# Errata for Expository Notes

This file contains miscellaneous errata and additional remarks for my expository notes that I haven't yet incorporated into the versions on the web.

Corrections for SVI, SVH, and MOT from Bhupendra Nath Tiwari

No known errors.

## Motives --- Grothendieck's Dream v2.04, 2012

p.10, line 2: should read "it is possible" (the "is" is omitted).

p.15, Literature: the book of Murre et al. has now been published:
Murre, Jacob P.; Nagel, Jan; Peters, Chris A. M. Lectures on the theory of pure motives. University Lecture Series, 61. American Mathematical Society, Providence, RI, 2013.

## Introduction to Shimura Varieties (as posted on my site)

p.11. From Alexey Beshenov: On p. 11 you give example of P^1 (C) as a Hermitian symmetric space and you say "reflection along a geodesic through a point is a symmetry". I think you actually want to give a symmetry with isolated fixed points, and it could be rather the rotation of the sphere by \pi along the axis, which leaves the poles fixed.

p. 23, line 13: t is missing on the right hand side
p. 26, line 23: the pair (r,s) should be (p,q) (Christian Weiss)
p. 60, in the first display, z/\bar{z} and \bar{z}/z should be interchanged.

Lemma 5.22 This is misstated: in general, T(Q) is not closed in T(A_f) (unless (G,X) satisfies SV5) and so T(Q)\T(A_f) is not Hausdorff (hence not compact). The last step of the proof "An arbitrary torus ..." fails when T(Q)\T(A_f) is not compact. The proof of the finiteness of T(Q)\T(A_f)/nu(K) needs to be rewritten. (Bas Edixhoven)

p72. Lucio Guerberoff points out that the uniqueness assertion in Proposition 8.14 fails and that the condition (**) in Theorem 8.17 is inadequate. He writes (slightly edited): this.

Poonen has sent me a long list of comments and corrections. These will be incorporated into a new version later this year (2016).

## Shimura Varieties and Moduli

A few minor misprints were fixed in the published version. In Definition 3.7, delete "algebraic group!of"

No known errors.

## The Work of John Tate

### The following were corrected on the version on my website 03.12.12

1968a, p63. In fact, linearity fails even for two finite potent operators on an infinite-dimensional vector space. See: A Negative Answer to the Question Of the Linearity of Tate's Trace For the Sum of Two Endomorphisms, Julia Ramos Gonzalez and Fernando Pablos Romo

#### From Matthias Schuett

p.31 at least in my pdf retrieved from arXiv, there is the same k for k and its alg closure [The bars are there, but a bit weird; they are better on the copy on my website.]
p.37 in footnote 92, Carayol... is missing a comma
5.3 ( [Don't know where this got lost; it's OK on the copy on my website.]
p.45, footnote 123 the -> that
p.60 This is not [a] major result.
1966f refers to 1963a which does not exist

### The following were corrected on all versions 23.09.12.

p. 13, eq. (8), should read H^{2-r}(G, M') (Timo Keller)

p16, in the second line after the heading The Tate module\ldots'', there is a $k$ missing from one of the $A(k^{sep})$'s.

p.22, Section 2.5. The Mumford-Tate group need not be semisimple, only reductive.

There is also the following article, which, not being clairvoyant, I didn't know about: Tate, John, Stark's basic conjecture. Arithmetic of L-functions, 7 31, IAS/Park City Math. Ser., 18, Amer. Math. Soc., Providence, RI, 2011.

I should have mentioned the work of Tate on liftings of Galois representations, as included in Part II of: Serre, J.-P. Modular forms of weight one and Galois representations. Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 193--268. Academic Press, London, 1977.

Also: An oft cited (1979) letter from Tate to Serre on computing local heights on elliptic curves. arXiv:1207.5765 (posted by Silverman).

From Matthias Künzer
p. 3, l. -17: are classified by subgroups of ray class groups ?
p. 6, l. -3: x + Z
p. 8, l. -10: \sum_{\sigma\in G}
p. 9, l. -10: \hat H^r(G,C(\phi))
p. 11, l. 16: the composite I can derive from the ses in l. 14 is trivial - or?
p. 13: it seems that in the displays in l. 24, 27, 34, some modules M should be M'
p. 14, third display, 9-term exact sequence in (9): last but one term should contain H^2, not H^0
p. 17, (14) and p. 20, (17): A', not A^t
p. 17, l. -16: \phi_f
p. 18, l. 1: definition of h_{T,q(P)}(t) ?
p. 23, l. -24: "Much is known about the conjecture." - Which one?
p. 26, l. -12: months
p. 31, l. -3: a great
p. 32, (26): bracket missing on lhs
p. 37, l. -9: "Hodge 1950" - reference missing
p. 39, l. 2: endomorphism f of F
p. 39, l. -7: q is a power of p ?
p. 41, l. 5: space that is
p. 42, l. 17: natural to replace
p. 42, l. -13: groups generalize
p. 42, l. -11: an n-dimensional
p. 42, l. -11: n-dimensional commutative formal Lie group (cf. p. 42, l. -6)
p. 42, l. -1: of a p-divisible group
p. 48, l. 22, the relation for the commutator: x_{il}(rs) on the rhs
p. 49, l.1: the free abelian group ?
p. 49, l. 3,4,5: brackets {,}, not (,)
p. 49, l. -13: if a > 0 or b > 0
p. 52, l. -20: extension of fields
p. 53, l. -3: power of A(\chi,f) is in Q
p. 54, l. -18: and a character
p. 56, §9.1: What motivates the definition of a regular algebra? If I'm interested in Azumaya algebras (cf. l. -23), what leads me to regular algebras?
p. 62, l. 1, 2: "resp.", not "or" (I'd write)
p. 62, l. 9, 25: p-subgroup
p. 68, l. -14: K_2
p. 70, l. 10: GL_n