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Section 5.7 What is ... a Shimura Variety? (Angus)

Motivation.

We began by studying modular curves e.g. \(Y_0(N) = \Gamma_0(N)\backslash \mathcal H\) Aash proved

\begin{equation*} Y_0(N) = \Gamma_0(N) \backslash \SL_2(\RR) / \specialorthogonal_2(\RR)\text{.} \end{equation*}

Consider \(\adeles = \prod'_v \QQ_v\) the adele ring of \(\QQ\text{.}\) Let

\begin{equation*} K_0(N) = \{ \begin{pmatrix} a\amp b \\ c \amp d \end{pmatrix} \in \GL_2(\widehat \ZZ) : c \equiv 0 p\mod N \}\text{.} \end{equation*}

We will generalise this final viewpoint for general \(G\text{.}\)

Last time \(X = \) conjugacy class of morphisms

\begin{equation*} h \colon \mathbf S \to G\text{ for } G/\RR \end{equation*}

an algebraic group s.t.

  1. \begin{equation*} h(\RR^\times) \subseteq Z(G(\RR)) \end{equation*}
  2. The hodge structure on \(\Lie(G)\) induced by \(\ad \circ h\) is of type \(\{(-1,1), (0,0), (1,-1)\}\text{.}\)

We also began studying Cartan involutions. Take an involution \(\sigma\) of \(G\) and define

\begin{equation*} G^\sigma(A) = \{ g\in G(A \otimes \CC : \sigma(g) = \bar g\} \end{equation*}

this \(G^\sigma\) is another algebraic group \(/\RR\text{.}\)

Remark 5.7.3.

\(G^\sigma\) is a real form of \(G\text{,}\) i.e. \(G^\sigma \otimes \CC \simeq G \otimes \CC\text{.}\)

Example 5.7.4.

\(G =\GL_n\text{,}\) \(\sigma(g) = (g^\transpose)\inv\) then \(G^\sigma = U(n)\text{.}\)

Recall the definition of a Cartan involution.

For \(C \in G(\RR)\) s.t. \(C^2 \in Z(G(\RR))\) then

\begin{equation*} \sigma \colon g \mapsto CgC\inv \end{equation*}

is an involution.

When is it Cartan?

Definition 5.7.5.

An \(\RR\)-representation \(V\) of \(G\) is \(C\)-polarizable if there exists a \(G\)-invariant bilinear form

\begin{equation*} \Psi \colon V \times V \to \RR \end{equation*}

s.t.

\begin{equation*} \Psi(x,Cy) \end{equation*}

is symmetric and positive definite.

\(\Rightarrow\text{.}\) Assume \(G^\sigma \) is compact. Let \(V\) be a faithful \(\RR\)-representation of \(G^\sigma (\RR)\text{.}\) From last time, there exists a \(G^\sigma(\RR)\)-invariant positive definite symmetric form

\begin{equation*} \Psi\colon V\times V\to \RR \end{equation*}

consider \(\Phi(u,v) = \Psi(u, C\inv v)\text{.}\) Then \(\Phi(x,Cy)\) is positive definite and symmetric so \(\Phi\) is a \(C\)-polarization.

\(\Leftarrow\text{.}\) Let \(V\) be \(C\)-polarizable so we have \(\Psi \colon V\times V \to \RR\text{.}\) Then \(\Psi_\CC \colon V_\CC \times V_\CC \to \CC\) is symmetric bilinear \(G\)-invariant. Let \(H(u,v) = \Psi_\CC(u,\bar v)\) consider \(H^\sigma(u,v) = H(u,Cv)\text{.}\)

\(H^\sigma\) is \(G^\sigma(\RR)\)-invariant positive definite, Hermitian. From last time \(G^\sigma\) is compact.

Now introduce polarizations on Hodge structures.

Definition 5.7.7.

A polarization on a weight \(k\) Hodge structure

\begin{equation*} h\colon \mathbf S \to \GL(V) \end{equation*}
\begin{equation*} V_\CC = \bigoplus_{p+q = k} V^{p,q} \end{equation*}

is a bilinear form \(\Psi\colon V\times V \to \RR\) s.t.

  1. \(\Psi\) is (symmetric/alternating) if \(k\) is even / odd.
  2. Letting
    \begin{equation*} H\colon V_\CC \times V_\CC \to \CC \end{equation*}
    be given by
    \begin{equation*} H(u,v) = i^k \Psi(u, \bar v) \end{equation*}
    then the
    \begin{equation*} V^{p,q} \end{equation*}
    are orthogonal with respect to \(H\) and \(H|_{V^{p,q}}\) has sign \(i^{p-q- k}\text{.}\)

Why polarize?

Recall: the set of polarized Hodge structures on \(\RR^{2g}\) of type \(\{(-1,0), (0,-1)\}\) is the Siegel upper half space \(\mathcal H_g\text{.}\)

\(\Leftarrow\) in Jared's notes.

\(\Rightarrow\text{,}\) we want to show

\begin{equation*} \Psi(h(z) v, h(z) w) = |z|^{-k} \Psi(v,w) \end{equation*}
\begin{equation*} \Psi(h(z) v, h(z) w) = i^{-k} H(h(z) v, \overline{h(z) w}) \end{equation*}
\begin{equation*} = i^{-k} H(h(z) \sum v_{p,q}, \overline { \sum h(z) w_{pq}}) \end{equation*}
\begin{equation*} = \cdots \end{equation*}
\begin{equation*} = |z|^{-k} \Psi(v,w) \end{equation*}

using orthogonality.

Let \(V \) be a faithful representation of \(G \) s.t. for all \(h\in X\) we get a Hodge structure on \(V\text{.}\)

Call \(V\) polarizable if in the weight decomposition

\begin{equation*} V = \bigoplus_k V_k \end{equation*}

each \(V_k\) admits a bilinear form \(\Psi_k\) s.t. \(h \in X\) gives a polarized Hodge structure on \(V_k\text{.}\)

To define the adjoint group, take the adjoint representation

\begin{equation*} \ad \colon G_1^{\ad} = \ad(G_1) \end{equation*}

if \(G_1\) is connected then \(G_1^{\ad} = G_1 /Z(G_1)\text{.}\)

\(\Rightarrow\) Let \(G_2 \subseteq G_1\) be the smallest subgroup containing \(h(U^1)\) for all \(h \in X\) where \(U^1 = \{|z| = 1\} \subseteq \CC^\times\text{.}\) Then \(G_1\) is generated by \(G_2\) and \(h(t)\) for all \(t \in \RR^\times, h \in X\text{.}\) Since \(h(t)\) is always central have \(G_1^{\ad}= G_2^{\ad}\text{.}\) By the previous lemma, \(\forall z \in U^1\)

\begin{equation*} \Psi(h(z), v, h(z) w) = \Psi(v,w) \end{equation*}

so \(\Psi\) is \(G_2\) invariant. Father \(\Psi(v, h(i) w)\) is symmetric positive definite for all \(h \in X\text{.}\) So conjugation by \(h(i)\) is a Cartan involution on \(G_2\) so on \(G_2^{\ad} = G_1^{\ad}\text{.}\)

Definition 5.7.10. Shimura data.

A Shimura datum is a pair \((G,X)\) where

  1. \(G/\QQ\) is a reductive algebraic group.

  2. \(X\) is a \(G(\RR)\)-conjugacy class of morphisms \(h\colon \mathbf S \to G_\RR\) s.t.

    1. \(\forall h \in X\) the Hodge structure on \(\Lie (G_\RR)\) induced by \(\ad \circ h\) is of type \(\{(-1,1), (0,0), (1,-1)\}\text{.}\)
    2. The involution \(\ad h(i)\) (i.e. conjugation by \(h(i)\)) is a Cartan involution on \(G^{\ad}\text{.}\)
    3. \(G\) has no \(\QQ\)-factor on which the projection of \(h\) is trivial.
Definition 5.7.11.

Let \(\adeles^\infty = \prod_{p\ne \infty}' \QQ_p\) be the ring of finite adeles of \(\QQ\text{.}\) Let \(K \subseteq G(\adeles^\infty)\) be a compact open subgroup. The shimura variety of level \(K\) then \({\operatorname{Sh}}_K(G,X)\) is given by

\begin{equation*} {\operatorname{Sh}}_K(G,X) = G(\QQ) \backslash X\times G(\adeles^\infty) / K \end{equation*}

The shimura variety at infinite level is

\begin{equation*} {\operatorname{Sh}}(G,X) = \varprojlim_K G(\QQ) \backslash X \times G(\adeles^\infty) / K = G(\QQ) \backslash X \times G(\adeles^\infty) \end{equation*}
Example 5.7.12. \(\GL_2\).

\(X = \) conj. class containing \(h \colon (a+bi) \mapsto \begin{pmatrix} a\amp b \\ -b \amp a\end{pmatrix}\) \(\leftrightarrow \CC \smallsetminus \RR\) \(\leftrightarrow \{\) complex structures on \(V = \QQ^2\}\text{.}\)

\begin{equation*} h \mapsto i \end{equation*}

Let \(E/\CC\) be an elliptic curve. We have the full Tate module

\begin{equation*} TE = \varprojlim_N E[N] \simeq \widehat\ZZ^2 \end{equation*}

We have the full rational Tate module

\begin{equation*} V^\infty E = TE \otimes \adeles^\infty \simeq (\adeles^\infty)^2 \end{equation*}
\begin{equation*} \simeq H^1(E,\QQ) \otimes_\QQ \adeles^\infty \end{equation*}
Remark 5.7.14.

An isogeny is \(f \in \Hom(E, E') \otimes \QQ\) sending \(\eta \mapsto \eta'\) \({\operatorname{Sh}}(\GL_2, X)\) has components indexed by \(\widehat \ZZ^\times\text{.}\)