% List of Errata for Advanced Topics in the Arithmetic of Elliptic Curves
%  by Joseph H. Silverman
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\def\versiondate{Thursday, June 5, 1997}

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{\tenit Advanced Topics in the Arithmetic of Elliptic Curves}
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\centerline{Errata List for {\it Advanced Topics in the 
Arithmetic of Elliptic Curves}, GTM 151}
\centerline{by Joseph H. Silverman}
\centerline{Version \versionnumber --- \versiondate}
\bigskip
\noindent
The author  would like to thank the following people for their
assistance in compiling this errata sheet:
Andrew Baker,   % Glasgow University
Brian Conrad,   % Princeton
Guy Diaz,       % St. Etienne
Lisa Fastenberg % Yale
Benji Fisher,   % Columbia
Joan-C. Lario,  %
Ken Ono,        % Penn State
Michael Reid,   % Brown
Ottavio Rizzo,  % Brown
David Rohrlich, % Boston University
Samir Siksek,   % Exeter
Horst Zimmer.   % Saarbrucken

\medskip\noindent
Material referred to as being on ``attached pages'' is not yet available 
for distribution.

\bigskip
% -------------------------------------------------------------------------

\\ix\\Second Paragraph\\
Tate's unpublished manuscript (Tate [9]) has now appeared.

\\11\\Figure 1.2\\
\item{(1)}
There is too much white space above this figure.
\item{(2)}
The fundamental domain should be labeled using a script $\cal F$ instead of
a Times Roman F.

\\15\\Figure 1.3\\
\item{(1)}
The caption is missing. It should read\par
\centerline{The geometry of $\Gamma(1)\backslash{\bf H}$}
\item{(2)}
The typeface for the words ``Figure 1.3'' is too large. 
\item{(3)}
The fundamental domain should be labeled using a script $\cal F$ instead of
a Times Roman F.

\\18\\Lines 11,12\\
The four occurences of $V_1$ should all be $U_1$. Thus the displayed
equation and the following line should read
$$
  \kappa=\kappa(U_1)=
  \sup_{\textstyle{\tau\in U_1\atop\gamma\in\Gamma(1)}}
  \mathop{\rm Im}(\gamma\tau)
  =\sup_{\textstyle{\tau\in U_1\atop
    \left({a\atop c}\,{b\atop d}\right)\in\Gamma(1)}}
    {\mathop{\rm Im}(\tau)\over|c\tau+d|^2}
$$
is finite. (Note that if $\tau=s+it\in U_1$, then $s$ and $t$ are bounded, so

\\28\\Line $-1$\\
In the diagram, the left vertical arrow should be labeled $g_x$ instead of
$g\!\!\!_{_{\textstyle x}}$.

\\31\\Theorem 3.10(c)\\
Change $\displaystyle\left[{k\over6}\right]$ to $[k/6]$ (twice).

\\32\\Line 13\\
All four occurences of ``$i$'' in this displayed equation should be changed
to ``$\rho$'', since $G_4$ vanishes at $\rho$, not at $i$. It should thus
read
$$
  \Delta(\rho)=\bigl(60G_4(\rho)\bigr)^3-27\bigl(140G_6(\rho)\bigr)^2
  =2^43^35^27^2G_6(\rho)^3\ne0.
$$

\\42\\Figure 1.5\\
The origin should be $0$, not $0^2$

\\45\\Line 2\\
``$\sigma$ has a simple pole at $\pm{1\over2}\omega$.'' should be
``$\sigma$ has a simple zero at $\pm{1\over2}\omega$.''

\\46\\Second displayed equation\\
Sign error, the exponent should be $z-{1\over2}b$. Thus the RHS of the
equation should read $\pm e^{\eta(b)(z-{1\over2}b)}$.

\\55\\Lemma 7.1.1 Displayed Equation\\
There should be a factor of $1/(2k-1)!$ in the righthand side of the
displayed equation. It should thus read
$$
  \sum_{n\in\ZZ}{1\over(\tau+n)^{2k}}
  ={(2\pi i)^{2k}\over(2k-1)!}
  \sum_{r=1}^\infty r^{2k-1}e^{2\pi ir\tau}.
$$

\\59\\Displayed equation in middle of page\\
The formula for $E_6$ should use $\sigma_5$, not $\sigma_4$. Thus
$$
  E_6(\tau)=1-504\sum_{n\ge1}\sigma_5(n)q^n.
$$

\\60\\Line -6 (and elsewhere)\\
``Apostel'' should be ``Apostol'', here and elsewhere
(including page 62, line 10; page 67, lines 9 and 14; page 484,
lines $-14$, $-2$, $-1$.

\\61\\Conjecture 7.6\\
Could also describe the conjecture of Atkin and Serre that for
every $\epsilon>0$,
$$
  \tau(n)\gg_\epsilon \sigma_0(n)n^{(9/2)-\epsilon}.
$$
See Serre, J.-P., Divisibilit\'e de certaines fonction arithm\'etiques,
{\it L'Ens.\ Math.} {\bf22} (1976), 227--260, especially equation~4.11.

\\64\\Lines 3--12 (second paragraph plus one line)\\
Replace this material, which reads ``Now it's time to \dots to compute''
with the following material:
\par
In order to compute the values of the derivatives in this expression, we
take the transformation formula~(5.4c) for~$\sigma(z,\tau)$ and
differentiate it with respect to~$z$. This gives
$$
  \sigma'(z+\omega,\tau)=\psi(\omega)\eta(\omega)
    e^{\eta(\omega)(z+{1\over2}\omega}\sigma(z)
    +\psi(\omega)e^{\eta(\omega)(z+{1\over2}\omega}\sigma'(z)
  \qquad\hbox{for all $\omega\in\ZZ\tau+\ZZ$.}
$$
Now put $z=0$ and use the fact that $\sigma(0)=0$ and $\sigma'(0)=1$ to get
$\sigma'(\omega)=\psi(\omega)e^{\omega\eta(\omega)/2}$. Taking $\omega=1$,
$\omega=\tau$, and $\omega=\tau+1$ in succession yields
$$
  \sigma'(1)=-e^{\eta(1)/2},\qquad
  \sigma'(\tau)=-e^{\tau\eta(\tau)/2},\qquad\hbox{and}\qquad
  \sigma'(\tau+1)=-e^{(\tau+1)\eta(\tau+1)/2}.
$$
Next we use Legendre's relation (5.2d), which in our situation reads
$\tau\eta(1)-\eta(\tau)=2\pi i$, to eliminate~$\eta(\tau)$. After some
algebra we obtain
$$
  \sigma'(1)=-e^{{1\over2}\eta},\qquad
  \sigma'(\tau)=-e^{{1\over2}\eta\tau^2}q^{-{1\over2}},\qquad\hbox{and}\qquad
  \sigma'(\tau+1)=-e^{{1\over2}\eta(\tau+1)^2},
$$
where to ease notation we write $\eta=\eta(1)$.
\par
The next step is to use the product expansion (6.4) for $\sigma$ to compute
$\sigma$ at the half periods. Thus


\\73\\Third displayed equation (line 12)\\
The $ad/a'd'$ should be flipped, so the full line should read
$$
  a=a'p,\qquad d=d's,\qquad
  ps{a'd'\over ad}=1,\qquad a,d,a',d'>0.
$$

\\77\\Line $-6$\\
$T_{12}\Delta$ should be $T_{12}(n)\Delta$.

\\81\\Line 16\\
Change ``distinct divisors of $n$'' to ``positive divisors of $n$''

\\81\\Remark 11.2.1\\
Need to specify that the cusp form is a normalized eigenform. So replace the
phrase 
\par
``The Fourier coefficients of a cusp form of weight $2k$ actually
satisfy the stronger estimate'' 
\par
with the phrase
\par
``Let $f(\tau)=\sum c(n)q^n$ be a normalized cusp form of weight $2k$ which is
a simultaneous eigenfunction for all Hecke operators $T_{2k}(n)$. Then the
Fourier coefficients of~$f$ actually satisfy the stronger estimate''

Need to specify that the cusp form is normalized (i.e., has leading
coefficient $c(1)=1$).

\\81\\Remark 11.2.2\\
There's a missing period after ``See exercise 1.24''

\\84\\Line $-8$\\
``fuctional'' should be ``functional''

\\93\\Exercise 1.27\\
Replace the first line with \par
``Let $f(\tau)$ be a cusp form of weight $2k$
with $k$ an even integer.'' 
\par\noindent
(If $k$ is odd, then $L(f,k)=0$ for the functional equation.)

\\93\\Exercise 1.28\\
The first displayed equation should be
$$
  \chi:(\ZZ/p\ZZ)^* \longrightarrow \CC^*
$$
and after the next phrase ``be a primitive Dirichlet character'', add
the words ``and extend~$\chi$ to~$\ZZ$ be setting $\chi(p)=0$.''

\\93\\Line $-1$\\
The functional equation should read (check this)
$$
  R(f,\chi,s)=(-1)^k\chi(-1)R(f,\bar\chi,2k-s).
$$
(The $\chi(-1)$ is missing.)

\\108\\Line 7\\
$[\QQ(\alpha):\QQ]$ should be $[\QQ(\beta):\QQ]$.

\\110\\Lines 3 and 8\\
Replace the script oh ($\cal O$) with an italic oh ($O$).

\\112\\Last displayed equation\\
The Galois group action is a left action, despite the fact that it's written
using superscript notation. So this displayed equation should read
$$
  (\sigma\tau)*E=E^{\sigma\tau}=(E^\tau)^\sigma
    =\bigl(F(\tau)*E\bigr)^\sigma
    =F(\sigma)*\bigl(F(\tau)*E\bigr)=\bigl(F(\sigma)F(\tau)\bigr)*E.
$$
Further, the line on the top of page 113 \par
``(Note that \dots abelian group.)'' \par
should be replaced with the remark that \par
``(Note that $\mathop{\rm Gal}(\bar K/K)$ acts on the left.)''

\\116\\Lines $-11$ to $-4$\\
Delete these lines ``Thus $\sigma_{\goth p}\in\mathop{\rm Gal}(L/K)$ is
uniquely \dots has positive $\goth P$-adic valuation'' and replace them with:
\par
Thus $\sigma_{\goth p}\in\mathop{\rm Gal}(L/K)$ is uniquely determined by
the condition
$$
  \sigma_{\goth p}(x) \equiv x^{{\rm N}^K_\QQ \goth p} \pmod{\goth P}
  \qquad\hbox{for all $x\in R_L$.}
$$


\\121\\Line $-6$\\
Add the sentence ``Note that the kernel of $F$ is actually a finite
quotient of $\mathop{\rm Gal}(\bar K/K)$, since any $E$ will be defined over
some finite extension $L/K$, and then $F(\sigma)=1$ for $\sigma\in\mathop{\rm
Gal}(\bar K/L)$.'' at the beginning of the line. (This is just before the
sentence ``Since ${\cal CL}(R_K)$ is an abelian group\dots''.)

\\124,125\\Reduction of maps\\
The book talks about the ``natural reduction map'' on isogenies, but
reduction is never really carefully defined. The proof just assumes that
there is a reduction map with various properties (group homomorphism,
compatible with composition and evaluating/reduction of points, pullback
of differential, action of Galois on points over unramified extensions and
action of Galois on points of reduction, compatibility with the Weil
pairing and formation of dual isogenies).
\par
There are two ways to do all of this carefully. The first is to say that
(almost) all of these properties are immediate consequences of the fact that
an elliptic curve with good reduction is already a N\'eron model. Then add a
short section in Chapter~IV proving the necessary properties of reduction,
and in Chapter~II add a brief comment detailing the difficulties and
referring the reader to Chapter~IV.
\par
The second approach is to do explicit computations on the minimal
Weierstrass equation. In some sense, this means proving some of the N\'eron
properties for~$E$, but much of it can be done completely explicitly. For
example, one can explicitly construct the functions used in the definition
of the Weil pairing and show that they reduce properly. Similarly, suppose
that~$E$ and~$E'$ have good reduction and that $\phi:E\to E'$ is an isogeny
over~$K$. Write~$\phi$ out explicitly using homogeneous polynomials with
coefficients in~$R$. Then it should be possible to show directly that the
reduction of~$\phi$ gives at least a rational map on the special fiber.
(Scheme theoretically, this reflects the fact that~$\phi$ is a rational map
$E/R\to E'/R$, and~$E$ and~$E'$ are regular, so~$\phi$ is defined off of a set
of codimension~2. In particular, it gives a well-defined rational
map~$\tilde\phi$ on the special fibers, and then~$\tilde\phi$ is a morphism
since the fibers are non-singular curves.)

\\135\\Proof of Theorem 5.6\\
I have made the implicit assumption that $L/K$ is abelian. This is certainly
not  clear. There are two alternatives. 
\par\noindent(1)\quad
Give a direct proof that $L/K$
is abelian. It's clear, for example, that $K(j(E),E[\goth c])$ is abelian
over $K(j(E))$, since the Galois group injects into $\mathop{\rm
Aut}(E[\goth c])\cong (R/\goth c)^*$. Give a similar proof for~$L/K$.
\par\noindent(2)\quad
In section II.3, state and give a reference
for the fact that an arbitrary field extension is determined by a density~1
subset of the split primes. Then in the proof of theorem~5.6, let 
$L'=K\bigl(j(E),E[\goth c]\bigr)$. Choose a prime $\goth P$ in $L$ 
lying over $\goth p$ and a prime $\goth P'$ in $L'$ lying over $\goth P$.
Then the Artin symbol notation will involve these primes.

\\149\\Lines 17 and 20\\
Remove line 17, which reads
\par
\centerline{${}={}$ the maximal abelian quotient of $I_v$.}
\noindent
(Be sure to put a period at the end line~16.)
\par
On line~20, replace\par
``$I_v$ acts through  its maximal abelian quotient $I_v^{\rm ab}$'' \par
with just\par
``$I_v$ acts throught the quotient $I_v^{\rm ab}$.''

\\168\\Line $-9$\\
(Definition of $W_m$) The condition on $s_{\goth p}$ should be
$s_{\goth p}\equiv1\pmod{mR_{\goth p}}$, not $\pmod{m\goth p}$.

\\168\\Line $-6$\\
Delete the words ``and has finite index''.

\\169\\Line 5\\
This condition should read 
``$({\rm N}_L^Kx)_{\goth p}\in (1+mR_{\goth p})\cap R_{\goth p}^*$ for all
$\goth p$'' instead of ``$\in 1+m\goth p$.''

\\169\\Line $-14$\\
Delete the words ``and has finite index''.

\\173\\Theorem 10.3\\
Replace the last sentence \par
``Then $L(s,\psi)$ has an analytic \dots $N=N(\psi)$.'' \par
with the following:\par
``Then $L(s,\psi)$ has an analytic continuation to the
entire complex plane. Further, there is a functional equation relating the
values of $L(s,\psi)$ and $L(N-s,\bar\psi)$ for some real number
$N=N(\psi)$.''

\\174\\Proposition 10.4\\
In line 5, the Gr\"ossencharacter takes its values in $\CC$. Replace
$\psi_{E/L}:{\bf A}_L^*\to K^*$ with $\psi_{E/L}:{\bf A}_L^*\to \CC^*$.

\\175\\Proposition 10.5(a)\\
The Gr\"ossencharacter takes its values in $\CC$. Replace
$\psi_{E/L}:{\bf A}_L^*\to K^*$ with $\psi_{E/L}:{\bf A}_L^*\to \CC^*$.

\\176\\Proposition 10.5(b)\\
The Gr\"ossencharacter takes its values in $\CC$. Replace
$\psi_{E/L'}:{\bf A}_{L'}^*\to K^*$ with $\psi_{E/L'}:{\bf A}_{L'}^*\to
\CC^*$.

\\176\\Corollary 10.5.1(ii)\\
In the functional equation for the case that $K\not\subset L$,
the exponent of the norm of the different should be $s/2$, not~$s$.

\\180\\Exercise 2.15\\
The field should be $K=\QQ\left(\sqrt{-2}\,\right)$, not
$K=\QQ\left(\sqrt2\,\right)$.

\\181\\Exercise 2.18(f)\\
There's a missing $\log(n)$. Thus the problem should be to prove that
$$
  \lim_{n\to\infty}{\log|\Phi_n|\over(\deg\Phi_n)(\log n)}=6.
$$

\\183\\Exercise 2.24(b)\\
The formula in this exercise is completely incorrect. The simplest example
is an isogeny of degree 2 as described in Example III.4.5 on page 74 of
[AEC]. For that map,
$$
  \phi(x,y)=(R(x),cyR'(x))\quad\hbox{with $R(x)=x+a+b/x$ and $c=-1$.}
$$
The discriminants are $16b^2(a^2-4b)$ and $256b(a^2-4b)^2$.
\par
Change this exercise to show that there is a commutative diagram
$$\matrix{
  \CC/\Lambda 
  & \mathop{\hbox to.5in{\rightarrowfill}}\limits_{z \mapsto c^{-1}z}
  & \CC/\Lambda' \cr
  \Big\downarrow & & \Big\downarrow \cr
  E(\CC) & \mathop{\hbox to.5in{\rightarrowfill}}\limits\limits^{\phi} 
  & E'(\CC) \cr
  }
$$
where the vertical maps are complex analytic isomorphisms and~$c$ is the
number from part~(a).

\\183\\Exercise 2.25\\
Given the correspondence between twists and cocycles in [AEC], it looks like
the relation should be
$$
  \psi_{E^\chi/L} = \chi^{-1}\psi_{E/L}.
$$
(Check this.)

\\183\\Exercise 2.27\\
The definition of the zeta function should not have a minus sign. Thus
$$
  Z(\tilde E/\FF_{\goth P})=\exp\biggl(
  \sum_{n=1}^\infty \#\tilde E(\FF_{\goth P,n}){T^n\over n}\biggr).
$$

\\184\\Exercise 2.30\\
New part (a). Prove that $\goth P$ is unramified in $L'$.
\par
Relabel parts (a),(b),(c) to be parts (b),(c),(d).
\par
Part (d) (old part (c)) is not quite correct. Replace it with the
following:
\par\noindent
(d) Let $\tilde E$ be the reduction of $E$ modulo $\goth P$, and let $p$
be the residue characteristic of $\goth P$. Prove that
$$
  \hbox{$\tilde E$ is $\left\{
    \vcenter{\openup1\jot\halign{#\hfil&\quad#\hfil\cr
      ordinary&if $\goth P$ splits in $L'$ and $p$ splits in $K$,\cr
      supersingular&if $\goth P$ is inert in $L'$ and $p$ does not split
          in $K$\cr
    } } \right.$}
$$

\\186\\Exercise 2.35(c)\\
The second displayed equation should have double brackets for the $R_{\goth
p}$. Thus it should read
$R_{\goth p}\bigl[\!\bigl[\mathop{\rm Gal}(L_{\goth p}/L')\bigr]\!\bigr]$
instead of
$R_{\goth p}\bigl[\mathop{\rm Gal}(L_{\goth p}/L')\bigr]$.
\par(Check this.)

\\186\\Exercise 2.36(d)\\
In the last sentence, there should be double brackets for the $\ZZ_p$. Thus
$\ZZ_p[\![\Gamma]\!]$ instead of $\ZZ_p[\Gamma]$.

\\195\\Line 5\\
There is a period missing after ``is a finite subgroup of $K^*/{K^*}^m$''

\\216\\Line $-5$\\
Replace $(x_1-x_2)^2$ with $\pi_t\bigl((x_1-x_2)^2\bigr)$.

\\227\\Line 12\\
There is extra space between the word ``infinite'' and the period at the end
of the sentence.

\\228\\Last two displayed equations\\
The points in $\phi^*((x,y),t)$ are in $\Gamma\times C$, so it makes no
sense to write them as $((x_i',y_i'),t)$. Instead just write them as
$(\gamma_i,t)$. There are three displayed equations (with one line in
between) that need to be changed. Thus the material at the bottom of
page~228 should read:
$$\matrix{
  \psi: & {\cal E}^0 & \longrightarrow & E_0\times C,\cr
  & \bigl((x,y),t\bigr) & \longmapsto 
  & \displaystyle\Bigl(\sum_{i=1}^m \phi\bigl(\gamma_i,t_0\bigr),t\Bigr)\cr
  }
$$
where the points $\gamma_i$ are determined by the formula
$$
  \phi^*\bigl((x,y),t\bigr)=\sum_{i=1}^m
    (\gamma_i,t).
$$
The displayed equation in the middle of page 229 should read:
$$
  \psi\bigl((x,y),t_0\bigr)
  =\Bigl(
  \sum_{(\gamma,t_0)\in\phi^*((x,y),t_0)}\mskip-12mu
  \phi(\gamma,t_0), t_0 \Bigr)
  =\bigl(m(x,y),t_0\bigr).
$$

\\229\\The five lines after the displayed equation\\
This argument is incorrect. We can't vary~$t_0$, since that would have the
effect of changing to a different map~$\psi$. Replace these five lines with
the following:
\smallskip\noindent
Thus~$\psi:{\cal E}_{t_0}\to E_0\times\{t_0\}$ is just the
multiplication-by-$m$ map on~$E_0$. In particular, since the
multiplication-by-$m$ map is surjective, we see that~$\psi({\cal E}^0)$
contains~$E_0\times\{t_0\}$.  This implies that the rational map \hbox{${\cal
E}\to E_0\times C$} is dominant, since otherwise the irreducibility of
$\psi({\cal E}^0)$ would imply that $\psi({\cal E}^0)=E_0\times\{t_0\}$,
contradicting the fact that
$\psi({\cal E}^0)$ maps onto~$C$ (i.e., $\psi({\cal E}^0)$ must contain at
least one point on each fiber of \hbox{$\Gamma\times C\to C$}).


\\242\\Figure 3.3\\
There is an extraneous ``3'' above the ``t''.

\\258\\Proof of Lemma 10.4\\
Delete the second sentence, which reads ``In other words, $D=\phi^*H$ and
$D=\psi^*H'$ for appropriatesly chosen hyperplanes in $\PP^r$ and $\PP^s$
respectively.''  This statement is incorrect, but its deletion will not
affect the validity of the remainder of the proof. Note that it is true that
there is a divisor  class associated to any morphism $\phi:V\to\PP^r$, but it
is not true that every divisor in that divisor class has the form
$\phi^*H$ for some hyperplane~$H$. This will only be true if~$\phi$
corresponds to a complete linear system. 

\\259\\Line $-7$\\
$A(T_0,\ldots,T_r)$ should be $A_j(T_0,\ldots,T_r)$ (missing subscript).

\\271\\Theorem III.11.4\\
This should read
$$
  \hbox{$\sigma_t:E(K)\to{\cal E}_t(\bar k)$ is injective for all
  $t\in C(\bar k)$ satisfying $h_\delta(t)\ge c$.}
$$
(The inequality on the $h_\delta(t)\ge c$ is reversed.)

\\281\\Exercise 3.15(a)\\
There is a missing period at the end of this part of the exercise.

\\290\\Line $-9$\\
Change ``covention'' to ``convention''

\\292\\Line 3\\
Change $\mu(O_G)=O_H$ to $\phi(O_G)=O_H$.

\\301\\Lines 7,8\\
There is a bad line break at $({\cal E}\times_C{\cal E})(k)$.

\\313\\Line 4\\
``is generated $x$, $y$, and 2'' should be
``is generated by $x$, $y$, and 2''

\\339\\Line 21\\
``a unformizer for $\Gamma$'' should be
``a uniformizer for $\Gamma$'' (missing i in uniformizer)

\\341\\Theorem 7.2(ii)\\
``with $D_1$ is linearly'' should be ``with $D_1$ linearly''

\\348\\Line 5\\
``we must to set'' should be ``we must set''

\\348\\Line $-5$\\
``along entire curve'' should be ``along the entire curve''

\\355\\Line $-5$\\
``so after we may also assume'' should be
``so after further relabeling we may also assume''


\\367\\Line $-11$\\
``If $P\pi\ne2$, then'' should be ``If $p\ne2$, then''.

\\383\\Lines $-12$ and $-11$\\
``Let $\ell$ be a prime dividing $m$'' should be
``Let $\ell$ be the largest prime dividing $m$'' (Otherwise
if $2\|m$ and we took $\ell=2$, then it would not be true that
$\ell'$ divides $m$.)

\\388\\Szpiro's Conjecture 10.6\\
The displayed equation should have a constant $c(K,\epsilon)$
instead of $c(E,\epsilon)$. Thus it should read
$$
  {\rm N}^K_\QQ({\cal D}_{E/K})\le c(K,\epsilon)
  {\rm N}^K_\QQ({\goth f}_{E/K})^{6+\epsilon}.
$$

\\397\\Exercise 4.6(b)\\
The reference should be to ``(2.9)'', not to ``(2.9b)''.

\\397\\Line $-2$\\
``as in $(b)$'' should be ``as in (b)''. (The ``b'' should be
in roman font, not italic.)

\\398\\Exercise 4.12(c)\\
It looks like these two schemes are isomorphic. Thus let
$G_0=\mathop{\rm Spec} R[x,y]/(x^2-1)$ with group law
$((x_1,y_1),(x_2,y_2))\mapsto(x_1x_2,x_1y_2+x_2y_1)$, and let
$\GG_1\times\mu_2=\mathop{\rm Spec} R[S]\times\mathop{\rm Spec} R[T]/(T^2-1)$. Then the maps
$(S,T)\mapsto(xy,x)$ and $(x,y)\mapsto(T,ST)$ give isomorphisms of group
schemes.

\\402\\Exercise 4.32\\
In the first line, there's too much space after the comma, before the word
``compute''
\par
Change the phrase ``deleting the last row and column'' to ``deleting a row
and column corresponding to a multiplicity-1 component''
\par
Change the phrase ``is equal to the number of'' to ``is equal to plus or
minus the number of''

\\404\\Line $-2$\\
``finitely many elliptic curve'' should be ``finitely many elliptic curves''

\\415\\Proposition 2.2\\
It should be ``{\tt$\backslash$BeginProclaimWithPart} {\bf Proposition 2.2}''
instead of ``{\tt$\backslash$BeginProclaim} {\bf PropositionWithPart 2.2}''.

\\418\\Line $-4$\\
The displayed equation should be
$$
  c_4(q)=u^4\bigl(12g_2(\tau)\bigr)\qquad\hbox{and}\qquad
  c_6(q)=u^6\bigl(216g_3(\tau)\bigr).
$$

\\418\\Line $-3$\\
Replace ``$u=2\pi i$'' with ``$u=(2\pi i)^{-1}$''. Also rather than saying
that this is irrelevant, might be better to say that the exact value of~$u$
doesn't matter, but it is important that~$u$ is independent of~$\tau$.

\\422\\Chapter V, Section 2\\
Add a Remark~2.5 mentioning Alling~[1] and the following two articles as
sources for further information about elliptic curves over $\RR$:
\par
Bochnak, J., Huisman, J., When is a complex elliptic curve the
product of two real algebraic curves?, {\it Math. Ann.} {\bf293} (1992)
469--474.
\par
Huisman, J., The underlying real algebraic structure of complex
elliptic curves, {\it Math. Ann.} {\bf294} (1992), 19--35.

\\423\\Theorem 3.1\\
In the first displayed equation, it should be ``$a_4(q)=-5s_3(q)$''.

\\424\\Remark 3.1.2\\
Replace with the following:\par
Theorem 3.1 is actually true for any field $K$ that is complete with respect
to a non-archimedean absolute value. The only time we will use the fact~$K$
is a finite extension of~$\QQ_p$ will be in the proof that the map~$\phi$
in~(3.1c) is surjective. (In fact, we will really only need the fact that
the absolute value is discrete, so our proof actually is valid somewhat more
generally, for example over the completion of $\QQ_p^{\rm nr}$.) For a proof
of Theorem~3.1 in the most general setting, using $p$-adic analytic methods,
see Roquette~[1].

\\427\\Line $-17$\\
The second equation in the display has a misplaced minus sign, the minus
sign should go outside of the parentheses. The formula should read
$$
  (x_2-x_1)y_3=-\bigl((y_2-y_1)+(x_2-x_1)\bigr)x_3-(y_1x_2-y_2x_1).
$$

\\430\\Line 9\\
Replace ``Robert [1] and Roquette [1]'' with 
``Robert [1], Roquette [1], and Tate [9].'' (Replace the old unpublished
manuscript Tate [9] by a reference to its published version.)

\\430\\Line $-10$\\
Might be a good idea to point out that the equation for~$E_q$ is minimal,
since its reduction modulo~$\goth M$ is $y^2+xy=x^3$. 

\\434\\Line $-9$\\
All we know is that $q/\pi^N$ is a unit, not that it is congruent to~1. So
this displayed equation should read
$$
  y_n^2+x_ny_n \equiv (-q/\pi^{2n}) \not\equiv 0 \pmod{\pi}.
$$

\\435\\Line 3\\
In the subscript on the union, make the inequality strict. Thus
$$
  E_q(K)=E_{q,0}(K)\cup W\cup
    \bigcup_{1\le n<{1\over2}\mathop{\rm ord}\nolimits_v q}(U_n\cup V_n).
$$

\\438\\Lemma 5.1\\
Replace the second sentence
\par
``Then there is a unique $q\in\QQ_p(\alpha)^*$ with $|q|<1$ such that
$j(E_q)=\alpha$.''
\par\noindent
with the sentence\par
``Then there is a unique $q\in\bar\QQ_p^*$ with $|q|<1$ such that
$j(E_q)=\alpha$. This value of~$q$ lies in $\QQ_p(\alpha)$.''

\\440\\Line 14\\ (Fourth line of proof) Change ``for some $u\in K^*$'' to
``for some $u\in \bar K^*$''.

\\441\\Theorem 5.3(a)\\
Replace this part with:
\par
``There is a unique $q\in\bar K^*$ with $|q|<1$ such that $E$ is isomorphic
over~$\bar K$ to the Tate curve~$E_q$. Further, this value of~$q$ lies
in~$K$.''
\par\noindent
The first line of the proof should also be changed.

\\447\\Line $-12$\\
``for all $\psi\in{\rm Gal}_{\bar K/K}$'' should be
``for all $\psi\in G_{\bar K/K}$''.

\\449\\Exercise 5.4(a)\\
Replace \par
``If $b<0$, prove that \dots'' \par
with\par
``If $\Delta(E)>0$, so in particular if $b<0$, prove that \dots''.
\par
Further, the homogeneous space~$C$ is incorrect, it should be
$$
  C:w^2=4bz^4-(1+az^2)^2.
$$

\\450\\Exercise 5.6(b)\\
The last line should be
$$
  \gamma(E/\RR)={\rm sign}(1-t).
$$
instead of ${\rm sign}(t-1)$. (Notice that $c_6=-1+504s_5(q)\to-1$
as $t\to\infty$.)

\\451\\Exercise 5.10(a)\\
``If $E_q$ and $E_q'$ \dots'' should be
``If $E_q$ and $E_{q'}$ \dots''. (The prime is on the $q$, not
on the $E$.)

\\451\\Exercise 5.10(b)\\
Replace\par
 ``homomorphisms from $K^*/q^\ZZ$ to $K^*/{q'}^\ZZ$''\par
with\par
 ``homomorphisms from $\bar K^*/q^\ZZ$ to $\bar K^*/{q'}^\ZZ$''.

\\451\\Exercise 5.11\\
Replace ``consider the quadratic extension'' with
``consider the field'', since could have $L=K$.

\\452\\Exercise 5.13(a)\\
Replace ``for each prime $\ell$ there is'' with
``for each prime $\ell\ne p$ there is''.

\\453\\Exercise 5.15(b)\\
In the Hint, replace \par
``Take Tate models $E=E_q$ and $E'=E_{q'}$'' \par
with\par
``Take Tate models $E_q$ and $E_{q'}$''.\par
(Note may need to go to the unramified quadratic extension of $\QQ_p$.)

\\454\\Introduction to Chapter VI\\
At the end of the introduction, add the sentence ``For further information
about local height functions, see for example Lang~[3] and Zimmer~[2].

\\463\\First paragraph of Section 3\\
Replace this paragraph with the following:
\smallskip\noindent
Let $K$ be a field which is complete with respect to an archimedean absolute
value $|\,\cdot\,|_v$, and let~$E/K$ be an elliptic curve. Then~$K$ is
isomorphic to either~$\RR$ or~$\CC$, and
$|\,\cdot\,|_v$ corresponds to some power of the usual absolute value.
(See~[** Get Reference**].) Thus in order to compute the local height
function~$\lambda$ over~$E(K)$, it suffices to consider the case that
$K=\CC$, so in this section we will derive explicit formulas for the local
height on elliptic curves over the complex numbers.

\\465\\Proposition 3.1(c)\\
Replace\par
``is real-analytic away from~$0$''\par 
with\par
``is real-analytic and non-vanishing away from~$0$.''

\\466\\Line 3\\
Replace\par
``real-analytic on $\CC$ away from $\Lambda$''\par 
with\par
``real-analytic and non-vanishing on $\CC\setminus\Lambda$''.

\\467\\Corollary 3.3\\
Add at the end of the statement of the Corollary:
\par\noindent\begingroup\sl
``{\rm(}Note that the quantity $\bigl(x(P)-x(Q)\bigr)^6/\Delta$ is
well-defined, independent of the choice of a particular Weierstrass model
for~$E$.{\rm)}''
\endgroup
\par\noindent
(Be sure to change the {\tt $\backslash$EndProclaimAtDisplay} and 
{\tt $\backslash$BeginProofAtDisplay} macros.)

\\470\\Remark 4.1.1\\
Replace\par
``which means that $E_0(K)=E(K)$ and $v(\Delta)=0$, then our proof will
show that the function''\par 
with\par
``then we can find a Weierstrass equation for~$E/K$ with $E_0(K)=E(K)$ and
$v(\Delta)=0$. In this situation, the proof of~(4.1) will show that the
function'' \par\noindent
Also replace\par
``under finite extension of the field~$K$ (1.1c),''\par
with\par
``under finite extension of the field~$K$ (1.1c) and is independent of the
choice of Weierstrass equation (1.1b),''

\\470\\Remark 4.1.2\\
The displayed equation should read
$$\displaylines{
  \omit\quad$\displaystyle
  {1\over2}\max\bigl\{v\bigl(x'(P)^{-1}\bigr),0\bigr\}
    +{1\over12}v(\Delta')$\hfil\cr
  \omit$\displaystyle
  \hfil{}={1\over2}\max\bigl\{v\bigl(u^{-2}x(P)^{-1}\bigr),0\bigr\}
    +{1\over12}v(u^{-12}\Delta).$\quad\cr
  }
$$
(There are two changes. The first $\Delta$ requires a prime, and the
second $\Delta$ has a $u^{-12}$ in front.)

\\472\\Line 1\\
Replace\par
``where $m$ is the slope of the tangent line to $E$ at $P$. Thus''\par
with\par
``where $m$ is the slope of the tangent line to $E$ at $P$. (Note
that $m\ne\infty$ since $[2]P\ne O$.) Thus''

\\472\\Line 13\\
The line
$$
  {}=0\qquad\hbox{since either $v\bigl(F_X(P)\bigr)=0$ or
   $v\bigl(F_Y(P)\bigr)=0$.}
$$
should be
$$
  {}=0\qquad\hbox{since either $v\bigl(F_X(P)\bigr)\le 0$ or
   $v\bigl(F_Y(P)\bigr)\le 0$.}
$$
(Change two $=$ signs to $\le$ signs.)

\\472\\Line $-16$\\
Replace the phrase\par
``If $E_0(K)=E(K)$, that is, if $E$ has good reduction, then''\par
with\par
``If $E_0(K)=E(K)$, that is, if $E$ has good reduction and we take a minimal
Weierstrass equation for~$E$, then''

\\472\\Line $-14$\\
Replace the phrase\par
``However, even if $E$ has bad reduction,''\par
with\par
``However, even if the Weierstrass equation for~$E$ has singular reduction,''

\\474\\Line 5\\
Replace \par
``This suggests that $v\bigl(\theta(u)\bigr)-{1\over2}v(u)$ would be''\par
with\par
``This suggests that
$v\bigl(\theta(u)\bigr)-{1\over2}v(u)-{1\over12}v(\Delta)$ would be''

\\479\\Exercise 6.8\\
The value of $\lambda(P)$ is off by  $(1/12)v(\Delta)$. Thus the
conclusion should read:
$$
  \lambda(P)=\cases{
    \ds-{1\over6}v\bigl(F_2(P)\bigr)+{1\over12}v(\Delta)
      &if $v\bigl(F_3(P)\bigr)\ge3v\bigl(F_2(P)\bigr)$,\cr
    \ds-{1\over16}v\bigl(F_3(P)\bigr)+{1\over12}v(\Delta)
      &otherwise.\cr
  }
$$

\\479\\Exercise 6.9(c)\\
The value of $\lambda(P)$ is off by $(1/12)v(\Delta)$. Thus the
conclusion should read:
$$\ds
  \lambda(P)={1\over2}\log\bigl|x(P)\bigr|
    -{1\over12}\log|\Delta|
    +{1\over8}\sum_{n=0}^\infty
    {1\over4^n}\log\bigl|z\bigl([2^n]P\bigr)\bigr|.
$$

\\480\\Chapter VI New Exercise\\
Let~$K$ be a locally compact field which is complete with respect to an absolute
value~$|\,\cdot\,|_v$. As described in~\S1, the topology on~$K$ can be used
to define a topology on~$E$.
\par\noindent(a)
Prove that $E(K)$ is a compact topological space. 
\par\noindent(b)
Prove that the negation map $E(K)\to E(K)$, $P\mapsto-P$, is a continuous map.
\par\noindent(c)
Prove that the group law
$$
  E(K)\times E(K)\longrightarrow E(K),\qquad(P,Q)\longmapsto P+Q
$$ 
is a continuous map.

\\480\\Chapter VI New Exercise\\ 
(Hard Exercise) 
The previous exercise says that $E(K)$ is a compact topological group. Recall
that on such a group one can construct a translation invariant measure~$\mu$,
called {\it Haar measure}. (See [***] for basic properties of Haar measure.)
Let$\lambda:E(K)\setminus\{O\}\to\RR$ be the local height function~(1.1).
Prove that
$$
  \int_{E(K)}\lambda(P)\,d\mu(P)=0.
$$ 
(This generalizes exercise 6.5(a).)

\\480\\Chapter VI New Exercise\\ 
Continuing with notation from the previous two exercises, normalize the
Haar measure on~$E(K)$ by the condition $\int_E(K)d\mu=1$.
Now fix a Weierstrass
equation for~$E/K$ with coordinates~$(x,y)$ and discriminant~$\Delta$.
Prove that~$\lambda$ is given by the integral formula
$$
  \lambda(Q) = -{1\over12}
    \int_{E(K)} v\left({\bigl(x(P)-x(Q)\bigr)^6\over\Delta}\right)\,d\mu(P).
$$
Exercise Reference: G.R. Everest and B. Ni Fhlathuin, The elliptic Mahler
measure, preprint 1994.
\par\noindent
Exercise Solution: Integrate the quasi-parallelogram law (exercise 6.3) and
use the translation invariance of $\mu$ to cancel most of the terms.

\\484\\Line 12\\
Change ``outwiegh'' to ``outweigh''

\\487\\New note\\
Add a note for exercise:\par
(5.10) See Tate [9], especially pages 176--177.

\\496\\References\\
Replace the reference Tate [9] with the following reference:
\par
\llap{[9]\quad} A review of non-archimedean elliptic functions. In {\it
Elliptic Curves, Modular Forms, \& Fermat's Last Theorem}, J. Coates and
S.T. Yau, eds., International Press, Boston, 1995, 162--184.

\\496\\References\\
In Vladut, change ``Jugentraum'' to ``Jungendtraum''

\\497\\References\\
Add Zimmer, H., reference [2]:\par
Quasifunctions on elliptic curves over local fields, {\it J.
reine angew. Math.} {\bf307/308}   (1979), 221-246;
Corrections and remarks concerning quasifunctions on
elliptic curves. {\it J.reine angew. Math.} {\bf343} (1983), 203-211.

\\503\\Line 13, Notation\\ The entry $\hat h\Bbb G_m$ should just read $\Bbb
G_m$.


\bye