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.
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.