2005a Introduction to Shimura varieties (Toronto long version)
Available at www.jmilne.org/math/
See xnotes for a corrected version of the article.

Erratum

These have been fixed on the version under "Expository Notes".

p9, top. It is an isomorphism of riemannian manifolds that is called an isometry, not a morphism.

p12, footnote 10. It was H. Cartan, not E. Cartan, who proved that the group of isometries of a bounded domain has a natural structure of a Lie group; then E. Cartan proved that the group of isometries of a symmetric bounded domain is semisimple (see Borel, Essays …, 2001, IV 6).

p30, proof of 2.14, just above the diagram 26. I write $G\overset{\mathrm{Ad}}{\longrightarrow}\mathfrak{g}{}$ where I should write $G\overset{\mathrm{Ad}}{\longrightarrow}\mathrm{GL}(\mathfrak{g}{})$ (from Bin Du).

p.51, l.5 ... depend on a (from Timo Keller).

p.59, Lemma 5.22 This is misstated: in general, $T(\mathbb{Q}{})$ is not closed in $T(\mathbb{A}_{f})$ (unless $(G,X)$ satisfies SV5) and so $T(\mathbb{Q}{})\backslash T(\mathbb{A}{}_{f})$ is not Hausdorff (hence not compact). The last step of the proof “ An arbitrary torus ...”\ fails when $T(\mathbb{Q}{})\backslash T(\mathbb{A}% {}_{f})$ is not compact. The proof of the finiteness of $T(\mathbb{Q}% {})\backslash T(\mathbb{A}{}_{f})/\nu(K)$ needs to be rewritten. (Bas Edixhoven)

p.67. In the display under SV1, interchange $z/\bar{z}$ and $\bar {z}/z$.

p80, 81. Lucio Guerberoff points out that the uniqueness assertion in Proposition 8.14 fails and that the condition (**) in Theorem 8.17 is inadequate. He writes (slightly edited):

In Theorem 8.17, you say that your condition (**) on the isomorphism $a$ is enough to guarantee that $ah_{A}$ belongs to $X$ ($h_{A}$ being the morphism defining the Hodge structure on $H_{1}(A,\mathbb{Q}{})$). However I believe that this only implies that $ah_{A}$ belongs to the Siegel double space of $(V,\psi)$, but not necessarily to the $G(\mathbb{R}{})$-conjugacy class $X$. More precisely, I'm not sure if I'm missing something in Proposition 8.14. I tried to reproduce all the relevant calculations, comparing with Kottwitz's JAMS paper, and my conclusion is that if $x$ is a morphism from $\mathbb{S}{}$ to $G_{\mathbb{R}{}}$, then $x$ belongs to $X$ if and only if two conditions hold:

1) $x$ lies in the Siegel double space of $(V,\psi)$, and

2) (fix one morphism $h$ in $X$) the two $B\otimes_{\mathbb{Q}}\mathbb{C}$ structures on $V\otimes_{\mathbb{Q}}\mathbb{R}$ (one is $x(i)$, other one is $h(i)$) are isomorphic.

Condition 1) only guarantees that they will be $\mathbb{C}$-isomorphic, not necessarily in a $B$-linear way. I don't see how this would be automatically implied from 1). In other words, your statement of Proposition 8.14 suggests that if $x$ and $h$ have target $G_{\mathbb{R}},$ satisfy 1) and are conjugate under $\mathrm{GSp}(\psi)(\mathbb{R})$, then they are also $G(\mathbb{R}% )$-conjugate, which doesn't seem to be the case. (For example, take a unitary group of signature $(r,s)$ over a CM extension $K/\mathbb{Q}{}$, $K$ quadratic imaginary say, and starting with the usual $h(z)=\mathrm{diag}(zI_{r},\bar{z}I_{s})$, consider $h^{\prime}(z)=h(\bar{z})$; then $h^{\prime}$ has target $G_{\mathbb{R}}$, and is obviously on the Siegel double space (to form $\psi$, use a trace zero element in $K$), but it's not $G(\mathbb{R})$-conjugate to $h$ unless $r=s$).

In the same vein, I'm seeing condition (**) as missing something. In the same example, suppose the hermitian space defining the unitary group is $(V,\langle,\rangle)$, and consider $(V,-\langle,\rangle)$, so it has signature $(s,r)$. Neither of the conditions on Theorem 8.17 care about whether you look at $\langle,\rangle$ or $-\langle,\rangle$, but the Shimura variety of $-\langle,\rangle$ should be the complex conjugate of the variety for $\langle,\rangle$.

p100, top line (proof of 11.2). Delete “ therefore ”from “ The map therefore factors through …” --- as Brian Conrad reminded me the group $\mathbb{A}{}_{E,f}^{\times}/E^{\times}$ need not be Hausdorff. In a detailed proof, one replaces $\mathbb{A}{}_{E,f}^{\times}/E^{\times}$ with a quotient $T(\mathbb{A}_{f})/T(\mathbb{Q}{})$, which is Hausdorff. Here $T$ is a certain subtorus of $(\mathbb{G}_{m})_{E/\mathbb{Q}}$. See my notes on Complex Multiplication for the details.

p124, top. Shenghao Sun has pointed out to me that the statement that pro-tori correspond to free $\mathbb{Z}{}$-modules with a continuous action of $\Gamma$ is contradicted on the next page where I show that the character group of $\mathbb{G}$ is $\mathbb{Q}$. The “free” should be “torsion-free”.