Errata

This file contains miscellaneous errata and additional remarks for my course notes that I haven't yet incorporated into the versions on the web.
Most are taken from e-mail messages -- I thank everyone who has contributed!

Fields and Galois Theory
Group Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Algebraic groups and arithmetic groups
Algebraic groups, Lie groups, and their arithmetic subgroups

Fields and Galois Theory v4.20 (FT)

p35, 3.2. As Maren Baumann pointed out, there is no need to assume that f is monic (here, and probably in other places).

From Kwangho CHOIY
p58, 5 lines below from Kummer theory; I think that, in the definition of 'exponent n', "the smallest positive integer for which this is true" can be removed.

Group Theory v3.01 (GT)

From Victor Petrov
Example 7.14 on p.95 isn't quite correct. Every left ideal in $M_n(F)$ is conjugate to $L(I)$ under the action of $GL_n(F)$, but isn't equal to $L(I)$ in general. Consider, say, the left ideal of matrices whose columns are all the same.
[Yes, I overlooked that imposing a linear relation on the columns of L(I) also gives a left ideal. Perhaps it is best to say that, given a left ideal A of End(V), there exists a basis for V relative to which A=L(I) for some I.]

Efthymios Sofos points out that in (5.19) on groups of order 60, s_2(A_5)=5 and s_2(A_5)=15 are contradictory. In fact, the case s_2=15 doesn't occur.

Algebraic Geometry v5.10 (AG)

p154, 9.20 Lars Kindler points out that, in the proof of 9.20, it is not obvious that the map \alpha\circ\pi is given globally by a system of polynomials (rather than just locally). It is in fact given globally, and this is not too difficult to prove: a regular map from a variety V to P^n corresponds to a line bundle on V and a set of global sections, and all line bundles on A^n are trivial (see, for example, Hartshorne II 7.1 and II 6.2). [I should include all this in the next version.]

From Soli Vishkautsan
Page 4 line -1, should be the intersection of all the ideals containing S (not A).
Page 8, In proof of Thm 1.8. there seems to be some confusion, The S_m are subsets of A[X_1,...,X_r] and not A, therefore the ideal a is of A[X_1,...,X_r] which you didn't prove is noetherian (yet), so you can't use lemma 1.7. Also you say f = g_1*f_1 + .... + g_s*f_s where g_i are in A, but they should be in A[X_1,...,X_r]. Hope I am not missing something. [Yes, the g_i should be in the polynomial ring.]

Algebraic Number Theory v3.02 (ANT)

No known errors.

Modular Forms and Modular Functions v1.10 (MF)

From Ulrich Goertz:
pdf (2 pages)

From Nousin Sabet:
I would like to make a comment about the Exercise 2.24, page 33, which is not correct. For example, if instead of $\Gamma(N)$, one takes $H=\Gamma_0(p)$ and $x_0=\infty$, then $[G:H]=p+1$ and $[G_\infty:H_\infty]=1$, and the formula of that exercise gives us $p+1$ for the number of inequivalent cusps for $\Gamma_0(p)$. However, we know that $\Gamma_0(p)$ has only 2 cusps.
[The exercise is correct if one requires H to be normal: Let Y be the set of orbits of H in X. Then G acts transitively on Y and the stabilizer of Hx_0 is HG_0.
Hence the number of orbits is (G:HG_0)=(G:H)/(HG_0:H)=(G:H)/(G_0:H_0).]

Abelian Varieties v2.00 (AV)

This draft is still very rough.

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 Storr's 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.

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..

Lectures on Etale Cohomology v2.10 (LEC)


p16. Proposition 2.1 should say that the projection map W-->V is etale, as is clear from the line 4th from the bottom of the proof. (Rex Cheung)
21.7. In the diagram, the arrow for i points the wrong way.

From Eric Moorhouse
p156 The expression appearing in the middle of the page (bounded by two powers of $q$) should be $|\alpha|$.

Class Field Theory v4.00 (CFT)

p2, line 4. I should have said that alpha is an algebraic integer (so f is a monic polynomial with coefficients in O_K. Then
disc(f)=disc(O_K[alpha]/O_K)=(O_L:O_K[alpha])^2 x disc(O_L/O_K)
(see 2.25). Therefore, if a prime is ramified in L, then it divides disc(f), but there may be primes dividing disc(f) that are not ramified in L. For example, let L be the the quadratic field generated by a square root of m. If m is congruent to 1 mod 4, then 2 divides the discriminant of f but is not ramified in L.
p119, Example 1.7, Chapter IV. I overlooked that imposing a linear relation on the columns of L(I) also gives a left ideal. It is best to say that, given a left ideal A of End(V), there exists a basis for V relative to which A=L(I) for some I. (See GT.)

Herve Jacquet has pointed out that Statement 3.3 page 106 does not (appear to) follow directly from the definition of the local Artin map and formula (30), as I claim, but that it can be proved using (3.6). I don't know right now whether it can be proved without using (3.6).

From Corinne Sheridan:
there is a mistake on page 154, in the paragraph before Theorem 3.5 (Reciprocity Law). On both lines 2 and 3, it says that a homomorphism admits a modulus if there exists a modulus m such that S(m) is contained in S .... It should say S(m) contains S. [Yes, if the map admits a modulus m and m|n, then it also admits the modulus n.]

From Jonah Sinick
p223 (first page of Chapter VIII): "On may hope..." should be "One may hope..."

From Yu Zhao
Page 97, 3rd line, $v_i \in U_K^{(i)}$ should be $v_i \in U_L^{(i)}$ because one wants an element in $U_L$ to get the surjectivity.

From Kwangho CHOIY

p3, 3 lines below from section "QUADRATIC EXTENSION OF Q"; I don't think "A prime number q(not equal to p) splits in Q[square root of p*] if and only if p* is a square module q" is true.
For p*=p=5 and q=2, 5 is a square modulo 2, however 2 does not split in Q[square root of 5]. In other words, the ideal (2) is ramified in Z[square root of 5] as (2, 1+square root of 5)^2, whereas (2) = (2) in Z[(1+square root of 5)/2], i.e. it is unramified(not split) in Z[(1+square root of 5)/2]. Because the minimal polynomial of (1+square root of 5)/2, X^2 -X^-1 is irreducible modulo 2, but but that of (square root of 5), X^2 -5 factors into (X^2 +1)^2 modulo 2.
Nevertheless, I think the statement, "A prime number q(not equal to p) splits in Q[square root of p*] if and only if p* is a square module q", holds if q is an "odd" prime.(We already assume that p is an odd prime as well as p is distinct from q). Because, in this case, "the minimal polynomial of (1+square root of p*)/2, X^2 -X^+(1-p*)/4 factors into 2 distinct linear factors modulo q" if and only if "the minimal polynomial of (square root of p*), X^2 -p* does", and of course, if and only if "p* is a square modulo q". This equivalence is based on the fact "p and q are distinct 'odd' primes".
p4, bottom line; totally real -> totally "positive" real
p5, line12; Theorem 0.2 -> theorem 0.4
p5, line15; m|m' -> m'|m
p7, 2 lines above from the section "L-SERIES"; /The Frobenius density theorem/ can be replaced by /The "Chebotarev" density theorem/ [True, but the Frobenius theorem is older and much more elementary than the Chebotarev theorem.]
p7, 4 lines above from the bottom; in the equation of L(s,kai), /kai(Na)^(-s)/ can be replaced by /kai"(a)"*N(a)^(-s)/
p7, 2 lines above from the bottom; /an ideal class group/ can be replaced by /the "ray" class group/.
p9, 4 lines above from the bottom; /in which every prime of K/ can be replaced by /in which every prime of K "outside S"/
p16, Theorem0.9; /L=K[alpha]/ can be replaced by /"B=A[alpha]", and we need to rearrange sentences in this Theorem(see ANTv.3.00 theorem3.41).
p19, 2 lines above from the bottom; /sigma(alpah)=(alpha)^q for all.../can be replaced by /sigma(alpah)=(alpha)^q "mod m_L" for all.../
p43, 4 lines below from a section "THE RAMIFICATION GROUPS..."; /i>=0/ can be replaced by /i>=1/.
p97, 6 lines below from the section "THE INVARIANT MAP"; "H^2(G,L*) -> H^2(G,Z)" instead of 1st cohomolgies, i.e. both H^1's must be H^2's.
p105, Theorem3.1; in the front H^r(Gal(L/K),L*), L*-->Z.
p108, proposition4.1; I think that we need to give a degree of the cyclic extension of K. So, I think that we can add "of degree dividing n" after "every cyclic extension of K" in the sentence.(I referred to Serre's Local Fields)
p109, 8 lines below from the section "THE GENERAL CASE"; you wanted to have the sequence 0 -> Z -> Z -> Z/nZ -> 0 where the map Z -> Z is 'n', instead of 0 -> Z/nZ -> Z -> Z -> 0.
p171, line2; "phi(id(a))" -> "psi(id(a))"
p173, bottom line; (idele) I_L -> (idele) I_K

p5, line 15; I^S(m) imbedding to I^S(m') can be replaced by I^S(m') imbedding to I^S(m) m'|m, but according to the other statements below this is correct, instead of m|m' -> m'|m.
p100, 3 lines from the bottom; L* --> Nm(L*)
p186, the bottom line; (zeta)_L,S(s) = (zeta)_K,T(s)^[L:K] --> (zeta)_L,T(s) = (zeta)_K,S(s)^[L:K]
p213, the first line below section "7 Application to the Brauer Group"; K^(a...) --> K^(al)*

The followings are things that you wanted to add "finite", I think;
p4, line1; An unramified abelian - > An "finite" unramified(abelian),
p4, theorem0.3; every unramified abelian - > every "finite" unramified (abelian),
p4, just below theorem0.3; an unramified abelian - > an "finite" unramified(abelian),
p5, theorem0.4; every abelian -> every "finite" abelian,
p153, Corollay3.4; For any abelian extension -> For any "finite" abelian extension,
p157, line2; the abelian extensions -> the "finite" abelian extensions
p157, Theorem3.16; a Galois extension -> a "finite" Galois extension.

p5, Example0.6; we should assume all (p_i)* is positive and also we should let -d=(p_1)*...(p_t)* instead of d=(p_1)*...(p_t)*, in order to say both that "K is ramified exactly at the prime ideals (p_1), (p_2)....,(p_t)" and that "K[sqrt (p_i)*] is unramified over K". But I don't think that this is a matter with the result about the narrow class group in the example.
p148, Example1.8(c); I think that in the first short exact sequence, the kernel {-1, 1} depends on the modulus (m). For example, if m=2, U_m,1 is equal to {-1, 1}, thus the kernel is {1}. Also, in the trivial case m=1 the kernel is equal to {1}. The case m=1 also belongs to the second exact sequence.
p185, Example2.13(b); in the formula, you wanted (zeta)_K instead of just (zeta).
p192, line 1 in the proof of Theorem4.9; you wanted i(K_m,1) instead of K*.
p196, (c); you wanted u_E/L instead of u_L/K.
p206, in the proof of Corollary4.8; you wanted P|p not in S.
p228,Remark3.2; you wanted "5^2, 7^2, 10^2, ....." instead of "2".(See Exercise5.3 pp360, Cassels-Frohlich)
p238, in the middle of the section THE POWER RESIDUE SYMBOL; you wanted (zeta)^((q-1)/n) instead of u^((q-1)/n).
p238, 2 lines above (5.1), (5.1) itself, (5.2); you wanted p not in S(a)

Algebraic groups and arithmetic groups v1.01 (AAG)


11.34. The last sentence should read: In nonzero characteristic, the only connected algebraic groups whose representations are all simple are the tori.
Section 23, p167. In the definition of Phi^+ and Phi^-, need to take the intersection with Phi.
In the definition of P^+, need to replace Phi with Phi^{\vee}

From Victor Petrov:
p110...the point-wise definition of normalizers and centralizers of algebraic subgroups as in Proposition 13.18 (p. 110) seems to be not quite correct. At least both Demazure, Gabriel in "Groupes algebriques" (Proposition II.1.3.5) and Demazure, Grothendieck in SGA 3 (Expose I, Definition 2.3.3) define the value of the normalizer of a subgroup $H$ in a group $G$ at a ring $R$ to be the set of all elements $g\in G(R)$ such that $gH(S)g^{-1}$ is contained in $H(S)$ for any ring extension $S$ over $R$ (and not just for $R$ itself as written in your notes).
The criticism is correct: the definition I give in 13.18 is only valid when H(k) is Zariski dense in H, which is true if H is connected, k is infinite, and either H is reductive or k is perfect (Borel 1991, 18.3). In general, H(k) might be too small, and so its centralizer and normalizer in G(k) might be too big.
p131. The claim that the map GL_2(R)->PGL_2(R) is surjective for all rings R is incorrect. The map is surjective if all f.g. projective R-modules of rank 1 are free, e.g., if R is local. However, let R be a Dedekind domain whose class number is divisible by 2. Then there is a f.g. projective R-module L of rank 1 such that L+L is free (cf. ANT 3.33), and the etale (or Amitsur) cohomology sequence of 1->Gm->GL_2->PGL_2->1 shows that GL_2(R)->PGL_2(R) is not surjective.

From Xiandong Wang:
The proof of LEMMA 18.3(b) (formula (68)) has some inaccuracy. This can be checked by using (65) and the definition directly.

Algebraic groups, Lie groups, and their arithmetic groups v1.01 (BAG)

From Darij Grinberg
Example 2.14. This needs to be fixed. For instance, over Z/4Z, the line (1,2,-2) has 2nd and 3rd elementary symmetric polynomials equal to 0, but is not monomial.
In Exercise 3-4, I would write (Z/pZ)_k instead of Z/pZ in accordance to your notations.
Example 4.1: "Let $\mathfrak{a}$ be an ideal $k[X_1,...,X_n]$" - there is an "in" missing here.
In Example 4.3, you are speaking of a regular map V -> A^1. You are probably abbreviating (V, O(V), alpha) by V here, but you only introduce this abbreviation further below.