ERRATA ET ADDENDA DIOPHANTINE APPROXIMATIONS AND VALUE DISTRIBUTION THEORY (SLN 1239)

Paul Vojta

1 July 2003; last revised 11 August 2017

Page iii, line 5 Change “number number field” to “number field.”

Page iii, line 6

Change “localized at a finite number of places” to “localized away from a finite number of places.”

Page iii, line 7

Change “ k-rational solutions of the system” to “ k-rational solutions of the system, for a number field k.”

Page iv, line -10 Change ℚ to \overline{ℚ}.

Page 2, equation (1.1.4) [NEW 5 March 2011] Change ∥x∥_{v} on the left-hand side to ∥x∥_{w}.

Page 4, line -5 Change “self intersection” to “self-intersection.”

Page 4, line -2 (Comment.) This is known as Kodaira’s lemma.

Page 7, line -14 Change |f_{i}/f_{j}|^{2} to |f_{i}/f_{j}|.

Page 8, line 11 Change λ_{D} ∘ f to λ_{D} ∘ ϕ and f^{∗}D to ϕ^{∗}D.

Page 8, line -1

[NEW 17 September 2006] This formula is correct (an earlier version of this list incorrectly stated that the formula held only for v ∤ ∞).

Page 9, line -7 Change “a hyperplane” to “the hyperplane at infinity.”

Page 9, line -6 Change \frac{1}{2[k:ℚ]} to \frac{[k_{v}:ℝ]}{2[k:ℚ]} .

Page 14, line -5 [NEW 11 August 2017] Change [L 7, Theorem 8.1] to [L 7, Ch. 2, Theorem 8.1].

Page 17, line 6 [NEW 11 April 2006] We assume n > 0 .

Page 17, line 7 [NEW 11 April 2006] The set S must contain all archimedean places.

Page 17, line -6

Change ∏_{v∈M_{k}} to ∏_{v∈S}.

Page 20, lines 5 and 6 Insert factors \frac{1}{[k:ℚ]} on the right-hand sides.

Page 22, lines 1–4 ρ should be taken relative to k.

Page 25, line -6

Delete the words “and n_{i} ≤ 0 otherwise.” (This condition is not used subsequently.)

Page 26, line 14 Change “max” to “lub” (since we are dealing with a partially ordered set).

Page 27, line -7

Change this line to read, “ P_{0} = [1 : 0 : 0 : 0] , P_{1} = [0 : 1 : 0 : 0] , and P_{2} = [0 : 0 : 1 : 0] .”

Page 34, line 6 [NEW 7 September 2007] In the right-hand column, change ord_{v} f to ord_{v} b.

Page 34, Definition of counting function in left-hand column The correct definition is

N(a, r) = ∑_{|w|<r} ord^{+}(f − a) log \frac{r}{|w|} .

Page 40, lines 4 and 11

Change ∫_{0}^{r} \frac{dt}{t} to ∫_{0}^{r} \frac{dt}{t^{2n−1}} .

Page 47, line -3 Missing “.” after “Conjecture.”

Page 54, line -5 Change CalO(1) to 𝒪(1) .

Page 57, line 18 Change “ramaining” to “remaining.”

Page 58, line 5

(Comment.) The bound (1 + log_{p}(n)) log p comes from Serre, Corps Locaux, Ch. III, Remark 1 following Prop. 13.

Page 62, line 16 [NEW 5 March 2011] Change “local on X ” to “local on V .”

Page 62, line -10 [NEW 5 March 2011] Change C[[x_{i}, …, x_{n}]] to C[[x_{1}, …, x_{n}]] .

Page 63–64, Conjecture 5.2.6

(Comment.) Isolated points of Z can be absorbed into the O(1) term, so we may assume that all irreducible components of Z have dimension > 0 .

Page 64, (generally)

(Comment.) For all smooth projective curves C there exists a reduced effective divisor D on C such that equality holds (up to O(1) ) in the General Conjecture for points P of bounded degree, of heights → ∞.

Proof. This is known if C ≅ ℙ^{1} , so we may assume that g(C) > 0 . Pick a nontrivial finite morphism f: C → ℙ^{1} and let B be the branch locus of f . Then B contains at least two points, which we may assume to be 0 and ∞. It is known that for sufficiently large S there are infinitely many S-units in k with heights → ∞. Then apply the contrapositive of the argument on page 63. □

Remark. Even in the split function field case, I cannot see how to get this with D = 0 . Indeed, we may assume that the base is ℙ^{1} ; then we want maps

\begin{matrix}C′ & ⟶^{p} & C \\ π & ↓ \\ ℙ^{1} \end{matrix}

such that π has large degree, p has small degree, and p is almost étale. But a naïve count of degrees of freedom belies this: If h′ = deg p, and C′ has s points of ramification, then g(C′) ≥ h′(g − 1) + s/2 . Also, the degree of the base locus on ℙ^{1} is 2g(C′) + 2 deg π − 2 , so the set of curves with a g_{deg π}^{1} is at most 2g(C′) + 2 deg π − 3 dimensional, and the set of curves C′ covering C is s-dimensional, which should add up to > 3g(C′) − 3 , but it doesn’t unless h′/ deg π is small, approaching 2/(g − 1) .

See also Xiangjun Song’s thesis.

Page 64, line 10

(Comment.) Note also that if (u, v) is a solution of u+v = 1 and is close to equality in the inequality h(P) ≤ d(P) + O(1) , then the binomial expansion of (u + v)^{n} = 1 gives a solution of u_{0} + ⋯ + u_{n} = 1 which is close to equality in the inequality h(P) ≤ (dim V)d(P) + O(1) for integral points on ℙ^{n} \ { (n + 2) hyperplanes}. (But, this is not a Zariski-dense example.) See Brownawell-Masser, Math. Proc. Camb. Phil. Soc. 100, page 427.

Page 65, line 12 This follows from van der Waerden, § 5.9.

Page 65, line 14 This was actually proved for S of arbitrary (finite) size.

Page 68, lines 17–18

Change “such that the intersection of the supports of the principal divisors (f_{i}) is equal to D ” to “such that the supports of the principal divisors (f_{i}) contain D for all i.”

Page 70, equation (5.4.3.2)

Change h(Q) to h_{π^{∗}𝒪(1)} (Q) (twice).

Page 70, line 10 Change d(Q) ≤ N(R, Q) ≤ 2h(Q) to

\frac{1}{2} log D_{k(Q)} = d(Q) ≤ N(R, Q) ≤ h_{π^{∗}𝒪(1)} (Q) .

Page 71, Example 5.5.1

Inequality (5.5.0.1) implies a bound on n for which there is a nontrivial rational solution [a : b : c] ; i.e., a solution whose coordinates are all nonzero.

Page 71, line -8 Change N = Conductor(abc) to N = Conductor(abc) .

Page 71, line -1 Change [\sqrt[4]{a} : \sqrt[4]{b} : \sqrt[4]{c}] to [ \sqrt[n]{a} : \sqrt[n]{b} : \sqrt[n]{c}] .

Page 80, Theorem 5.7.2

The divisor D doesn’t need to have normal crossings, and the (1, 1)-form ω is not holomorphic (which wouldn’t make sense), just C^{∞}.

Page 80, line -4 Add at the end of the sentence: “which has the usual continuity properties.”

Page 81, line 4 Change (−1)^{p+q} to (−1)^{p+q+1} .

Page 83, line -4 [NEW 2 June 2017] Change 5.7.7 to (5.7.7).

Page 84, line -4

Change ∑_{\begin{matrix}v∈S \\ v(abc)>0 \end{matrix}} to ∑_{\begin{matrix}v∉S \\ v(abc)>0 \end{matrix}} .

Page 85, line -6

This line should read, “Then for all m, n, x, y, z ∈ ℤ with (x, y) = 1 and satisfying … ”

Page 86, line -11 Change “For an elliptic curve” to “For a stable elliptic curve.”

Page 88, line -2 Change “applies” to “implies.”

Page 89, Chapter 6

Somewhere in Chapter 6 I should say something about Plücker formulas (Griffiths-Harris, pp. 269 and 270).

Page 90, Section 6.1 This section should be done using general convex symmetric bodies.

Page 91, line 3 Change [Bo-V] to [B-V].

Page 91, line 5 [NEW 4 July 2019] Add space after “however,”.

Page 91, line 14 Change “volume 2^{n} ” to “volume 2^{n}d(Λ) .”

Page 92, lines -13 and -12 [NEW 4 July 2019] Change 𝒪_{k,s} to 𝒪_{k,S} (twice).

Page 93, line 1 [NEW 4 July 2019] Change “ S-integers” to “algebraic integers.”

Page 94, line 14 [NEW 4 July 2019] Change (8) to (6.1.6).

Page 94, lines -12 and -11 Change [Bo-V] to [B-V].

Page 94, line -2 Change N(𝒟_{v}^{−1})^{r/2} to N(𝒟_{v}^{−1})^{n/2} .

Page 95, line 5 Change [Bo-V] to [B-V].

Page 95, line 7 [NEW 4 July 2019] Change “ v ∈ S ” to “ v ∉ S.”

Page 96, lines 4–7

These two paragraphs should not be italicized; they are not part of the statement of Theorem 6.1.11.

Page 97, equation (6.2.5)

[NEW 5 March 2011] Change c_{1}λ_{1}ρ_{i} to c_{1}λ_{i}ρ_{i}.

Page 98, line 11 Change x to x (four times).

Page 103, line -1

[NEW 24 April 2006] In the upper-right entry of the big matrix on the left-hand side, B′ should be B.

Page 104, line 17 Change ϵ ∈ 0 to ϵ > 0 .

Page 104, line -11 Change ∥L_{v,i}(w_{j})∥_{v} to ∥L_{v,i}(w_{j})∥_{v}.

Page 104, line -11 change 0 ≤ 1 ≤ n to 0 ≤ i ≤ n.

Page 104, lines -9 – -7

This paragraph should not be italicized; it is not part of the statement of Theorem 6.4.1.

Page 104, line -1 [NEW 8 August 2020] Change . (period) to “for all 1 ≤ j ≤ n;”.

Page 105, line 13 Change x · b_{i} = 0 to x · b_{i} = 0 .

Page 108, line 9 Change ρ_{1} to ρ_{i}.

Page 111, line -7 Change (6.5.10) to (6.4.10).

Page 117, lines -6 – -1

Remark 6.5.12 looks at it wrong! We should actually look at it this way: (x(t^{2}))′ = x′(t^{2}) · 2t; (x(t^{2}))′′ = x′′(t^{2}) · 4t^{2} + 2x′(t^{2}) ; etc.; the lower derivatives of x(t^{2}) go away when you wedge, so the contribution of the ramification point is (e−1)(n+1)(n+2)/2 . I don’t see as much of a connection with the factor dim V anymore.

8 PAUL VOJTA

Page 120, line -9

Change “exists” to “exist.” Page 120, line -1

Change λ_{1} to λ_{i}. Page 123, Lemma 6.7.1

We may assume that L_{1}, …, L_{n} are x_{1}, …, x_{n}, respectively. In order for this to make sense we really should be considering x ∈ ℂ[t]^{n}; in that case the successive minima are all constants, so λ_{1} = ⋯ = λ_{n}. So the lemma is vacuous and should be omitted. Page 124, [Ar 1]

This reference was in volume 5, not volume 35 (the original Russian article was volume 35, pp. 1269–1293). Page 125, [G-K]

Change “146–220” to “145–220.” Page 128, [R]

This reference has a correction: Mathematika 2, p. 168. Page 128, [Sil 3]

This reference has appeared in Invent. Math. 81 (1985) 341–346; corrected Invent. Math. 84 (1986) 223. Page 129, [Sil 4]

This reference has appeared in Math. Ann. 279 (1987) 193–216. Page 129, [Sz 1]

Change 1971 to 1981. Page 129, [Vo 2]

Cornell-Silverman is not in gtm. Page 131, column 1, line -13

[NEW 2 June 2017] Change log^{+} to log^{+} (wrong font).

Changes Since the Original Version of This List (These changes also appear in the above list)

ERRATA ET ADDENDA 9

08/07/2005: Page 111, line -7

Change (6.5.10) to (6.4.10).

09/17/2006: Page 8, line -1

This formula was correct after all.

09/07/2007: Page 34, line 6

In the right-hand column, change ord_{v} f to ord_{v} b.

03/05/2011: Page 2, equation (1.1.4)

Change ∥x∥_{v} on the left-hand side to ∥x∥_{w}.

Page 62, line 16 Change “local on X ” to “local on V .”

Page 62, line -10 Change C[[x_{i}, …, x_{n}]] to C[[x_{1}, …, x_{n}]] .

Page 97, equation (6.2.5) Change c_{1}λ_{1}ρ_{i} to c_{1}λ_{i}ρ_{i}.

06/02/2017: Page 83, line -4

Change 5.7.7 to (5.7.7).

Page 131, column 1, line -13 Change log^{+} to log^{+} (wrong font).

08/11/2017: Page 14, line -5

Change [L 7, Theorem 8.1] to [L 7, Ch. 2, Theorem 8.1].

07/04/2019: Page 91, line 5

Add space after “however,”.

10 PAUL VOJTA

Page 92, lines -13 and -12

Change 𝒪_{k,s} to 𝒪_{k,S} (twice). Page 93, line 1

Change “ S-integers” to “algebraic integers.” Page 94, line 14

Change (8) to (6.1.6). Page 95, line 7

Change “ v ∈ S ” to “ v ∉ S.” 08/08/2020: Page 104, line -1

Change . (period) to “for all 1 ≤ j ≤ n;”.