ v(f)(x) = f\left( \left( \begin{matrix} \pi _ v^{-1} & 0 \\ 0 & \pi _ v^{-1} \end{matrix} \right)x\right)$

Notations used: here $\pi _ v \in O_ v$ is a uniformizer

$M_ v = \left\{ h\in Mat(2\times 2, O_ v) | \det h = \pi _ vO_ v^*\right\}$

and $dg =$ is the Haar measure on $\text{GL}_2(K_ v)$ with $\int _{\text{GL}_2(O_ v)} dg = 1$. Explicitly we have

$T_ v(f)(x) = f\left( \left( \begin{matrix} \pi _ v^{-1} & 0 \\ 0 & 1 \end{matrix} \right) x\right) + \sum _{i = 1}^{q_ v} f\left(\left( \begin{matrix} 1 & 0 \\ -\pi _ v^{-1}\lambda _ i & \pi _ v^{-1} \end{matrix} \right) x\right)$

with $\lambda _ i\in O_ v$ a set of representatives of $O_ v/(\pi _ v)=\kappa _ v$, $q_ v = \# \kappa _ v$.

Eigenforms. An eigenform $f$ is an unramified cusp form such that some value of $f$ is a unit and $T_ vf = t_ vf$ and $U_ vf = u_ vf$ for some (uniquely determined) $t_ v, u_ v \in \Lambda$.

Theorem 63.31.2. Given an eigenform $f$ with values in $\overline{\mathbf{Q}}_ l$ and eigenvalues $u_ v\in \overline{\mathbf{Z}}_ l^*$ then there exists

$\rho : \pi _1(X)\to \text{GL}_2(E)$

continuous, absolutely irreducible where $E$ is a finite extension of $\mathbf{Q}_\ell$ contained in $\overline{\mathbf{Q}}_ l$ such that $t_ v = \text{Tr}(\rho (F_ v))$, and $u_ v = q_ v^{-1}\det \left(\rho (F_ v)\right)$ for all places $v$.

Proof. See [D0]. $\square$

Theorem 63.31.3. Suppose $\mathbf{Q}_ l \subset E$ finite, and

$\rho : \pi _1(X)\to \text{GL}_2(E)$

absolutely irreducible, continuous. Then there exists an eigenform $f$ with values in $\overline{\mathbf{Q}}_ l$ whose eigenvalues $t_ v$, $u_ v$ satisfy the equalities $t_ v = \text{Tr}(\rho (F_ v))$ and $u_ v = q_ v^{-1}\det (\rho (F_ v))$.

Proof. See [D1]. $\square$

Remark 63.31.4. We now have, thanks to Lafforgue and many other mathematicians, complete theorems like this two above for $\text{GL}_ n$ and allowing ramification! In other words, the full global Langlands correspondence for $\text{GL}_ n$ is known for function fields of curves over finite fields. At the same time this does not mean there aren't a lot of interesting questions left to answer about the fundamental groups of curves over finite fields, as we shall see below.

Central character. If $f$ is an eigenform then

$\begin{matrix} \chi _ f : & O^*\backslash \mathbf{A}^*/K^* & \to & \Lambda ^* \\ & (1, \ldots , \pi _ v, 1, \ldots , 1) & \mapsto & u_ v^{-1} \end{matrix}$

is called the central character. If corresponds to the determinant of $\rho$ via normalizations as above. Set

$C(\Lambda ) = \left\{ {\text{unr. cusp forms } f \text{ with coefficients in }\Lambda } \atop {\text{ such that } U_ v f = \varphi _ v^{-1}f\forall v} \right\}$

Proposition 63.31.5. If $\Lambda$ is Noetherian then $C(\Lambda )$ is a finitely generated $\Lambda$-module. Moreover, if $\Lambda$ is a field with prime subfield $\mathbf{F} \subset \Lambda$ then

$C(\Lambda )=(C(\mathbf{F}))\otimes _{\mathbf{F}}\Lambda$

compatibly with $T_ v$ acting.

This proposition trivially implies the following lemma.

Lemma 63.31.6. Algebraicity of eigenvalues. If $\Lambda$ is a field then the eigenvalues $t_ v$ for $f\in C(\Lambda )$ are algebraic over the prime subfield $\mathbf{F} \subset \Lambda$.

Proof. Follows from Proposition 63.31.5. $\square$

Combining all of the above we can do the following very useful trick.

Lemma 63.31.7. Switching $l$. Let $E$ be a number field. Start with

$\rho : \pi _1(X)\to SL_2(E_\lambda )$

absolutely irreducible continuous, where $\lambda$ is a place of $E$ not lying above $p$. Then for any second place $\lambda '$ of $E$ not lying above $p$ there exists a finite extension $E'_{\lambda '}$ and a absolutely irreducible continuous representation

$\rho ': \pi _1(X)\to SL_2(E'_{\lambda '})$

which is compatible with $\rho$ in the sense that the characteristic polynomials of all Frobenii are the same.

Note how this is an instance of Deligne's conjecture!

Proof. To prove the switching lemma use Theorem 63.31.3 to obtain $f\in C(\overline{\mathbf{Q}}_ l)$ eigenform ass. to $\rho$. Next, use Proposition 63.31.5 to see that we may choose $f\in C(E')$ with $E \subset E'$ finite. Next we may complete $E'$ to see that we get $f\in C(E'_{\lambda '})$ eigenform with $E'_{\lambda '}$ a finite extension of $E_{\lambda '}$. And finally we use Theorem 63.31.2 to obtain $\rho ': \pi _1(X) \to SL_2(E_{\lambda '}')$ abs. irred. and continuous after perhaps enlarging $E'_{\lambda '}$ a bit again. $\square$

Speculation: If for a (topological) ring $\Lambda$ we have

$\left( {\rho : \pi _1(X)\to SL_2(\Lambda ) \atop \text{ abs irred}} \right) \leftrightarrow \text{ eigen forms in } C(\Lambda )$

then all eigenvalues of $\rho (F_ v)$ algebraic (won't work in an easy way if $\Lambda$ is a finite ring. Based on the speculation that the Langlands correspondence works more generally than just over fields one arrives at the following conjecture.

Conjecture. (See ) For any continuous

$\rho : \pi _1(X)\to \text{GL}_ n(\mathbf{F}_ l[[t]])$

we have $\# \rho (\pi _1(X_{\overline{k}}))<\infty$.

A rephrasing in the language of sheaves: "For any lisse sheaf of $\overline{\mathbf{F}_ l((t))}$-modules the geom monodromy is finite."

Proof. See . $\square$

Proof. See . $\square$

It turns out the conjecture is useful for something. See work of Drinfeld on Kashiwara's conjectures. But there is also the much more down to earth application as follows.

Theorem 63.31.10. (See [Theorem 3.5, dJ-conjecture]) Suppose

$\rho _0: \pi _1(X)\to \text{GL}_ n(\mathbf{F}_ l)$

is a continuous, $l\neq p$. Assume

1. Conj. holds for $X$,

2. $\rho _0 |_{\pi _1(X_{\overline{k}})}$ abs. irred., and

3. $l$ does not divide $n$.

Then the universal deformation ring $R_{\text{univ}}$ of $\rho _0$ is finite flat over $\mathbf{Z}_ l$.

Explanation: There is a representation $\rho _{\text{univ}}: \pi _1(X)\to \text{GL}_ n(R_{\text{univ}})$ (Univ. Defo ring) $R_{\text{univ}}$ loc. complete, residue field $\mathbf{F}_ l$ and $(R_{\text{univ}}\to \mathbf{F}_ l)\circ \rho _{\text{univ}}\cong \rho _0$. And given any $R\to \mathbf{F}_ l$, $R$ local complete and $\rho : \pi _1(X)\to \text{GL}_ n(R)$ then there exists $\psi : R_{\text{univ}}\to R$ such that $\psi \circ \rho _{\text{univ}}\cong \rho$. The theorem says that the morphism

$\mathop{\mathrm{Spec}}(R_{\text{univ}}) \longrightarrow \mathop{\mathrm{Spec}}(\mathbf{Z}_ l)$

is finite and flat. In particular, such a $\rho _0$ lifts to a $\rho : \pi _1(X) \to \text{GL}_ n(\overline{\mathbf{Q}}_ l)$.

Notes:

1. The theorem on deformations is easy.

2. Any result towards the conjecture seems hard.

3. It would be interesting to have more conjectures on $\pi _1(X)$!

[1] This is likely nonstandard notation.

Comment #2990 by Wen-Wei Li on

In the Theorem 53.115.10: the "universal deformation ring" instead.

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