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Section 5.8 Canonical models (Alex)

Recall we defined Shimura varieties given a Shimura datum \((G,X)\) and a compact open \(K \subseteq G(\adeles^\infty)\) as

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

a quasi-projective variety, and more generally the infinite level version

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

which is a pro-variety and in fact a scheme.

These are varieties over \(\CC\text{,}\) we might hope to define them over a number field or even a ring of integers, so that we can do number theoretic things (look locally prime by prime for instance, or identify special rational points).

In the case where our Shimura variety is a natural moduli space (modular curves) might expect that this is indeed possible.

There will be two words in this talk, special and canonical that already have a vague meaning, we will be giving them a precise meaning in this talk for once!

Subsection 5.8.1 Galois descent

Say you have a variety over \(\CC\text{.}\) Is it really a \(\CC\)-variety, or is it a \(k\) variety for \(k \subseteq \CC\) that has been base-changed to \(\CC\text{?}\)

Question 5.8.1.

Given \(X/\CC\) a variety, is there a subfield \(k \subseteq \CC\) and an \(X_0/k\) with

\begin{equation*} X \simeq X_0 \times_k \CC \end{equation*}

we then say \(X\) descends to \(k\text{,}\) and that \(X_0\) is a model of \(X\) over \(k\text{.}\)

Preview: some examples of curves.
Example 5.8.2.

Let

\begin{equation*} C \colon x^2+ y^2 = \pi/\CC \end{equation*}

is there \(C_0/\QQ\) s.t. \(C_0\times_\QQ \CC \simeq C\text{.}\) Yes, we have

\begin{equation*} C \simeq x^2 + y^2 = 1/\CC \simeq (x^2 + y^2 = 1/\QQ )\times_{\QQ} \CC\text{.} \end{equation*}
Example 5.8.3.

Let now

\begin{equation*} E \colon y^2 = x^3 + ix + 1/\CC \end{equation*}

is there some \(E_0/\QQ\) such that

\begin{equation*} E_0\times_{\QQ} \CC \simeq E\text ? \end{equation*}

If there was such the following would be true: For any \(\sigma\in \Gal{\CC}{\QQ}\) we have

\begin{equation*} \underbrace{E^\sigma}_{y^2 = x^3 + \sigma(i)x + 1} \xleftarrow{f^\sigma, \sim} E_0 \times_\QQ \CC \xrightarrow{f,\sim} E \end{equation*}

but two elliptic curves are isomorphic (over \(\CC\)) if and only if they have the same \(j\)-invariant.

\begin{equation*} j(E) = 1728 \frac{4i^3}{4i^3 + 27\cdot 1^2} = 1728\left(1+ \frac{ - 27}{-4i + 27}\right) \end{equation*}
\begin{equation*} j(E^\sigma) = 1728 \frac{4\sigma(i)^3}{4\sigma(i)^3 + 27\cdot 1^2} = \sigma(j(E)) \end{equation*}

so these curves are not isomorphic over \(\QQ\text{,}\) no way does it come from a \(\QQ\)-curve.

This example suggests another interesting behaviour, the curve over \(\CC\) could come from a \(k\)-curve in multiple ways, which are non-isomorphic over the base.

Example 5.8.4.

Let now

\begin{equation*} E \colon y^2 = x^3 + x + 1/\CC \end{equation*}

we have

\begin{equation*} E_0 \colon y^2 = x^3 + x + 1/\QQ \end{equation*}

duh... but also

\begin{equation*} E_0' \colon 2 y^2 = x^3 + x + 1 \simeq y^2 = x^3 + 4 x + 8/\QQ\text{,} \end{equation*}

both are isomorphic to \(E\) over \(\CC\) but are not isomorphic to each other over \(\QQ\text{.}\)

Coming back to our necessary condition:

Question 5.8.5.

If for all \(\sigma \in \Gal{\CC}{k}\) we have some

\begin{equation*} f_\sigma \colon X\xrightarrow{\sim} X^\sigma \end{equation*}

does \(X\) descend to \(k\text{?}\)

For elliptic curves we have \(j(E^\sigma) = \sigma(j(E))\) so \(E \simeq E^\sigma\) for all \(\sigma\) implies \(j(E) \in k\) and hence there is an elliptic curve \(E_0/ k\) with \(j(E_0) = j(E)\) hence they are isomorphic over \(\CC\text{,}\) explicitly:

\begin{equation*} y^2 + xy = x^3 - \frac{36}{j(E) - 1728} x - \frac{1}{j(E) - 1728} \end{equation*}

when \(j\ne 0,1728\text{.}\)

So our necessary condition is sufficient for genus 1 (exercise: genus 0).

Now I will subtly switch to quasiprojective-variety-land.

Notice however that given

\begin{equation*} X_0/k \end{equation*}

so that we have natural

\begin{equation*} (X_0 \times_k \CC) \xrightarrow{f_\sigma,\sim} (X_0 \times_k \CC)^\sigma \end{equation*}

various isomorphic curves, we have the relation

\begin{equation*} f_\sigma^\tau f_\tau = f_{\sigma\tau}\text{.} \end{equation*}

This condition sounds like it could be irritating to check, fortunately we have the following:

Remark 5.8.7.

If \(X/\CC\) has no automorphisms (i.e. a generic genus \(g \ge 3\) curve) then the cocycle condition is trivial and we just need the isomorphisms as in our first necessary condition. This is as \(f_{\sigma\tau}\inv f_\tau ^\sigma f_\sigma\) is just some automorphism, we want it to be the identity.

Unfortunately many curves of interest have a lot of automorphisms however. Like superelliptic curves/cyclic covers.

This motivates the following definition:

Definition 5.8.8. Field of moduli.

The field of moduli of \(X/\CC\) is the fixed field of

\begin{equation*} \{\sigma \in \Gal{\CC}{\QQ} : X^\sigma \simeq X\}\text{.} \end{equation*}

It would be great if every curve could be defined over its field of moduli.

“You can't always get what you want, but if you try sometimes, you might find, you get what you need” - The philosopher Jagger.

Example 5.8.9. Shimura.

Let \(m\) be odd and define a hyperelliptic curve of genus \(m - 1\) (which is even) as

\begin{equation*} X\colon y^2 = a_0 x^m + \sum_{r= 1}^m (a_r x^{m+r} + (-1)^r a_r^\rho x^{m-r}),\,a_i \in \CC,\,a_m = 1,\,a_0 \in \RR \end{equation*}

\(\rho\) is complex conjugation, then we have an isomorphism

\begin{equation*} \mu\colon X \to X^\rho,\,\mu(x,y) = (-x\inv, i x^{-m} y) \end{equation*}
\begin{equation*} \mu^\rho \mu \colon (x,y)\mapsto (x,-y) \end{equation*}

so the field of moduli is contained in \(\RR\text{.}\) As long as we pick all \(a_i,a_i^\rho\) algebraically independent over \(\QQ\) there are no automorphisms except \(\pm 1\text{.}\) Exercise, in this case \(X\) has no model over \(\RR\text{.}\)

Warning even though trivial automorphism group is best, it is not really the case that more automorphisms is worse for you.

What does help is points

Goal.

Identify a special set of points, and some field \(L\) as above where we “know” the galois action.

Subsection 5.8.2 Reflex fields

First we define a field based on a Shimura datum, this will (eventually) be the field we hope to descend the associated Shimura variety to.

Definition 5.8.11. Algebraic tori.

An algebraic torus over a field \(k \) is an algebraic group \(T\) such that \(T_{\bar k} \simeq (\mathbf G_m)^n\text{.}\)

Let \(G/\QQ\) be reductive, \(k \subseteq \CC\) and let

\begin{equation*} C(k) = G(K) \backslash \Hom(\mathbf G_m, G_k) \end{equation*}

be the set of conj. classes of cocharacters \(/k\text{.}\)

For \((G,X)\) a Shimura datum we can take

\begin{equation*} X \ni x \mapsto \mu_x(z) = h_{x\CC} (z,1) \in C(\QQ^\alg) \subseteq C(\CC)\text{.} \end{equation*}

So think of \(c(X) \in C(\QQ^\alg)\)

Definition 5.8.12. Reflex fields.

The reflex field, denoted \(E(G,X)\) is the field of definition of \(c(X)\) inside \(\QQ^\alg\text{.}\)

Subsection 5.8.3 Special points

In the theory of modular curves and the upper half plane there are certain points that play an important role, imaginary quadratic integers in \(\mathbf H\text{.}\)

Why are these points special? They are fixed points: if we try and solve for \(z \in \mathbf H\)

\begin{equation*} z= \overbrace{\begin{pmatrix} a\amp b \\ c \amp d\end{pmatrix}}^{\in \SL_2(\ZZ)} z = \frac{az + b}{cz+d} \end{equation*}

we get

\begin{equation*} cz^2 + (d - a) z - b = 0 \end{equation*}

which has discriminant \((d-a)^2 + 4cb = d^2 - 2ad + a^2 + 4bc = (a+d)^2 - 4(ad - bc) = \trace^2 - 4\det\) so \(z\) is an eigenvalue of this matrix. (note that a matrix must be elliptic to have fixed points in the upper half plane).

In fact this is a general phenomenon:

Definition 5.8.14. Special points.

\(x \in X\) is a special point if there is a \(\QQ\)-torus \(T \subseteq G\) s.t.

\begin{equation*} h_x(\CC^\times) \subseteq T(\RR) \end{equation*}

we also say \((T,x)\) is a special pair.

Remark 5.8.15.

\((T,x)\) special means \(T(\RR)\) fixes \(x\text{.}\)

Conversely if \(T\) is a maximal torus of \(G\) with \(T(\RR)\) fixing \(x\) then \(h_x(\CC^\times)\) is in the centraliser of \(T(\RR)\) inside \(G(\RR)\) which is itself \(\implies (T,x)\) is special.

I said this generalises CM points, how?

Example 5.8.16.

Let \(G = \GL_2\) and \(\mathbf H_1^\pm = \CC \smallsetminus \RR\) then we have our old friend the \(G(\RR)\) action

\begin{equation*} \begin{pmatrix} a\amp b \\ c \amp d\end{pmatrix} z = \frac{az + b}{cz+d} \end{equation*}

so if \(z \in \CC \smallsetminus \RR\) generates an imaginary quadratic field \(E/\QQ\) (which is a 2-d \(\QQ\)-vector space) we can embed

\begin{equation*} E \hookrightarrow \Mat_2(\QQ) \end{equation*}

using basis \(\langle 1, -z\rangle\) for \(E\text{.}\)

So we get a maximal subtorus \(T = \Res_{E/\QQ}(\mathbf G_m) \subseteq G\text{.}\)

Now

\begin{equation*} E\otimes \CC = \langle 1\otimes 1, 1 \otimes (-z)\rangle \end{equation*}

and we can map

\begin{equation*} E\otimes \CC \to \CC \end{equation*}
\begin{equation*} e\otimes z \mapsto ez \end{equation*}

we have a kernel of dimension 1

\begin{equation*} \langle z \otimes 1 + 1\otimes (-z) \rangle = \langle \begin{pmatrix} z \\ 1 \end{pmatrix}\rangle \end{equation*}

exercise check \(\Res_{E/\QQ}(\mathbf G_m)(\RR)\text{.}\)

Subsection 5.8.4 Canonical models

Given a special pair \((T,x) \subset (G,X)\) we have a cocharacter \(\mu_x\) of \(T\) defined over \(E(x)\) we can form the map

\begin{equation*} r_x \colon \adeles^\times_{E(x)} \xrightarrow{ P \mapsto \prod_{\rho \colon E(x) \to \QQ^\alg} \rho(\mu_x(P)) } T(\adeles_\QQ) \to T(\adeles_f) \end{equation*}

the last map just forgets the infinite components.

We have the artin map from CFT

\begin{equation*} {\operatorname{art}}_{E(x)} \colon \adeles^\times_{E(x)} \twoheadrightarrow \Gal{E(x)^\ab}{E(x)} \end{equation*}
\begin{equation*} r_x \colon \adeles^\times_{E(x)} \to T(\adeles_f)\text{.} \end{equation*}

Call \(\lb x,a\rb_K\) the point of \({\operatorname{Sh}}_K(G,X)\) represented by \((x,a)\text{.}\)

Definition 5.8.17. Milne 12.8.

Let \((G,X)\) be a Shimura datum, and let \(K\) be a compact open subgroup of \(G(\adeles_f)\text{.}\) A model \(M_K(G,X)\) of \({\operatorname{Sh}}_K(G,X)\) over \(E(G,X)\) is a canonical model if, for every special pair \((T, x) \subseteq (G,X)\) and \(a \in G(\adeles_f)\text{,}\) \(\lb x,a\rb _K\) has coordinates in \(E(x)^\ab\) and

\begin{equation*} \sigma [x,a]_K = [x, r_x(s) a]_K \end{equation*}

for all

\begin{equation*} \sigma \in \Gal{E(x)^\ab}{E(x)} \end{equation*}
\begin{equation*} s\in \adeles^\times_{E(x)} \end{equation*}
\begin{equation*} {\operatorname{art}}_{E(x)}(s) = \sigma \end{equation*}

In other words, \(M_K(G,X)\) is canonical if every automorphism \(\sigma\) of C fixing \(E(x)\) acts on \(\lb x,a\rb_K\) according to the above rule, where \(s\) is any idele such that

\begin{equation*} {\operatorname{art}}_{E(x)}(s) = \sigma| E(x)^{\ab}\text{.} \end{equation*}
Example 5.8.18.

\(T \) an algebraic torus over \(\QQ\) and

\begin{equation*} h \colon \mathbf S \to T_\RR \end{equation*}

then \((T,h)\) is a Shimura datum \(E = E(T,h)\) is the field of definition \(\mu_h\) in this case

\begin{equation*} {\operatorname{Sh}}_K(T,h) = T(\QQ) \backslash \{h\} \times T(\adeles_f) / K \end{equation*}

is a finite set, defines a continuous action of

\begin{equation*} \Gal{E^\ab}{E} \acts {\operatorname{Sh}}_K(T,h)\text{,} \end{equation*}

this action defines a model of \({\operatorname{Sh}}_K(T, h)\) over \(E\) which by definition is canonical.

Some references:

  1. Weil’s Galois Descent Theorem; A Computational Point Of View - Ruben A. Hidalgo And Sebastian Reyes-carocca
  2. On the field of moduli of superelliptic curves - Ruben Hidalgo and Tony Shaska
  3. Varieties Without Extra Automorphisms I: Curves - Bjorn Poonen
  4. Lecture On Shimura Curves 6: Special Points And Canonical Models Pete L. Clark http://math.uga.edu/~pete/SC7-CMpoints.pdf (Shimura curves only but still)
  5. Shimura Varieties and Canonical models (slides) - Brian Smithling http://www.math.mcgill.ca/goren/Montreal-Toronto/Brian.pdf.
  6. https://tlovering.wordpress.com/2014/09/03/galois-descent-for-transcendental-extensions/.
  7. Canonical models of Shimura curves - J.S. Milne (a great article I found after the talk...)