Arithmetic and geometry, Vol. I, 239--265, Progr. Math., 35, Birkhäuser Boston, Boston, MA, 1983.

Comments

In 1981 it was generally considered that the problem of proving the existence of canonical models for the Shimura varieties not treated in Deligne's Corvallis article (that is, for those not of abelian type) was beyond reach. This perception changed with Borovoi's manuscript “Canonical models of Shimura varieties, 26.5.81”. There he uses the idea (which he credits to Piatetski-Shapiro) of embedding the Shimura variety in a larger Shimura variety that contains many Shimura subvarieties of type $A_{1}$ to attempt to prove the existence of canonical models. He doesn't succeed in his attempt, and in fact no one has succeeded in making this idea work directly except for the uninteresting case of Shimura varieties defined by adjoint groups, but the idea is a fruitful one. I used it to extend Shih's and my proof of Langlands's conjugacy conjecture, which has the existence of canonical models as a consequence, to Shimura varieties not of abelian type. I talked on this at a conference at the University of British Columbia in August 1981 (manuscript 26/11/81). At the time, my proof wasn't completely general --- in some cases I needed to assume that the congruence subgroup problem had a positive solution for the group. A suggestion of Deligne in early 1982 (as I recall, that a connected Shimura variety behaves as though it were simply connected) allowed me to remove the restriction, and I completed my proof (and manuscript) in early 1982. After talking to Deligne, I wrote to Borovoi (through Zarhin) offering him joint authorship. He declined, and instead announced his own proof (The Langlands conjecture on the conjugation of Shimura varieties, Functional Anal. Appl., 16 (1982), pp.~292--294).

I submitted my paper to the Shafarevich volume in April 1982, and it was published in 1983.

In May 1983, about 12 months after he had received my manuscript, Borovoi submitted his own manuscript, which was published in 1984¹. However, his proof in that manuscript is incomplete since it relies on a statement “Theorem 3.21” which “will be proved in another paper by the author”. As far as I know, “Theorem 3.21” is still unproven² but Borovoi proved a weaker statement sufficient for the application to Shimura varieties in October 1986³.

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1. Langlands' conjecture concerning conjugation of connected Shimura varieties, Sel. Math. Sov., 3 (1983/84), pp. 3--39). ↩
2. In a letter to me dated July 3, 1988, Borovoi said that he thought it was his “duty to prove it”.↩
3. On the groups of points of a semisimple group over a totally real field, Problems in Group Theory and Homological Algebra, Yaroslavl 1987, pp. 142-149. ↩

Erratum

p. 239 Replace [8, pp. 222-223] with [8, pp. 232-233].

p. 239 In the last displayed map, omit the second copy of $G_{\mathbb{A}^{f}}^{\prime}$.

p. 250 In Lemma 3.8, replace $G$ with $Z$ in all the cohomology groups.

p. 256. First line: $(T,h)\overset{i}{\hookrightarrow}(G,X^{+})$.

Nonerratum

p.262. Theorem 7.2 asserts that canonical models exist for all Shimura varieties as a consequence of the proof of Langlands's conjugation conjecture. The original source for this implication is Langlands's Corvallis article, pp 233--234, which sketches the derivation of a descent datum from the conjugation conjecture and says that “one applies the Weil criterion for descent of the field of definition”. In Milne and Shih 1982, Conjugates..., the sketch is made more detailed. In neither reference is it checked that the descent datum satisfies the continuity condition necessary before one can apply Weil's criterion, although there is no reason to think that Langlands did not in fact check the continuity. The fact that the continuity was not explicitly checked anywhere in the literature has achieved a certain notoriety with authors stating “this argument is flawed” and referring to a “gap in the argument” and to the need to “correct the proof”. One author went so far as to claim as a “new result” the statement that had been proved 17 years earlier. In fact the descent datum does satisfy the continuity condition, and it is relatively easy to verify this. Moreover, the proof requires nothing that was not available in 1977 when Langlands wrote his article. See my article in the Michigan Math. J. 1999 and notes.

Reply (June 14, 2025) to an e-mail from Richard Taylor

Rather than revisiting the ad hoc methods I use in Section 6 of my 1983 paper to prove compatibility for different special points, I think one should instead use the following beautiful result of Borovoi.

Theorem 1. Let $G$ be a simply connected semisimple algebraic group over a totally real algebraic number field $F$. Assume that $G$ has an anisotropic maximal torus $T$ that splits over some totally imaginary quadratic extension $K$ of the field $F$. Let $\Pi$ be a base of the root system $R=R(G_{K}% ,T_{K})$. Then $G(F^{\mathrm{rc}})$ is generated by the subgroups $G_{\alpha }(F^{\mathrm{rc}})$, $\alpha\in\Pi$ (here $F^{\text{rc}}$ is a totally real closure of $F$).

Theorem 2. Under the conditions of Theorem 1, assume that $G$ is a geometrically simple group of totally hermitian type that is not totally compact. Then $G(F^{\mathrm{rc}})$ is generated by the subgroups $G_{\alpha }(F^{\mathrm{rc}})$, $\alpha\in R^{\mathrm{rtc}}$.

In his 1983/84 paper, Borovoi made use of a stronger statement, which still hasn't been proved, but later he did prove Theorems 1 and 2 with the help of his Russian colleagues. See,

The group of points of a semisimple group over a totally real closed field Borovoi, M. V., Selecta Math. Soviet. 9 (1990), no. 4, 331--338.

In the last two sections of my 1988 Inventiones paper, I gave two proofs of the compatibility, one with and one without Borovoi's statement.

But, as I mentioned briefly at the [Tate 100] conference, I think the whole business of canonical models (in the general case) needs to be rethought.

At present we

1. use Kazhdan's theorem that conjugates of arithmetic varieties are arithmetic varieties (arithmetic variety = bsd/arithmetic group);
2. deduce the conjugation theorem for Shimura varieties (Borovoi-Milne);
3. prove the conjugation theorem for the standard principal bundle (Milne, 1988, Inventiones).

What Nori and Ragunathan (1993) show is that Kazhdan asked the wrong question. Let $D$ be a bounded symmetric domain and $G$ a real algebraic group such that $G(\mathbb{R}{})$ acts transitively on $D$ with compact isotropy groups. From an arithmetic $\Gamma$ we get a system $$ (Y,P,\nabla,D^{\vee},\gamma),\qquad Y\leftarrow (P,\nabla)\xrightarrow{\gamma} D^{\vee}, $$ where $Y=\Gamma\backslash D$, $P$ is the principal bundle $\Gamma\backslash D\times G(\mathbb{C})$, $\nabla$ is a flat connection, $D^{\vee}$ is the compact dual, and $\gamma\colon P(\Gamma)\rightarrow D^{\vee}$ is defined by the Borel map. This system is algebraic, and the correct question to ask is that the conjugate of such a system be again such a system. Nori and Raghunathan characterize such systems and show that the characterizing properties are preserved under conjugation. This is much much simpler than Kazhdan.

From a Shimura datum $(G,X)$, we get a similar system $$ (S,P,\nabla,X^{\vee},\gamma),\qquad S\leftarrow (P,\nabla)\xrightarrow{\gamma}X^{\vee} $$ and I think, similarly, that one should work directly with such systems instead of just the Shimura variety. This should make everything simpler --- the conjugation conjecture, canonical models, and even integral canonical models. I talked about this at the Borel conference at Hangzhou in 2004, but haven't worked out the details.

A little history. Langlands made little progress in understanding the zeta functions of Shimura varieties until Deligne explained to him his axioms (especially $h$!) and that he should think of them as moduli varieties of motives. Langlands stated his Corvallis conjecture in order to understand the factors of the zeta function at infinity, and his conjecture with Rapoport to understand the factors at finite places. When I asked Langlands how he came up with the “cocycle” for the conjugation conjecture, he just said that it was the only thing he could think of. When he explained it to Deligne, Deligne realized that it conjecturally gave an explicit description of the Taniyama group, something that he and others had been searching for.

(Edited for clarity.)