Marc Hindry and Joseph H. Silverman

Last updated: 24 Januray, 2016

Disclaimer: Many of these corrections have been sent to us, but we have not checked that they are correct.

**Page 27, Line -4, after Proposition A.1.4.6. **:
"identity" should be "identify".

**Page 30, Exercise A.1.6**:
The cubic surface $V$ given in the text is singular along the line
$x_2=x_3=0$, and the image of the map $\phi\times\psi$ is a curve in
$\mathbb P^1\times\mathbb P^1$, it is nto a birational map.

**Page 45**:
In the statement of Lemma A.2.3.1. there is missing "dim".

**Page 51, Theorem A.3.1.6(ii)**:
"Morphisms" should be "Rational maps"

**Page 66 Exercise A.3.13**:
"to to the free sheaf" should be "to the free sheaf".

Further, the statement that ${\cal F}\otimes\check{\cal F}$ is
isomorphic to ${\cal O}^{r^2}$ is incorrect in general except if $r=1$.

**Page 66, Exercise A.3.14**:
As with the previous exercise, it is not true in
general that $\check E\otimes E$ is a trivial
bundle except if $r=1$

**Page 67**:
In Section A.4, it may be necessary to assume that the ground
field is perfect.

**Page 75, Theorem A.4.3.1**:
The alignment on (i) and (ii) is incorrect.

**Page 84, Remark A.4.6.3.3**:
$m\deg(D)=1-g$ should be $m\deg(D)+1-g$.

**Page 88, Exercise A.4.7(e)**:
This exercise might be too hard. Does one need to know, for
example, some material on fundamental groups
from SGA1?

**Page 161, definition at bottom of page**:
It should be the *geometric* fiber
that has only ordinary double points.

**Page 161, Theorem A.9.3.2**:
It might be worth noting that the extension $L/K$ may
be taken to be separable.

**Page 162, Line 1**:
"Artin-Winter" should be "Artin-Winters". Also the proof in [1] may
only be for curves of genus $g\geq2$.

**Page 163 middle**:
The text says
"We recall that after a finite extension of the base field, any
commutative affine group is isomorphic to a product of additive
groups and multiplicative groups." This is not correct, for example
commutative Witt groups.

**Page 164, Items (2) and (3)**:
In this section $S$ has been global, but these statements are local.
Also, in the first statement of (2), one needs
to assumed that the gcd of the multiplicities of the components is 1.

**Page 167, Exercise A.9.13**:
Is it necessary to assume that $\dim R=1$?

**Page 170, Last displayed equation**:
The image of $\text{ord}_p$ is the integers, so this should read
\[ \text{ord}_p : \mathbb Q^* \longrightarrow \mathbb Z \]

**Page 177, Proof of Theorem B.2.3**:
We take $k=\mathbb Q(x)$ and $v\in M_k$ and then write $|x_i|_v$
where $x_i$ is a conjugate of $x$. So $x_i$ might not be in $k$.
One easy solution is to instead take $k$ to be the Galois closure
of $\mathbb Q(x)$ over $\mathbb Q$. Then it will contain
$x_1,\ldots,x_d$.

**Page 178, proof of B.2.3.1**:
We say in the first paragraph "First suppose that every $x_i$ is a
root of unity." But we are also allowing $x_i$ to be 0, so it should
say "First suppose that every $x_i$ is a root of unity or 0." Also
change "then $|x_i|=1$" "then $|x_i|=1$ or 0".

**Page 179, line -3**:
"$(n+1)$-tuple" should be "$(m+1)$-tuple".

**Page 180, Remark B.2.7**:
There are parts (i), (ii), and (2). It should be (i), (ii), and (iii).

**Page 181, next to last displayed equation**:
There is a superscript $n$ and a subscript $v$ (twice),
that are both supposed to be a superscript $n_v$. So this should read
\[
\prod_{v\in M_k} \epsilon_v(r)^{n_v}
\prod_{v\in M_k^\infty} r^{n_v} = r^{[k:\mathbb Q]}.
\]
Also note that $n_v$ is the local degree $[k_v,\mathbb Q_v]$, while $n$
is the number of variables.

**Page 183, Line -3**:
In the formula for $g_i$, the lower index for $j$ should be $0$. So
it should read
$$ g_i=\sum_{j=0}^N b_{ij}h_j,\quad 0\le i\le m, $$

**Page 183, Statement and Proof of Theorem B.3.1**:
The symbol $V$ is used as a projective variety, but in the proof
$V$ and $V'$ are also used to denote vector subspaces of $L(D)$. Change
the latter $V$ to $W$.

**Page 184, Theorem B.3.2(b)**:
As it is formulated, we need to
assume that $\phi(V)\not\subset D$. However, it is always possible
to replace $D$ by a linearly equivalent divisor for which this
is true, and (d) says that this only affects the height by a bounded
amount.

**Page 185, Theorem B.3.2(g)**:
It might be good to mention the stronger result (due to Northcott)
that for every $d\ge1$ and every constant $B$, the set
\[ \{ P\in V(\bar k) : h_{V,D}(P)\le B~\text{and}~[k(P):k]\le d \} \]
is finite.

**Page 190, First diplayed equation**:
This is not correct. It should read
\[
h_{V,D} = h_{V,\phi^*H} + O(1) = h_{P^n,H} o \phi + O(1) = h + O(1),
\]

**Page 194, line -1**:
"a smoothness hypotheses" should be "a smoothness hypothesis."

**Page 195, Theorem B.4.1**:
The variety $V$ should be assumed to be projective.

**Page 198, Proposition B.4.2(b)**:
The proof of Proposition B.4.2(b) is missing. Here is the proof: (b)
From (a), the preperiodic points satisfy $\hat h_{V,\phi,D}(P) = 0$.
Hence $h_{V,D}(P)=O(1)$, so the set of $k$-rational preperiodic points
is contained in a subset of $V(K)$ having bounded $h_{V,D}$ height.
The finiteness property for (ample) heights tells us that such sets
are finite, which completes the proof of (b).

**Page 199, Paragraph 2**:
The definition of antisymmetric is missing a minus sign. Thus
it should read "an antisymmetric divisor $D$ on $A$ (i.e., $D$
satisfies $[-1]^*D\sim-D$),"

**Page 199–200, Proof of Theorem B.5.1**:
It has been suggested that since (b) follows immediately from (d), a
more natural order of proof is (a),(c),(d),(b),(e). But pedagogically
it may make sense to prove (b) first as warm-up for the rest of
the proof.

**Page 207, Last two lines of the proof of (d)**:
There are three errors. The two heights should be for the variety $B$,
not $A$, and "in A" should be "in (a)". So these two lines should read
"has all of the properties to be $\hat h_{B,\phi^*D}$, so by the
uniqueness assertion in (a), it is equal to $\hat h_{B,\phi^*D}$."

**Page 213, Theorem B.6.3**:
It says $\Gamma \subset \text{rank } A(k)$, but it should
be $\Gamma \subset A(k)$.

**Page 220, line 11**:
It says $u \ge v$, but it should be $u \le v$.

**Page 228, Third displayed equation**:
Replace $\displaystyle\max_{i=1}^r$ with $\displaystyle\max_{1\le i\le r}$.

**Page 228, Line 2 of second displayed equation**:
The $r$ should have an exponent $n_v$. So this line should read
\[
{}\le \prod_{v\in M_k^\infty} \left(r^{n_v}\max_{1\le i\le r}
\bigl\{1,|f_i|_v^{n_v} \bigr\} \right)
\]

**Page 228, Proposition B.7.3 (and elsewhere)**:
"Gelfand's inequality" should be "Gelfond's inequality"

**Page 252, Exercise B.5(b,c)**:
The exercise asks you to prove that two maps $\phi$ and $\psi$ that
have a common eigen-divisor class $D$ satisfy: $\hat h_{\phi,D}=\hat
h_{\psi,D}$ as functions on $V(\bar k)$ if and only if
$\phi\circ\psi=\psi\circ\phi$. One direction is easy, but the other is
probably not quite true, even for $\mathbb P^1$. What is true is that
$\hat h_{\phi,D}=\hat h_{\psi,D}$ implies that
$\text{PrePer}(\phi)=\text{PrePer}(\psi)$, which (at least for
$\mathbb P^1$) implies (over $\mathbb C$) that $\phi$ and $\psi$ have
the same Julia set.

**Page 259**:
There is a reference to Knapp [1], but the reference is missing in the
bibliography. It should be *Elliptic Curves*, Anthony W. Knapp,
Mathematical Notes (Book 40), Princeton University Press, 1992

**Page 262**:
The second displayed equation is missing an
equals sign. It should read
$(\sigma(y)-y)-(\sigma(y')-y') = \sigma(y-y')-(y-y') = 0.$

**Page 264, Line 2**:
"be finite a place" should be "be a finite place"

**Page 264, Proposition C.1.6**:
"postive" should be "positive"

**Page 280, last line of the proof**:
"a book such as of Hilton-Stammbach [1]" should be "a book such as
Hilton-Stammbach [1]"

**Page 282, Proposition C.4.2**:
The last line reads "the Selmer group if finite."
It should be "the Selmer group is finite."

**Page 295, Exercise C.14**:
The statement of this exercise is meaningless because a part of the
sentence is left out. It should read:
"The intersection of $\text{ker}(\rho_p)$ with the set of matrices of
finite order is trivial" or alternatively
"The kernel $\text{ker}(\rho_p)$ contains no torsion element other than
the identity." (When $p=2$, there are some elements of order 2.)

**Page 301, Remark D.1.1.1**:
It says to "see Exercise D.3 and the references cited there",
but Exercise D.3 cites no references.

**Page 308, Line 7**:
"Sterling's formula" should be "Stirling's formula"

**Page 318, line -1**:
The reference Anderson and Masser [1] does not appear in the bibliography.
Also, it says to "see Exercise D.9 and the references cited there",

**Page 319, Lemma D.4.2**:
As was done in Lemma D.4.1, it would be a good idea to say what $N$
and $M$ are. So replace "Assume that $dM\lt N$" with
"Let $N$ and $M$ be positive integers satisfying $dM\lt N$."

**Page 321, Last two displayed equations**:
$P_{i_1,\ldots,i_m}$ should be $\partial_{i_1,\ldots,i_m}P$.

**Page 328, Last displayed equation**:
This inequality is reversed, and it is the reverse inequality that is
needed. We are given that $\delta\lt 1$ and $\epsilon\lt
\delta/22$. An easy algebra calculation then shows that
\[
\left[ \left(1+\frac{\delta}{2}\right) - (1+\epsilon) \right] -
\frac{\delta}{4}
= \frac{\delta}{4} - 4\epsilon - \frac{3\delta\epsilon}{2}
= \frac{\delta-22\epsilon}{4} + \frac{3}{2}(1-\delta)
\gt 0.
\]

**Page 336**:
In the definition of $W$ second line, the sum term of $\phi_r$, the
parentheses are not balanced.

**Page 348, Third displayed equation**:
In the estimate for $\|Y\|_w$, the middle two products should have
maxs. The line should read
$$
\|Y\|_w = \max_{v\in S}\|Y\|_v \ge
\left(\prod_{v\in S} \max\bigl\{\|Y\|_v,1\bigr\}\right)^{1/s}
=
\left(\prod_{v\in M_K} \max\bigl\{\|Y\|_v,1\bigr\}\right)^{1/s}
= H_K(Y)^{1/s}.
$$

**Page 351, Third displayed equation**:
In Siegel's identity, the $W_2$ in the second
denominator should be a $W_3$. So it should read
\[ \frac{W_1-W_2}{W_1-W_3}+\frac{W_2-W_3}{W_1-W_3}=1. \]

**Page 361, Exercise D.1 (Remark)**:
The conjecture of Lang in the remark is not correct. Here is a
counterexample: Let $\alpha$ be algebraic, choose $a_n/q_n$ with
$|a_n/q_n-\alpha|<1/q_n^2$ and $q_n\ge 2^n$. Then let $F(q)=q^2$ if
$q=q_n$ for some $n$, and otherwise let $F(q)=q^3$. Then $\sum q/F(q)$
converges, but $\alpha$ is in the set in (b). Probably okay if one
requires that the function $F(q)$ be an increasing function.

**Page 373, line -8**:
There is extra space between the word
"vary" and the variables $d_1, d_2, d$.

**Page 376, line 1**:
The constant $s$ should be $M$

**Page 378, line -2**:
The constant $s$ should be $M$

**Page 379, Line 15**:
"can never by nearly parallel"
should be
"can never be nearly parallel"

**Page 392, line 9**:
The period should go inside the parenthesis.

**Page 408, Exercise F.4(d)**:
"The elliptic the curve"
should be
"The elliptic curve"

**Page 435, middle of page**:
$nx\in G_0$ should be $nx\in\Gamma_0$.

**Page 439**:
In this brief history, should mention that Coleman found a
gap in Manin's proof and subsequently fixed it.

**Page 441, Corollary F.1.2.3**:
Delete the second $k$ and add the conclusion "is finite."
So the full statement should read:
**Corollary F.1.2.3.** Let $X$ be a general curve of genus $g$,
let $d\lt g/2+1$, and let $k$ be any finitely generated field over which
$X$ is defined. Then $X^{(d)}(k)$ is finite.

**Page 448, Second displayed equation**:
There is a missing right parenthesis in the bounds of integration. It should
read
$$\|\alpha\|^2=\frac{1}{(2\pi)^g}\int_{A(\mathbb C)}|\alpha\wedge\bar\alpha|\,.$$

**Page 449, Fourth displayed equation**:
There is an extraneous symbol in the log Norm term. It should read
$\log\text{N}_{k/\mathbb Q}\Delta_{E/k}$

**Page 452, mid-page**:
"for which the the absolute value"
should be
"for which the absolute value"

**Page 453, Conjecture F.3.2(a,b) and first displayed equation**:
$\log\mathcal F_{E,k}$ should be $\log\text{N}_{k/\mathbb Q}(\mathcal
F_{E,k})$. (Three times)

**Page 453, line -9**:
"height of rational of $C(k)$"
should be
"height of rational points of $C(k)$"

**Page 454, First displayed equation**:
The height should be with respect to $D$, so it should read
$\hat h_D(P)\ge ch(A)$.

**Page 456, End of Proof of Theorem F.3.6**:
The last inequality should be reversed. It should read
$2g-2\geq d-\text{card} S$.

**Page 458, Theorem F.4.1.1(ii)**:
$E(\mathbb Q)_{\text{tors}}$ should be $E(k)_{\text{tors}}$

**Page 459, line 12**:
We need to choose a basis for $\hat A(k)/\text{(torsion)}$,
not for $\hat A(k)$.

**Page 460 line 2**:
On the right-hand side of the formula, the norm of $u_1$ should
probably appear to the $i$'th power. Thus this formula should probably
read
\[
\|\boldsymbol u_i\| \le \left(\frac43\right)^{r(r-1)/2(r-i+1)}
\left(\frac{\text{Vol}(L)}{\|\boldsymbol u_1\|^i}\right)^{1/(r-i+1)}
\quad\text{for $1\le i\le r$.}
\]

**Page 472, middle of the page**:
"due de Diego" should be "due to de Diego".

**Page 473, Conjecture F.4.3.3 (ii)**:
The $c_2$ should just be $c$, so it should read
$\#C(k)\le c^{1+\text{rank }\text{Jac}(C)(k)}$.

**Page 476, Lemma F.5.1.2(i)**:
Some further condition is needed on the map $f$, for example
generically finite. Otherwise a counterexample is
$\text{Proj}_2:\mathbb P^1\times C\to C$, where $C$ is a curve
of positive genus.

**Page 480, line -7**:
"can reduced" should be "can be reduced"

**Page 480, Remark 5.2.4(b)**:
The hypothesis needed by both Noguchi and Moriwaki is that the
cotangent bundle is an ample vector bundle. This is stronger than
saying that the canonical bundle is an ample line bundle, which is the
condition stated in the book.

**Page 486, Line 2**:
"normal crossing condition" should be "normal crossings condition".

**Page 494, line -12**:
"they prove the" should be "they prove the asymptotic estimate"

**Page 496**:
The "counterexample" of Colliot-Thélèene, Swinnerton-Dyer and
Skorobogatov is actually a counterexample to the first version of
Mazur's conjecture that the closure of rational points is open in the
real locus, or equivalently equal to a union of connected
components. It is not to a counterexample to the version stated that
the closure is semialgebraic.

**Page 496, Line 1**:
"is Zariski in $X$" should be "is Zariski dense in $X$"

**Page 499, Exercise F.8**:
"genus at least 2 Prove that"
should be
"genus at least 2. Prove that"
(missing period)

**Page 506**:
The Bosch reference is not in alphabetical order, and it is
lacking initials.

**Page 515**:
Reference to Poizat is mis-typeset.

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