p. 17. In the proof of Theorem 4.9, "Let $c$ be an irreducible element" should be "Let $a$ be an irreducible element" (Brian Rushton).
p. 21, Example 5.7, 4th line it should be: A prime ideal is disjoint from $S_p$ if and only if it is contained in $p$. The 'in' is missing.
p. 41, under the section Tensor products of algebras second paragraph last line $(f,g)\circ i=f$ (and not $\alpha$) and $(f,g)\circ j=g$ (and not $\beta$). (Chirantan Mukherjee)
p. 62. The folklore in the footnote is apocryphal --- the Rabinowitsch of Rabinowitsch's trick is not the Rabinowitsch who became Rainich. For the history of Hilbert's Nullstellensatz, see arXiv:2309.14024. Rabinowitsch's proof is exactly our proof, but is less detailed (it is only 11 lines).
p. 81. The polynomial $F-F(x)$ (12th line from bottom) may not have coefficients in $A$ as claimed. Instead write $F(X)=\sum a_i X^i$, write $F(x)=\sum b_i x^i$ with $b_i\in \mathfrak{p}$, and consider $\sum(a_i - b_i)X^i$ instead of $F-F(x)$ (Yutaka Matsuura).
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.
Corrections from Jungin Lee for 2017 version.
In Theorem 1.16, $G$ is a connected algebraic group over $\mathbb{R}$, not $k$ (Haohao Liu)
In 5.30, I should have said that the local rings of $S$ are noetherian and regular (instead of that $S$ is locally noetherian and regular) (Qiu, Congling).
p.20 Paragraph 3, for a proof that $\Gamma$ is a lattice if $\Gamma\backslash D$ is an algebraic variety, see Proposition 2.8 of my manuscript Kazhdan's theorem on arithmetic varieties www.jmilne.org/math/articles/1984T.pdf
p.69 In Theorem 1.16, the condition should be $Z(\mathbb{Q})$ is discrete in $Z(\mathbb{A}_f)$, not $Z(\mathbb{R})$. (Condition SV5 of Introduction to Shimura varieties.)
p.23, Proof of 2.16. The pair doesn't satisfy the conditions of 2.14 (Kapil Paranjape). Assume, for simplicity, that $B$ is finite (otherwise need to pass to limits). Let $\omega^{\prime}$ be the functor \[ (X,Y)\mapsto\omega(X)\otimes_k\omega(Y)\colon\mathrm{Comod}_{B}\times\mathrm{Comod}_{B} \rightarrow\mathrm{Vec}_{k}. \] First show that there is a canonical isomorphism $\mathrm{End}(\omega^{\prime})\simeq(B\otimes_k B)^{\vee}$ (cf. Saavedra 1972, II, 2.5.1.1). Now let $\phi$ be a functor $\mathrm{Comod}_{B}\times\mathrm{Comod}_{B}\rightarrow\mathrm{Comod}_{B}$ such that $\phi(X,Y)=X\otimes_{k}Y$ as a $k$-vector space. Such a $\phi$ defines a homomorphism $\mathrm{End}(\omega)\rightarrow\mathrm{End}(\omega^{\prime})$, i.e., a homomorphism $B^{\vee}\rightarrow(B\otimes_k B)^{\vee}$, and hence a homomorphism $u\colon B\otimes_k B\rightarrow B$. Now $\phi\mapsto u$ is inverse to $u\mapsto \phi^{u}$.
p.30. The sentence "For any $R$-algebra ..." in the proof of Theorem 3.2 requires clarification (Parul Keshari).
Remark 6.19 should read: The proposition shows that the category of Artin motives over $k$ is equivalent to the category of sheaves of finite-dimensional $\mathbb{Q}$-vector spaces with finite-dimensional stalk... (Julian Rosen).
p.67. In the definition of fibred category, $F$ should be in the category $F_{V}$, not $F_{U}$ (Parul Keshari).
In 3.1, replace $|j|<1$ with $|j|>1$ (twice).
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