・Added an Appendix to the paper concerning the portion of the theory of complex multiplication that occurs in
[Mzk20], Remark 3.8.1
・Corrected a misprint in \S I1 ("philsophical" ---> "philosophical")
・Replaced the word "homeomorphism" by the word "self-homeomorphism" (4 instances)
・Deleted the wording "Finally," from the final paragraph of \S I1 and added a paragraph to \S I1 concerning the Appendix
・In the first and fifth paragraphs of \S I3, replaced inclusions of the positive real numbers into the multiplicative
group associated to a complex archimedean field by surjections onto the positive real numbers (3 instances)
・Slightly modified the second sentence of the third paragraph of \S I3
・In \S I4, put the fifth occurrence of the term "observable" in boldface
・Added a supplementary explanatory phrase concerning the phrase "isomorphism of underlying fields" to the discussion
of contact structures in \S I4
・Corrected a misprint towards the end of \S I5 ("from point of" ---> "from the point of view of")
・Corrected a minor oversight in condition (a) of the definition of a "nexus" in \S 0 (added the word "nonempty")
・Corrected a misprint in \S 0 ("strictly speaking" ---> "strictly speaking,")
・Added a sentence ("Let D be a divisor on X.") to the beginning of Proposition 1.1
・Added a copy of the initial portion of Proposition 1.2 to the initial portion of Proposition 1.3
・In Definition 1.5, moved the definition of the subgroup H from (a) to the initial portion of Definition 1.5
・Corrected slight inaccuracies in Remark 1.5.3, (i), (ii), concerning \mu(k)
・Corrected two misprints in Remark 1.5.3, (ii) (", and, moreover," ---> "and, moreover,"; "prime of residue" --->
"prime of k_0 of residue")
・Corrected a minor oversight in the first sentence of the statement of Proposition 1.8 ("1.6" ---> "1.6, (iii)")
・Added a supplementary explanation concerning the term "cuspidal principal divisors" at the beginning of Proposition 1.8
・Corrected a minor oversight in the statement of Proposition 1.8, (i) (by introducing a positive multiple of \eta)
・Corrected a misprint ("reconstructions" ---> "reconstruction") in the introductory sentence preceding Theorem 1.9
・Slightly modified the statement of Theorem 1.9, (a) ("(a), (b)" ---> "(a), (b), (c)")
・Slightly modified the final portion of the first sentence of Corollary 1.10, (a)
・Slightly modified the final portion of Remark 1.10.3, (i)
・Corrected an inaccurate expression ("closure of" ---> "closed subgroup topologically generated by") in Theorem 1.11, (a)
・Slightly modified the final portion of the third sentence of Definition 2.1, (i)
・Corrected a misprint in Definition 2.1, (i) (", and, moreover," ---> "and, moreover,")
・Corrected a misprint in the final sentence of Definition 2.1, (iii) ("X^top" ---> "Z")
・In Corollaries 2.4, (b); 2.7, (d), added supplementary references to Corollary 2.3, (i)
・Corrected a misprint in Remark 2.4.1 ("is" ---> "arises from")
・Corrected a misprint at the beginning of the statement of Proposition 2.5 (switched "parallelograms" and "squares")
・Slightly strengthened the initial portion of the statement of Proposition 2.5 ("abstract topological space" --->
"abstract set"; added a brief supplementary explanation)
・Slightly modified the wording of Corollary 2.7, (b), immediately following the display
・Corrected a slight omission in Corollary 2.7, (d) ("that fixes p")
・Deleted 3 (unnecessary) instances of the word "nonconstant" from the statement of Corollary 2.8
・Corrected some notational oversights in the final portion of the statement of Corollary 2.8, (b); Remark 2.8.2
・Slightly reworded the initial portion of Remark 2.8.3
・Corrected a misprint in Definition 3.1, (i) ("ind-monoids" ---> "ind-topological monoids")
・Throughout the paper (starting with Definition 3.1, (ii)), replaced the term "arithmetic datum" by the term
"arithmetic data" (in order to avoid the confusion arising from the issue of deciding which term one should use)
・Slightly modified the phrasing of the explanations following the final displays of Definition 3.1, (iv), and
Definition 4.1, (iv)
・In the statement of Proposition 3.2, (i), replaced the "---" surrounding "where" by "[ ]" (to avoid confusion)
・In the statement of Proposition 3.2, (v), deleted the first sentence, replaced the "MLF"'s in the display by
"MLF-sB" (3 instances), and added a brief supplementary explanation
・Corrected a misprint in the proof of Proposition 3.2, (iv) ("Proposition 2.3, (i), (ii)" ---> "Proposition 2.3, (ii)")
・Added some supplementary explanatory phrases to the final sentence of the proof of Proposition 3.2, (iv)
・Corrected a minor technical oversight in Definition 3.5, (ii), (b) (i.e., added a condition concerning pre- and post-
composition)
・Corrected a minor technical oversight in Definition 3.5, (iv), (b)
・Slightly modified the reference to Proposition 3.2, (v) ("3.2, (v)" ---> "3.2, (iii), (v)") in the statement of Corollary 3.6
・Added a reference to Proposition 3.2, (ii), (iii), in the statement and proof of Corollary 3.6, (ii)
・Shifted the position of the dots on the right-hand side of the third to last displays of item (ii) of Corollaries 3.6, 4.5, 5.5
・Slightly modified the second to last display in the statement of Corollary 3.6, (ii) (as well as the corresponding
portion of the proof of Corollary 3.6, (ii))
・Corrected a misprint in the statement of Corollary 3.6, (iii) (", and, moreover," ---> "and, moreover,")
・Slightly modified the wording of the proof of Corollary 3.6, (ii) ("observe that the evident" ---> "observe that the")
・In Remarks 3.6.1, (i); 4.5.1, (i), slightly modified the wording of the explanation following the first display
("where" ---> "for")
・Slightly modified the wording of the first sentence of Remark 3.6.2, (i)
・Added a reference to Proposition 3.3 in Remark 3.6.4
・Corrected misprints in the definition of D^\ast in Corollary 3.7, (ii)
・Slightly modified the second to last display in the statement of Corollary 3.7, (ii)
・Corrected a misprint in the statement of Corollary 3.7, (iii) (", and, moreover," ---> "and, moreover,")
・Corrected a misprint in the notation at the end of the second sentence of Corollary 3.7, (v) ("double dagger" --->
"asterisk")
・Slightly modified an expression in Remark 3.7.3, (i) ("everything we do" ---> "everything that we do")
・Corrected a misprint in Remark 3.7.4 ("tries to tries to" ---> "tries to")
・Corrected slight notational inconsistencies in the first display (and accompanying explanation) of Remark 3.7.5, (i),
as well as in Remark 3.7.5, (ii), (1)
・Slightly modified the wording of the display (and the explanation immediately following the display) of
Remark 3.7.6, (iii)
・Put the phrase concerning "basepoints" in Remark 3.7.7, (i), in boldface
・Slightly modified the first display of Remark 3.7.7, (i)
・In Definition 4.1, (i), added a reference to the definition of the term "elliptically admissible"
・Corrected a misprint in Definition 4.1, (iii) ("for subcategories determined" ---> "for the subcategories determined")
・Corrected a misprint in Definition 4.1, (iv) ("composing the natural map ..." ---> "compositing the composite of
natural maps ...")
・Slightly modified the final sentence (by adding "="'s) of the statement of Proposition 4.2, (i); made corresponding
modifications to the proof of Proposition 4.2, (i), as well as to the first display and the explanation following the
second display of Corollary 4.5
・Corrected a misprint in the statement of Proposition 4.2, (ii) ("determine" ---> "determines")
・Corrected slight inaccuracies in the phrasing of the portion of the proof of Proposition 4.2, (i), concerning
categories of localizations and cores
・Slightly modified the statement of Lemma 4.4 (modified the definition of \alpha)
・Slightly modified the wording of the explanation following the second display of Corollary 4.5, (ii)
・Slightly modified the second to last display in the statement of Corollary 4.5, (ii)
・Corrected a misprint in the statement of Corollary 4.5, (iii) (", and, moreover," ---> "and, moreover,")
・Added a supplementary reference fo Definition 4.1, (i), (ii), (iii), to the explanation following the first display
of the proof of Corollary 4.5; slightly modified a similar reference to Definition 4.1 in Remark 4.5.1, (ii)
・Added a supplementary explanation concerning the item (4) in the proof of Corollary 4.5, (iii)
・Corrected a misprint in the final portion of the proof of Corollary 4.5, (iv) ("\zeta'_2=id" ---> "\zeta'_1=id")
・Corrected a misprint in Definition 5.1, (ii) (missing accent over the "e" of "etale")
・Corrected a misprint in Definition 5.1, (ii) ("of F]," ---> "of F];")
・In Definition 5.1, (ii), modified the statement and subsequent explanation of the "assumption on X"
・Corrected a misprint in Definition 5.1, (ii) ("Gal(F/Q)" ---> "Aut(F)")
・In Definition 5.1, (ii), slightly modified the explanation of the notation "{\overline k}^\times_{NF}(\Pi_X)"
・Throughout the paper (starting with Definition 5.1, (iii)), replaced the term "open immersion" (in the case of a
homomorphism of profinite groups) by the term "open injection"
・Throughout the paper (starting with Definition 5.1, (iii)), replaced the terms "reference [iso]morphism" and
"reference morphism" by the term "reference isomorphism" (in order to avoid the confusion arising from the issue
of deciding which term one should use)
・Added missing parentheses in the fourth display of Definition 5.1, (iii)
・Added a reference to [Mzk20], Theorem 2.6, (v), in Definition 5.1, (iv)
・Modified the first arrow in the display of the statement of Corollary 5.2, (iii) (from an inclusion to an isomorphism)
・Corrected a misprint in Remark 5.4.3 ("a ...-line bundle" ---> "an ...-line bundle")
・Slightly modified the second to last display in the statement of Corollary 5.5, (ii)
・Added references [Kobl], [Lang-CM], [Milne-CM] to the Bibliography