Erratum for Elliptic Curves 2006 (J.S. Milne)

In the blurb and introduction, I should have noted that the group is commutative.

**p28.** The third cubic curve should be
\[
\ell(R,Q)\cdot\ell(P,Q+R)\cdot\ell(PQ,O)=0
\]
This is the product of the three horizontal lines. (Dmitriy Zanin)

**p36.** In the definition of $k[C]_{\mathfrak{p}{}}$, the condition on
$h$ should be $h\notin\mathfrak{p}$ (Jochen Gerhard).

**p39.** In the definition of a regular map between projective plane
curves, $a_{m}$ should read $a_{2}$ (Rankeya Datta).

**p100, 3.23b.** The sign is wrong: it should read $4d-c^{2}\geq0$. As
PENG Bo pointed out to me, I forgot to include the proof. Here it is.

Let
\[
X^{2}+c^{\prime}X+d^{\prime}=\det(X-n\alpha|T_{\ell}E).
\]
By linear algebra, we see that $c^{\prime}=nc$ and $d^{\prime}=n^{2}d$. On
substituting $m$ for $X$ in the equality, we find that
\[
m^{2}+cmn+n^{2}d=\det(m-n\alpha|T_{\ell}E).
\]
According to Proposition 3.22, the right hand side equals the degree of
$m-n\alpha$. Therefore
\[
m^{2}+cmn+n^{2}d\geq0
\]
for all $m,n\in\mathbb{Z}{}$, i.e.,
\[
r^{2}+cr+d\geq0
\]
for all $r\in\mathbb{Q}{}$. The minimum value of $r^{2}+cr+d,$ $r\in
\mathbb{R}{}$, is $(\frac{c}{2})^{2}+c(-\frac{c}{2})+d=-\frac{c^{2}}{4}+d$,
and so $4d\geq c^{2}$.

Happily, this is how I used it on p150 in the proof of
the congruence Riemann hypothesis.

**p107, line 2** (exact sequence of cohomology groups): a bracket
"$)$" is missing: $H^{1}(G,\mu(k^{al}))$
instead of $H^{1}(G,\mu(k^{al})$ (Michael Mueller).

**p148, 9.1b.** Should read: The Frobenius map acts as zero...
*not* as zero acts; at least I not think).

**p150, 9.5.** Taylor et al. prove the conjecture of Sato and Tate only
for elliptic curves that do not have potential good reduction at some prime
$p$.

**Bibliography:** Fulton's book, Algebraic Curves, is now freely
available on his website pdf

**From Stefan Müller: **

page 7, line -7: the coordinates should be small $x$ and $y$

page 9, line -13: $k[X,Y]$ square brackets also inside the set definition

page 33: in my class I used $K_{C}$ instead of $W$, since it is "the" usual
notation, of course the letter $K$ can be confused with the field $K$

page 36, line 18: $h$ not in $\mathfrak{p}{}$, instead of non-zero.

page 37, section on Riemann-Roch: in contrast to the rest of the book the
algebraic closure here is $\bar{k}$ not $k^{\mathrm{al}}$.

page 39, line -6: delete word before $\mathbb{P}{}^{2}$.

page 51, line -12: in my opinion $c$ must be $u_{1}/u_{2}$ not $u_{2}/u_{1}$.

page 66, line -8: it is Corollary 4.2 not Prop. 4.2 (perhaps also at other places)

page 100, Corollary 3.23: In (b) the inequality sign seems wrong, at least it
contradicts what you use of it later. The sign of the term $c\alpha$ seems
also wrong, at least contradicts the proof. The proof of (b) is completely
missing, but it is very important in the applications (Hasse-Weil). [See above.]

page 104, proof of Cor. 1.4: in my opinion it must be $\sigma c/c$ not
$c/\sigma c$. At the blackboard I was fighting with this problem for about 10
minutes, still not sure.

page 105, footnote: element not elements

page 149, Thm. 9.4: square root of p ! Proof refers to Cor 3.23 (see above).

page 157, line 6: inverse roots not roots

**From Nicholas Wilson:**

On page 167, line -17, there is written "Coatesand Wiles (1977)...", which I
believe should read "Coates and Wiles (1977)..

**From Enis Kaya**

Page 109: definition of nth Selmer group is wrong, you missed the index 'n' for the groups on the RHS in kernel.

Page 110: in the fundamental exact sequence, you have nth Tate-Shafarevich group but you have not defined this group.

Page 114: Definition of nth Selmer group is wrong because of the same reason before.

Page 127: In the last paragraph, you said '... for a single elliptic curve over a $\mathbb{Q}$, and the ...'. I think the second 'a' is incorrect.