Errata for Course Notes - J.S. Milne, Top
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!

Group Theory
Fields and Galois Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Complex Multiplication
Algebraic Groups
Basic Theory of Affine Group Schemes
Lie Algebras, Algebraic Groups, and Lie Groups
Reductive Groups
Algebraic groups, Lie groups, and their arithmetic subgroups

## Group Theory v3.13 (GT)

p.40, Exercise 2-5. $Q_n/Z(Q_n)$ =‹$a,b | a^{2^{(n-1)}}, (a^{2^{(n-2)}})b^{-2}, abab^{-1} \textrm{ and } a^{2^{(n-2)}}$›
=‹$a,b | a^{2^{(n-2)}}, b^2, abab$›
$=D_{2^{(n-2)}}$ (not $D_{2^{(n-1)}}$)
or, order on left side is $(2^n)/2=2^{(n-1)}$ while that on right side is $2^n$ so they don't fit. (Yunghyun Ahn)

No known errors.

No known errors.

## Algebraic Number Theory v3.06 (ANT)

On p.118 of Algebraic Number Theory (not Class Field Theory) two lines from the bottom of the page it says $(\sum a_i)/(p-1) \leq (log n)/(log p)$. I think you need to add 1 to the right hand side to make it correct. For example, take $p=2$, $n=7=111_2$ in base 2, then left hand side of the inequality is $3$ while right hand side is $log 7/log 2 = 2.8$. (Yogesh More)

From Enis Kaya
Page 118, in the tenth line from the top, '=' should be '$\equiv$'. Also, in example 7.54, I'm implicitly assuming that $K$ is nonarchimedean.
Page 137, 'Let | | be the normalized absolute value on L corresponding to | |.' should be 'Let || ||...'
Page 157, in the solution of problem 7.5, '... quadratic extensions of $Q_2$ are $Q[\sqrt{a}]$' should be '... $Q_2[\sqrt{a}]$'.
In the definition of Newton polygon there is no $a_0$, but $P_0=(0,ord(a_0))$.
Proposition 7.64: '... finitely many totally ramified extensions of $Q_p$' should be '... of $K$'. The proof should read: '... only finitely many totally ramified extensions ... '
On page 119, in the third line from from the bottom, you used '='.
In remark 8.12, you said '... for any lifting $h_1$ of $g_1$ to $Z[X]$.'
On page 143, there are two mistakes: in the fifth line from the bottom, the expression $(-1/p)^{(p-1)/2}$ should be $(-1/q)^{(p-1)/2}$. Also in the second line from the bottom, you mean the value of the function is $-1$ if $p\equiv -1 mod 4$.
On page 145, the factorization of $f$ should be '$f(X)=f_1(X)...f_s(X)$' (you should correct both)
On page 151, in the proof of theorem 8.42, you used the square in a wrong place.
On page 147, in corollary 8.32, I think you meant '... primes of $K$ that split in $L$'.

## Modular Forms and Modular Functions v1.30 (MF)

p6 you said '... trivally on on $\mathbb{H}$ (Enis Kaya).

p10 The equality "$\Lambda(\tau)=\Lambda(\tau')$" should be interpreted as saying that the two lattices are equal up to multiplication by a nonzero complex number --- see here.

p.14, proof of 1.2, third paragraph
As Keenan Kidwell and Tony Feng have pointed out, I muddle the proof. I should say: $G$ is a union of the interiors of the sets $gV$, $g$ in $G$; fix a countable base; the sets from the countable base contained in the interior of some $gV$ form a countable cover of $G$; now we only need to take enough $g$'s to get each of the sets in the countable cover at least once.

Keenan Kidwell points out, the sets $gV$ are not open, hence not unions of open subsets. Fortunately this is easily fixed. As he writes:

In the proof of Proposition 1.2 in your exposition of modular forms and modular functions (beginning near the bottom of page 14), is it really possible to assume that the set $V$ is both open and compact? Initially I assumed that, because you referred to $V$ as a neighbourhood of e, you meant that it was a set which contained e in its interior, but later you say that each set $gV$ is a union of open sets in the countable base, which implies that $gV$ is in fact open. For a general LCH group, we have a base of opens around each point consisting of sets with compact closure, but not necessarily open compact sets, right? But I think this is fixed fairly easily, since g is in the interior of $gV$, we have an open cover $G=\bigcup (gV)^{int}$, and by second-countability, this has a countable subcover, and so in particular we can write $G=\bigcup g_n V$ for some sequence $g_n$, and then the proof goes through unchanged. So I think the only problem is when you say that each $gV$ is a union of open sets.

Tony Feng points out:
In the third paragraph of the proof of Proposition 1.2, I was thinking that not every open set in the countable base need be contained in a single $gV$, but of course we can throw away such open sets from the base.
p.15, lemma 1.3 From Tony Feng:
In the proof of Lemma 1.3, I am not sure why $V_1 \cap U_1$ would be compact. I agree that it is relatively compact/closed, which is enough to get the desired $U_2$.

From Francesc Gispert Sánchez
Page 17 In paragraph 1.7, $U$ should be an open set of $X$ (not of the complex plane). Moreover, the codomain of the function $f$ should be the complex plane (the $C$ which appears on the notes is not the symbol $\mathbb{C}$).

Page 33 In the proof of theorem 2.12, where the case $c \geq 2$ is discarded, it should say "because $y^2 \geq 3/4$" (instead of 3/2), because the minimum imaginary part is $\sqrt{3}/2$ (attained at $\rho$).

There is a sentence which I do not understand and I suspect that there might be a mistake. In the second paragraph of remark 2.25 (page 39), you say that, since the surface $X(2)$ has genus $0$ and three (non- equivalent) cusps, it is isomorphic to the complex plane minus two points. But, since it is a compact surface and has genus 0, isn't it isomorphic to the whole Riemann sphere? Maybe you meant $Y(2)$ (the surface without the cusps), which then would be isomorphic to the Riemann sphere removing three points. --- Yes, I meant $Y(2)$.

The letter 'H' used to describe the upper half-plane is not $\mathbb{H}$ in the following places: in page 36, in the second sentence after the section title ("Write $p$ for the quotient map $\mathbb{H} \to \Gamma(1)\backslash H$"; this last 'H') and in the next-to-last sentence ("Alternatively, we can consider ..."); in page 37, in the first paragraph (after "Clearly"); in page 38, in the first sentence ("... let $\phi$ be the map ...").

In page 33, after the proof of lemma 2.13, I think that the position of the two QED symbols is wrong: one of them is at the end of the next-to- last sentence of the lemma. One should be after the last sentence in the lemma (closing the proof of the lemma) and the other one immediately afterwards (closing the proof of theorem 2.12).

In the statement of corollary 3.6 (page 43), a ring $R$, which is not defined before (unless I am missing something; maybe it is just a standard notation which I am not aware of), is mentioned. From the proof, I deduce that it should be $\textrm{End}(\mathbb{C}/\Lambda)$. Is my guess correct? --- Yes.

In page 45, I think it could be helpful to other readers to rephrase the first paragraph of the section "The addition formula". In particular, the part "... and therefore it is a rational function of $\wp$ and $\wp'$.", since this is proved later in the text (in page 46). Something along the lines "... and we will prove later that ..." would do the job.

In remark 4.8 (page 51), I think that the sentence "Thus $\omega$ is invariant if and only if $f(z)$ is a meromorphic differential form of weight 2" should be changed to "... is a meromorphic modular form of weight 2" (because f is a function, but not a differential, and we want it to satisfy the modularity condition).

In page 56, when each term in the expression of $z\cot(z)$ is expanded as a geometric series (i.e., when the index $k$ is introduced to obtain a double sum), the sign is wrong. That is, $1 - \sum_{n=1}^{\infty}\frac{2t_n}{1-t_n} = 1 - 2\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} t_n^k$ where $t_n$ would be $\frac{z^2}{n^2\pi^2}$. (Notice the minus sign before the 2.)

In definition 4.23 (page 59), the symbol $\mathbb{H}$ is used and I am not sure if this is what was intended. In the previous discussion (the paragraphs immediately preceding the definition), there is a Riemann surface $H$ (not necessarily the upper half-plane), and I guess that the definition of automorphy factor is valid in this situation in general.

At the end of page 59, when the "chain rule" is stated, the "subindex" (the point where the tangent map is computed) $\alpha(n)$ should be $\alpha(m)$.

In page 65, the Poincaré series of character 0 is defined as $\phi_0$, but in the rest of the section it is referred as $\varphi_0$ (a different symbol).

In page 72, there is a sentence which seems confusing to me: "because $\Lambda/mn\Lambda$ is the direct sum of a group of order $m$ and a group of order $n$". I think this is wrong, for it is a group of order $(mn)^2$ and not $mn$.

In the statement of corollary 5.17, part (a) (page 76), $\gamma(1)=c(m)$ should be $\gamma(1)=c(n)$ ($n$ is used for the index of the Hecke operator).

At the end of page 81, in the right-hand side of the definition of $\gamma(m)$, there is a pair of parenthesis which are probably too small (or the fraction inside is too big) and, just after this, there is a comma misplaced (there is a space before the comma but not after it; it should be the other way round).

In page 84, where the action of a matrix on a lattice is defined in terms of a basis, there is a closing parenthesis which was not opened before: "... and of the choice of a representative for the double coset $\Gamma\alpha\Gamma$.)"

1) On page 58 at the end of the proof of Theorem 4.21, change "z = 1" to "z = i."
2) On page 64, on the line after the proof of Lemma 4.32, the "2k" needs k in italics.
3) On page 84 line 7 put "ad" in italics.

From Tony Feng: I think I've identified a small typo in your modular forms notes that was tripping me up for a while.
On page 65, in the right hand side for the inner product $\langle f, \varphi_n\rangle$ I think the $\exp(-2 \pi i n z/h)$ in the integrand should be $\exp(2\pi i n (-x+iy)/h)$, or something equivalent: it's the complex conjugate of $\exp(2\pi i n z/h)$, but $z$ is complex!

## Abelian Varieties v2.00 (AV)

This draft is still very rough. Some proofs have been fixed in the corrected version of my 1986 article on Abelian Varieties (xnotes).

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)

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.]

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)
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 v2.21 (LEC)

From Section 9 on, I often abbreviate $H^i(X_{et},F)$ to $H^i(X,F)$ (Hongbo Yin).

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?

## Class Field Theory v4.02 (CFT)

p.2, l7, Replace $X^2+m$ with $X^2-m$ (Jack Petok).

p. 20, towards the bottom of the page, when talking about transitivity of norms, it is stated that $Nm_{L/L} = Nm_{L/K}\circ Nm_{L/L}$. The left hand side should be $Nm_{L/K}$. (Andrew Kirk)

p.154. Theorem 3.6 (Existence theorem). The theorem should say that there exists an $L/K$ unramified at the primes not dividing $m$ etc., otherwise the statement doesn't really make sense. (Li Jianing)

From Yogesh More
p.4 Theorem 0.3 ..abelian extension of $K$ arises as the class group of some subgroup of $C$ group should be field
p.6 Example 0.6 A couple of lines later you use the term totally ramified . It would be nice to recall the definition --- on p. 2 you defined ramify, but not totally ramified.
p.7 line 1 $\sigma \alpha \cong \alpha^q$ it would be good to indicate that the congruence is mod $P$
p.146 proof of Prop 1.6 nonzero $c \in \mathfrak{c} \cap K^S$ should $K_S$ be $K_{m, 1}$? I m not a 100% sure, but I think so to create $C_{m,1}$ you mod out by $K_{m,1}$
Top of p.151 On applying log to the equality in (a) I guess you are also using the approximation $-log(1-x) \sim x$? If so, add that to help the reader.
p.155 Example 3.11 We compute the conductor of $K=Q[\sqrt{d}]$ earlier on the page you define the conductor as something assigned to an extension $L/K$. So what is $L$ in this example? [Should be the conductor of $Q[\sqrt{d}]/Q$.]

From Kiran S. Kedlaya
On page 115 of version 4.02, I found footnote 4 a bit confusing: it seems to be arguing that given a family of sets with empty intersection, the images under a map also have empty intersection. Of course this works in this case because of the topology in the picture, but I found it clearer to reinterpret this argument without taking quotients: since $I$ is open and the intersection of the compact sets $N \cap U_K$ is contained in $I$, the sets $(N \cap U_K) - I$ themselves form a family of compact sets with empty intersection. Hence some finite subfamily has empty intersection, and so on.

In II, Proposition 1.34, p.69, there is no explicit mention of the restrictive hypothesis on the vanishing of the $H^i(H,M)$, which might be confusing for the reader who has never seen a spectral sequence before. It of course gets used to deduce that $0\to M^H \to M_*^H \to M_{\dagger}^H \to H^1(H,M) = 0$ is exact, in order to finish the dimension shift.

No known errors.

## Algebraic Groups v1.00 (iAG)

19.56 is misstated: It should say that every split reductive group... T is a split torus.

## Basic Theory of Affine Group Schemes v1.00 (AGS)

p.30. The definition of a sub-coalgebra doesn't make sense unless, e.g., k is a field, because in general D\otimes D is not a submodule of C\otimes C. (Roland Loetscher).

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 Bhupendra Nath Tiwari: list of misprints for AGS, LAG, RG odt; doc

## Lie Algebras, Algebraic Groups, and Lie Groups, v2.00 (LAG)

p.15. In the final paragraph before "The isomorphism theorems", it states that if a is a characteristic ideal of g, then every ideal in a is also an ideal in g. It should read: If a is an ideal in g, then every characteristic ideal in a is also an ideal in g (which is characteristic if a is characteristic).

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".

## Reductive Groups, v1.00 (RG)

p.14, following the statement of 1.29: The centre (and radical) of GL_n consists of the scalar matrices, not the diagonal matrices. (Justin Campbell)

From Timo Keller
p. 21: ... with [A,B] = AB - BA,and <- space missing
p. 25, l.-3: ) missing at the end of the equation

## Algebraic groups, Lie groups, and their arithmetic groups v3.00 (ALA)

p. 340 footnote 4. From Brian Conrad: you point out that a root datum is really an ordered 6-tuple, not an ordered 4-tuple. In case it may be of interest, the last bit of information in the 6-tuple (the bijective map a |---> a^{\vee} from roots to coroots) is uniquely determined by everything else, due to the requirements in the axioms of a root datum -- see Lemma 3.2.4 in the book "Pseudo-reductive groups".

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.

p. 372 footnote 16. Chapter II?? should be Chapter III.
p. 172, l.1: "and so is split be a finite" should be "and so is split by a finite" (Timo Keller).