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
Basic Theory of Affine Group Schemes
Lie Algebras, Algebraic Groups, and Lie Groups
Reductive Groups
Algebraic groups, Lie groups, and their arithmetic subgroups

No known errors.

## Fields and Galois Theory v4.40 (FT)

p45/46. The definition of the discriminant of a polynomial on p45 is only for monic polynomials. On p46 I apply it incorrectly to aX2+bX+c --- this polynomial has discriminant b2-4ac, as everyone learnt in high school. (Geir Arne Magnussen)

## Algebraic Geometry v5.22 (AG)

p66. In the statement of Proposition 3.22 on page 66, the order of the composition of f with phi should be reversed. (Cody Gunton).

p83. For the Examples 4.28 (a) and (b), the inclusions $U_0\cap U_1 \hookrightarrow U_i$ are reversed. (Felipe Zaldivar)

p84. The "above diagram" refers to the top diagram. The \phi in the second diagram should be \varphi (Isac Hedén).

No known errors.

No known errors.

## 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).

From Bhupendra Nath Tiwari: For AV, CFT, and CM

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)

p26. now shows that dim(H^1(A,IF_l)) >= 2g: ')' missing (Timo Keller).

From Tobias Barthel See pdf file (2 pages)

From Shaul Zemel In the proof of Theorem 10.15, p49, concerning the map from Hom(A,B) \tensor Z_l to the module 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.

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 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)
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..

No known errors.

No known errors.

No known errors.

## 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.261. For an improved exposition of the proof of Theorem 5.1, see RG I, 1.29.

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

No known errors.

## 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).