p.29. In the proof of Theorem 1.64, I'm using that the characteristic polynomial has distinct roots. Specifically, an nxn matrix with coefficients in a field k is similar to a diagonal matrix if its characteristic polynomial has n distinct roots in k.
The e-reader version 3.11 is not being corrected to updated.
p.179, the second line of the proof of Lemma 8.20
should send $(f,g)$ to the product
$f|_{U'} \cdot g\circ\varphi|_{U'}$.
Also, in the bottom row of the commutative diagram there is a missing product symbol
$\prod_{i,j}$.
p.201, in Example 9.13 (b), flat maps of varieties not are
defined until 9.22, two pages later.
p.208, I think Aside 9.39 begs for some extra explanations
for the quotient of an affine variety by a finite subgroup of
its automorphisms, for example that the quotient map is
finite and its fibers are the orbits of the action.
(Felipe Zaldivar)
In Example 5.14, it is claimed that 22 is not a square modulo 29,
however $14^2=196=22+6\times 29$.
Instead argue as follows.
As $K$ is a subfield of $\mathbb{R}$
the first and last possibilities
$\frac{\alpha-1}{\alpha+2}$,
$-\alpha\frac{\alpha-1}{\alpha+2}$ are eliminated as they are negative
hence not square.
Take $\mathfrak{q}=(29,\alpha-2)$, the third $\alpha\frac{\alpha-1}{\alpha+2}$ is
excluded.
Take $\mathfrak{q}'=(29,\alpha-3)$, the second $-\frac{\alpha-1}{\alpha+2}$ is excluded. (Haohao Liu, Jingxin Wang)
In Remark 7.17, I say that "according to 7.3, the absolute values of $K$ are discrete", but in 7.3, I only prove that they are nonarchimedean. Sometime, I'll add a proof (not difficult) that they are, in fact, discrete (Haohao Liu).
p79 The claim in the proof of Proposition 5.26 that $E_2$ and $E_3$ generate
the algebra of modular forms with integer coefficients $a_n$ is incorrect because the
modular form $(E_2^3 - E_3^2)/1728$ has integer coefficients,
but it is not a polynomial in $E_2$ and $E_3$ with integer
coefficients.
The Eisenstein series do not form an
integral basis of modular forms. For example, in the book
Modular Forms by Cohen and Stromberg (section 10.6) they use a basis that
also involves $(E_2^3 - E_3^2)/1728$. (Xevi Guitart)
p118, l. -7, "a \equiv 3 mod 3" doesn't look right (Timo Keller).
p1. In equation (1), $aXZ$ should be $aXZ^2$ (Helge Øystein Maakestad).
p2. The claim in the footnote that every abelian surface is a Jacobian variety is not quite true. See the preprints of E. Kani, "The moduli spaces of Jacobians isomorphic to a product of two elliptic curves" and "The existence of Jacobians isomorphic to a product of two elliptic curves". (Kuang-yen Shih)
p.113, for a finite group $M$, the Galois coverings with group $M$ are classified by the surjective homomorphisms $\pi_1(V,P)\to M$.
p. 136, l. 6: "there *is* an abelian variety ..." (Timo Keller).
p37, line 6. The regular map $\alpha$ should go from $T$ to the dual of $A$ (Bart Litjens).
From Bhupendra Nath Tiwari: For AV, CFT, and CM
From Tobias Barthel See pdf file (2 pages)
From Shaul Zemel In the proof of Theorem 10.15, p49, concerning the map from $\textrm{Hom}(A,B) \otimes Z_l$ to the module $\textrm{Hom}(T_l A,T_l B)$ (over $Z_l$), you start by proving that if $e_1,...,e_n$ are linearly independent over Z in Hom(A,B) then their images are linearly independent over Z_l in Hom(T_l A,T_l B). But this immediately proves that n cannot exceed the rank of the latter over Z_l, i.e., 4dimAdimB (as can be even more clearly seen in Hom(V_l A,V_l B) over Q_l, after tensoring the latter with Q_l). Hence you immediately obtain the finiteness of the rank of Hom(A,B), and the desired bound, without the need to involve decomposition into simple Abelian varieties, different topologies, and polynomials. This is in fact similar (as you have indicated there for something else) to the fact that showing that if a (clearly torsion-free) subgroup of a real vector space of dimension n is discrete then it's free of rank not exceeding n. [This is fixed in AVs I think.]
p47, Lemma 10.12. The statement should require that the degree of $f(xv+w)$ as a polynomial in $x$ is bounded, say, at most 2g (otherwise the first displayed sum in the proof may be infinite). (Angus Chung).
Timo Keller points out that, in the proof of Theorem 10.15, p49, M should be defined to be a Z-submodule of End^0(A), not End(T_l(A)) (the degree map P is defined on End^0(A), not on End(T_l(A)). Also, when I write "Now choose the e_i to be a Q-basis for End^0(A)." I seem to be assuming that End^0(A) is finite dimensional over Q, which is what I'm trying to prove. The proof should be replaced by this.
From Everett Howe In Prop. 13.2(b), I found a small typo, probably carried over from copying the result from [1986b] and not changing all of the notation: the "f" in the exponent should be an alpha.
Tim Dokchitser points out that I prove Zarhin's trick (13.12) only
over an algebraically closed field , and then immediately apply it in (13.13) over a finite field.
This is doubly confusing because (13.10) is certainly false over nonalgebraically closed field (over such a field
an abelian variety need not be isogenous to a principally polarized abelian variety).
However, I believe everything is O.K. Specifically, the proof of Zarhin's trick
requires only (13.8), and, because this holds over an algebraically closed field, it holds
over every perfect field (see my 1986 Storrs article Abelian Varieties 16.11 and
16.14).
From Sunil Chetty. Near the start of I 14 (Rosati involution): in $(\alpha\beta)^\dagger = \beta\alpha$ there should be a dagger on each of $\beta$ and $\alpha$.
p.154, IV. Theorem 7.3. The property in (a) of the statement only holds for the points P outside a closed subset of codimension 1. As Martin Orr writes:
[The theorem] asserts that the Siegel moduli variety M_{g,d} over the complex numbers satisfies: for every point P in M_{g,d}, there is an open neighbourhood U of P and a family A of polarized abelian varieties over U such that the fibre A_Q represents j^{-1}(Q) for all Q in M_{g,d} (I assume this should say "all Q in U" at the end).
I don't see how such a neighbourhood can exist around the elliptic point 0 in the j-line, because of the standard monodromy argument that there is no family of elliptic curves on all of M_{1,1}. Specifically, if we had such a U then the period mapping (which is just the inclusion U -> M_{1,1}) would lift to a map from U' to the upper half plane H, for some open neighbourhood U' of 0 (wlog lifting 0 to i). Then the image of this lifting contains an open neighbourhood V of i in H, and the map H -> M_{1,1} is not injective on V.
From Roy Smith (on proofs of Torelli's theorem III 13)
You ask on your website for advice on conceptual proofs of Torelli. ... here goes.
There are many, and the one you give there is the least conceptual one, due I believe to Martens.
Of course you also wanted short, ....well maybe these are not all so short.
The one due to Weil is based on the fact that certain self
intersections of a jacobian theta divisor are reducible, and is
sketched in mumford's lectures on curves given at michigan. Indeed
about 4 proofs are sketched there.
The most geometric one, due to Andreotti - Mayer and Green is to
intersect at the origin of the jacobian, those quadric hypersurfaces
occurring as tangent cones to the theta divisor at double points, thus
recovering the canonical model of the curve as their base locus, with
some few exceptions.
To show this works, one can appeal to the deformation theoretic
results of Kempf. i.e. since the italians proved that a canonical
curve is cut out by quadrics most of the time, one needs to know that
the ideal of all quadrics containing the canonical curve is generated
by the ones coming as tangent cones to theta. the ones which do arise
that way cut out the directions in moduli of abelian varieties where
theta remains singular in codimension three.
But these equisingular deformations of theta embed into the
deformations of the resolution of theta by the symmetric product of
the curve, which kempf showed are equal to the deformations of the
curve itself. hence every equisingular deformation of theta(C) comes
from a deformation of C, and these are cut out by the equations in
moduli of abelian varieties defined by quadratic hypersurfaces
containing C. hence the tangent cones to theta determine C.
This version of Green's result is in a paper of smith and varley,
in compositio 1990.
Perhaps the shortest geometric proof is due to andreotti, who
computed the branch locus of the canonical map on the theta divisor,
and showed quite directly it equals the dual variety of the canonical
curve. this is explained in andreotti's paper from about 1958, and
quite nicely too, with some small errata, in arbarello, cornalba,
griffiths, and harris' book on geometry of curves.
There are other short proofs that torelli holds for general
curves, simply from the fact that the quadrics containing the
canonical curve occur as the kernel of the dual of the derivative of
the torelli map from moduli of curves to moduli of abelian varieties.
this is described in the article on prym torelli by smith and varley
in contemporary mat. vol. 312, in honor of c.h. clemens, 2002, AMS.
there is also a special argument there for genus 4, essentially using
zariski's main theorem on the map from moduli of curves to moduli of
jacobians.
There are also inductive arguments, based on the fact that the
boundary of moduli of curves of genus g contains singular curves of
genus g-1, and allowing one to use lower genus torelli results to
deduce degree torelli for later genera.
Then of course there is matsusaka's proof, derived from torelli's
original proof that given an isomorphism of polarized jacobians, the
theta divisor defines the graph of an isomorphism between their
curves.
For shortest most conceptual, I recommend the proof in Arbarello,
Cornalba, Griffiths Harris, i.e. Andreotti's, for conceptualness and
completeness in a reasonably short argument..
From Zheng Yang.
In S. 2, p. 20, there is a typo in the definition of an unramified morphism of schemes,
namely, it should be "$O_{X, \varphi(y)} \rightarrow O_{Y, y} ...$"
In S. 6, p. 45, under "The sheaf defined by a scheme $Z$"
the functor given is notated $\mathcal{F}$ but in subsequent appearances it is written
as $\mathcal{F}_Z$. It may be better to correct it for this first instance.
In S. 6, Example 6.13 (b) the stalks of finite type
schemes $Z$ over $X$ are computed.
It might be helpful to include a reference for this fact, for instance Lemma 3.3 in your book "Étale Cohomology."
In S. 10, Theorem 10.7 there is a typo in the statement of the Grothendieck spectral sequence,
namely $FG$ on the right should be $GF$.
In S. 13, p. 84, the very first sentence references "We saw in the last section..." but it should
instead reference S. 11.
In S. 14, p. 89, after "We saw in the last section..." the $0$-th cohomology group of $U$ should be
$\Gamma(U, O_U^{\times})$. (Which matches with the notation in Thm. 13.7).
Fix the following problems: the heading of S. 27 "Proof of the Weil Conjectures..." is off (p. 154) and there is some overlap with the section name of S. 29 "The Lefschetz fixed point formula..." with the page numbers.
(Bence Forrás)
(Apparently) I should use caron instead of breve in Cech.
In the diagram on page 22, the subscript $b$ should be before the last bracket.
In the proof of Proposition 8.12, $\mathcal{P}^*$ is not defined.
Define it to be $\prod i_{x*}i^{x*}\mathcal{F}$. There are canonical injective
homomorphisms $\mathcal{F}\to \mathcal{P}^*\to \mathcal{I}$.
The first sentence on page 118, should read "as it is in topology"
instead of "as it is topology".
The second word in Example 21.4 should be "hypothesis" instead of "hypotheses"?
p.56. "Grothendieck has banished [\'espace \'etal\'e] from mathematics." Grothendieck uses them in his 1957 Tohoku paper (p. 154-155) but (somewhat) banishes them in 3.1.6, p.25, of Chapter 0 in EGA I.
(Nikita Kozin)
p.78: "Proof: Let M be..." - change M (mathfrak?) to another font
p.80: "Remark 11.7: ... associated with the preseaf ... H^r" - must be "H^s"?
p.81: Diagram - must be "Sh(Y_et)" in left low corner?
p. 176. Lemma 30.2 is true for semisimple endomorphisms, but is false in general, as the following example illustrates. Let $K=\mathbb{Q}[a]$, where $a=\sqrt{2}$, and let \[ A=% \begin{pmatrix} a & 1 & 0 & 0\\ 0 & a & 0 & 0\\ 0 & 0 & -a & 0\\ 0 & 0 & 0 & -a \end{pmatrix} \in M_{4}(K). \] Then $A$ has characteristic polynomial $(X-a)^{2}(X+a)^{2}=(X^{2}-2)^{2}% \in\mathbb{Q}{}[X]$, and so, according to the lemma, $A=CBC^{-1}$ with $B\in M_{4}(\mathbb{Q}{})$ and $C\in\mathrm{GL}_{4}(K)$. Let $\sigma$ be the automorphism of $K$ sending $a$ to $-a$. Then $\sigma A=(\sigma C)B(\sigma C)^{-1}$, and so $\sigma A$ is similar to $A$. But \[ \sigma A=% \begin{pmatrix} -a & 1 & 0 & 0\\ 0 & -a & 0 & 0\\ 0 & 0 & a & 0\\ 0 & 0 & 0 & a \end{pmatrix}, \] which is not similar to $A$ by the uniqueness of Jordan forms. (The point is that the characteristic polynomial doesn't say anything about the Jordan blocks.)
From a different perspective, there are four isomorphism classes of $K[X]$-modules with characteristic polynomial $(X^{2}-2)^{2}$, but only two isomorphism classes of $\mathbb{Q}{}[X]$-modules with this characteristic polynomial. Thus, not every $K[X]$-module with characteristic polynomial $(X^{2}-2)^{2}$ can be obtained by extension of scalars from a $\mathbb{Q}% {}[X]$-module, contradicting the lemma.
This necessitates some later changes (I think minor). (From Vladimir Uspenskiy.)
p.32, line -5. Delete the third $\pi$: ... $=\pi^{r+1}Q(X_1\ldots)$ (David Craven).
p.41. In step 3, $\theta F_f(\theta^{-1}(X),\theta^{-1}(Y))$ has the properties characterizing $F_g(X,Y)$ (and not $F_f(X,Y)$) (Ashutosh Jangle).
IV, Example 1.11. Replace the last sentence with a reference to GT 7.19(a).
Proof of Theorem II.3.10. In the paragraph "To proceed
further", there is a bit of confusion created by the use of $H^r$
instead of $H^r_T$, which then propagates into the next
paragraph and the use of the Sylow theorems: the reduction to
the solvable case is only valid for $r>0$,
whereas you also need the case $r=0$ in order to carry out the
dimension-shifting argument to cover $r<0$.
This is of course easy to fix: one just has to observe that since
$H^0_T(G_p, M) = 0$, we can deduce directly that $H^0_T(G, M)$ is
killed by $[G:G_p]$. Hence again, varying over all $p$ tells us that
this group vanishes; and then the dimension-shifting argument kicks
in. (Kiran Kedlaya)
At the bottom of page 154 and the beginning of page 155, replace "$1-s$" with "$s-1$" (three times), (Yiming Tang).
From Raghuram Sundararajan
They are labelled as (typographical), (mathematical) and (suggestion) for typographical/grammatical errata, mathematical errata and suggestions to make proofs easier to follow, respectively.19.56 is misstated: It should say that every split reductive group... T is a split torus.
p.60, 5.3. This is completely muddled --- the homomorphism $i$ goes in the other direction and is surjective etc. I was probably thinking of a group $G$ over $k$ (not $k^{\prime}$) etc.... (Giulia Battiston)
p.261. For an improved exposition of the proof of Theorem 5.1, see RG I, 1.29.
p.259. There is a problem is with my definition of "almost-simple". Certainly "almost-simple" should imply "semisimple", so we should define an almost-simple algebraic group to be a noncommutative smooth connected algebraic group that is semisimple (so its radical over the algebraic closure is trivial) and has no nontrivial proper smooth connected normal algebraic subgroup. The quotient of such a group by its centre should be called simple. (The centre of a reductive group is the kernel of the adjoint representation on the Lie algebra. It suffices to check this over an algebraically closed field, see RG 2.12).
If we don't require that G be semisimple, then we get to the land of pseudo-reductive groups (Conrad-Gabber-Prasad), which is very complicated. Tits defines a pseudo-simple algebraic group to be a noncommutative smooth connected affine algebraic group with no nontrivial proper smooth connected normal subgroup. (Sebastian Petersen)
From David Calderbank
Page 52, Weyl's Theorem 5.20. Part (a) states "If ad is semisimple, then g is
semisimple". However, semisimplicity of the adjoint representation is
characteristic of reductive Lie algebras (Proposition 6.2), not just semisimple
ones. The proof uses more than semisimplicity of the adjoint representation.
Page 56, Proposition 5.29. In the non-algebraically closed case, I don't think it is necessarily true that a Lie algebra with all elements semisimple is commutative. Indeed, so(3) over R would be a counterexample.
Page 57, Proposition 6.4. I don't see how the proof establishes the implication "g has a faithful semisimple representation => g is reductive".
From Timo Keller
p. 21: ... with [A,B] = AB - BA,and <- space missing
p. 25, l.-3: ) missing at the end of the equation
I, Theorem 16.21. The centre of a reductive group is of multiplicative type, but need not be reduced (e.g., SL_p). The correct statement is that the radical of G (which is a torus) is the reduced connected group attached to the centre. (There may be other errors of this type, where I incorrectly translated from the characteristic zero case to the general case.) There will be a new version of the notes probably in March 2012.