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

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.

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 month (July, 2017).

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

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.

See also: Variations on a theorem of Tate. Stefan Patrikis. arXiv:1207.6724.

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