The following file contains reported typos to the first
edition, first printing, of Prolegomena to a Middlebrow
Arithmetic of Curves of Genus 2, by J.W.S. Cassels & E.V. Flynn.
Many thanks to those who have reported the typos to us.
The file was last updated on 20 April, 2022.
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Page ix, line 11. "as as" should be "as"
Page xiii, line 10. "," should be ";"
page 8, line 18. "specialize to F'(x)/2y, y + x F'(x)/2y"
should be: "specialize to F'(x)/2y, -y + x F'(x)/2y"
page 21, equation (3.2.6) should be:
\beta_0 G(X) + N = 0
page 26, line -12. At the end of the line it should read "2-division
points".
p.27 (last line). Replace: "There only" by: "There are only"
p.29, first lines.
For the classical result, replace "Jacobian of a curve"
by "Jacobian of a curve or product of two elliptic curves".
In fact, the statement is correct, but it's not classical.
p.33, Formula 4.2.6 should be:
eta_2 = -u0*v3-u3*v0+2*w0*w3 (the last term is missing)
eta_3 = -u1*v3-u3*v1+2*w1*w3 (the last term is missing)
p.35. In the last line of display (4.3.6),
2 w_0 w_3 should be 2 w_0 w_2.
p.37, THEOREM 4.5.1. "be projective over the ground field".
should read "be projectively equivalent over the ground field".
p.54, at the bottom the page, the speculations are correct. The genus 2
curve has a point defined over a quadratic extension, and you can subtract
multiples of it to get from an odd degree element of Pic to one of even
degree, which you already know are represented by rational divisors.
p.56, line -7. "Clearly Criterion (ii) of Lemma 6.5.1 is satisfied
exactly when h(X) has a root in k, which is easy to check."
should read"
"If h(X) is square-free, Criterion (ii) of Lemma 6.5.1 is satisfied
exactly when h(X) has a root in k, which is easy to check.
If h(X) is not square-free, one can continue to replace
the current F(X) by F(X+1), and hope that the new
induced h(X) is square-free; it is easy to show that h(X) is
guaranteed to be square-free within 45 iterations."
Note that this leads to the following algorithm (pointed out to me by
Michael Stoll) for determining the kernel of \mu:
Compute h(X).
Factor it over k.
If it has a simple root in k --> kernel of \mu = 2G
If it hasn't and is square-free --> [kernel of \mu : 2G] = 2
(supposing there is some point of odd degree on the curve)
If neither --> replace F(x) by F(x+1) and repeat.
p.70. On the line after (7.4.1), "log" should be "-log"
p.70. At the end of the corollary, add: "except for possible 2-torsion"
p.71: "the rational prime $p = \pi^n$, for some $n$,"
should be:
"the rational prime $p$ satisfies $p\cdot\fo = \pi^n\cdot\fo$, for some $n$,"
p.72, line 8. "isomorphic to ${\fraktur p}$" should be
"isomorphic to ${\fraktur p}$ when $g=1$ and
${\fraktur p}\times {\fraktur p}$ when $g=2$"
p.72, line 9. The display
$$ \# {\fraktur G} / 2 {\fraktur G} =
\# {\fraktur G} [2] \cdot \# {\fraktur p} / 2 {\fraktur p} $$
should read
$$ \# {\fraktur G} / 2 {\fraktur G} =
\# {\fraktur G} [2] \cdot \# {\fraktur H} / 2 {\fraktur H} =
( \# {\fraktur p} / 2 {\fraktur p} )^g $$
p.78, line 18, eqn (8.1.14),
C + D should be E + F
and:
Q=(-{2\over 7}+{3\over 7}\sqrt{-5}, {192\over 343}+{6\over 343}\sqrt{-5})
should read:
Q=(-{2\over 7}+{3\over 7}\sqrt{-5}, -{192\over 343}-{6\over 343}\sqrt{-5})
p.79, line 16. "contribute nine" should be "contribute 18"
p.82, line 5.
"is the only case when $\tilde \fA_1 \not\in \widetilde J(\bbF_p)$."
should read:
"is the only case when $\tilde \fA_1 \not\in 2\widetilde J(\bbF_p)$."
p.84, Display (8.3.12). "v-A" should be "v+A" and vice versa.
p.86, Display (8.3.23). "v-A" should be "v+A" and vice versa.
p.86, line 20. "Example 8.3.3." should be "Example 8.3.4."
p.86, last line of equation (8.3.27).
"(A_3(X))^2 + 16 X(X-1)^2"
should be
"(A_3(X))^2 + 16 X(X-1)"
p.87. Display (8.3.28) should end with "." not ","
p.87, equation (8.3.29). Each of the two occurrences of "A" should be "A_1";
that is: "(0,A(0))" should be "(0,A_1(0))"
and: "(1,A(1))" should be "(1,A_1(1))"
p.87, 2 lines before Display (8.3.31),
"and the genus 2 curve of Example 2"
should be:
"and the genus 2 curve of Example 3"
p.88, line -7. "when the restriction of the 2-Weil pairing
is trivial" should be: "when the restriction of the n-Weil pairing
is trivial"
p.97, line -8. (k^*/k^{*2})^3 should be (k^*/k^{*2})^{\times 3}
(meaning the product of k^*/k^{*2} with itself 3 times)
to be consistent with the later notation of (11.1.11).
p.98, line 3. (k^*/k^{*2})^2 should be (k^*/k^{*2})^{\times 2}
for the same reason.
p.104, line -4. "this must be a jacobian" should read
"this must be a jacobian, when simple".
In fact, it must be isomorphic to one of:
a jacobian of a curve of genus 3,
a produce of an elliptic curve and the jacobian of a curve of genus 2,
the product of 3 elliptic curves.
p.108, line 7. Should be:
\mu^\phi \bigl( (1,0) \bigr) = [-3, (-3)(2)] = [-3,-6], and
\mu^\phi \bigl( (0,0) \bigr) = [-2, 1]
p.108, line 9. Should be:
[-3,-6][-2,1] = [6,-6]
p.111, line 15. "Combining (4) and (5) with"
should be "Combining (3) and (5) with"
p.112, line 2. "have been found" should be: "has been found"
p.112, diagram (10.4.10). All 3 of the middle-height downward
arrows should be labelled $\hat\phi$, not just the middle one.
p.117, first line of the proof of Example 11.2.1.
"# \tilde E(F_5) = 4," should be:
"# \tilde E(F_5) = 8, (1,0) is not in 2\tilde E(F_5),
(4,0) is not in 2\tilde E(F_5), and (0,0) is not in 2\tilde E(F_7),"
p.121, equation (11.3.5), there should be a comma between \pm 6 and \pm 5.
p.123, the line immediately after (11.3.13), remove the word: "again",
and replace (iv) = (ii)\cdot (iii) by (iv) = (i)\cdot (iii).
p.126, equation 11.4.3,{(\sqrt{-2},0), (\sqrt{-2},0)\} should have
a minus sign in front of the second \sqrt{-2}.
p.127, line -8.
"M \cap (Q_3^*)^2 = {1,-2}" should be "Q(S) \cap (Q_3^*)^2 = {1,-2}"
p.128, line 9. \mu^{\hat\phi} should be \mu_3^{\hat\phi}
p.128, line 16. "M \cap (Q_3^*)^2 is the trivial group"
should be "Q(S) \cap (Q_2^*)^2 is the trivial group"
p.129, line 1. "This mapes to [6,2]" should be "This maps to [-3,-2]"
p.129, line 3. "<[6,-3],[-3,3],[6,-6],[6,3],[6,2]>"
should be "<[6,-3],[-3,3],[6,-6],[6,3],[-3,-2]>"
p.131, line 10. "certainly certainly" should be "certainly"
On p.132, it is not always so easy to "cheat" because in general one
will not be able to find a Jacobian OVER Q which is isogenous to the
product of two given elliptic curves E1, E2 over Q. It is possible,
however, for instance, when E1 and E2 are both isomorphic to an
elliptic curve E such that j(E) is not 1728, and E has a rational 2-torsion
point. I don't remember what the largest Mordell-Weil rank known
for such curves is.
Replace
"Care should be taken to distinguish genuine genus 2 examples"
by
"Care should be taken to distinguish genuine dimension 2 examples"
Replace
"Then we could find a curve of genus 2 whose jacobian is isogenous
to E x E (how to do this for any E_1 x E_2 is described in
Chapter 14, Section 3), which will have rank at least 42."
by
"Then E x E will be an abelian variety, defined over Q, with
rank at least 42."
p.146, lines -7,-6 (in the statement of Theorem 13.1.1.)
each | c_l |_v should be replaced by | c_l |_p
each | c_j |_v should be replaced by | c_j |_p
each | x |_v should be replaced by | x |_p
p.149, line 5. "For each~${\eufm A} \in {\eufm U}$"
should be: "For each~${\eufm A} \in {\cal U}$"
(i.e. the font of U on line 5 should be the same as on line 4)
p.150, line 1. $n \cdot {\cal E}$ should be $n \cdot {\eufm E}$
(i.e. the font of E on line 1 should be the same as in (13.2.18))
p.150, line 5. "choices for $\eufb A$" should be "choices for $\eufm A$"
(i.e. A is in the correct font, except that it should be be bold)
p.151, line 9. There should be an extra comment here:
"Note that the leading term of the power series \theta is
(constant)n^6 over Z_5 (using (13.2.19) and the fact that the power series
in n giving t_1,t_2 have no constant terms), so that we can
take out a factor of n^6. Combining this with (13.2.20) and Strassman's
Theorem, we see that n=0 (repeated 6 times) is the only solution".
Note that (13.2.20) on its own is not sufficient
justification for taking out a factor of n^6.
p.151, line 12.
"(-1/2,-3/4),(-1/2,-3/4)}" should be "{(-1/2,-3/4),(-1/2,-3/4)}"
p.151, line 13. "((-1/2,\pm 3/4)" should be ""(-1/2,\pm 3/4)"
p.154. Introduction. It should be stated that there are examples of
reducible jacobians not of the type discussed in this chapter towards the
end of Kuhn (1988).
p.157, line 6. "Weierstrass points" should be "points of order 2"
p.158, eqn (14.4.4), the coefficient of X should be $b_p - 2p \over p$,
not $b_p^2 - 2p \over p$.
p.164, eqn (15.2.4) is missing the parameter `c'. Also, there
should not be a mixture of upper case `Y' and lower case `y'.
The 3-parameter family of curves C_{bcd} should read:
Y^2 + \bigl( X^3 + X + 1 + cX(X^2 + X) \bigr) Y
= b + (1+3b)X + (1-bd+3b)X^2 + (b-2bd-d)X^3 - bdX^4
p.182: Equation (17.3.5) should be
-t^2 + (2f_5s^2 + 4s^3)t + ...
(that is, 2f_5s^2 instead of 2f_5t^2).
p.189, the LHS of (18.7.11) should be T^2S^4. The rank of (11) is 2.
A search on the Kummer of (11) using PARI gives the point
R=(9+i)/5, T=3(254-603i)/125, S=1. This with its conjugate over Q gives
a rational divisor which is in a different class modulo 2 from the
obvious divisors.
Back cover: Remove "Edited by" on line -11 !!!
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Updatings of Bibliography:
p.207. The Adleman-Huang book is called
"Primality testing and abelian varieties over finite fields"
p.208. Brumer, K should be Brumer, A (twice).
p.209. The preprint of Coray and Manoil has now appeared in print. Acta
Arithmetica 76(1996), 165-189.
p.209 The title of de Weger's paper has the word "Diophantine" between
"hyperelliptic" and "equation".
de Weger, B. M. M.(NL-ROTT-EM)
Correction to the paper: "A hyperelliptic Diophantine equation
related to imaginary quadratic number fields with class number $2$"
[J. Reine Angew. Math. 427 (1992), 137--156; MR 93d:11034]. (English)
J. Reine Angew. Math. 441 (1993), 217--218.
p.209. The Faddeev reference has "the equation" in the title
before $x^4+y^4=1$.
p.209. Flynn, E.V. (1991). "433-422" should be: "433-442".
p.210. Flynn (1995c) has appeared:
Flynn, E.\ V., On a theorem of Coleman,
Manuscr. Math. {\bf 88} (1995), 447--456.
p.211. Grant (1995) "A proof of quintic reciprocity using the
arithmetic of $Y^2=... " has appeared in Acta Arith 75 (1996), 321-337.
p.215. There is a corrigenda page for the Merriman and Smart paper:
Corrigenda: {\em Math. Proc. Camb. Phil. Soc.} {\bf 118} (1995), 189.
p.217. Schaefer (1996a) has appeared:
Schaefer, Edward F. Class groups and Selmer groups.
J. Number Theory 56 (1996), no. 1, 79--114.
There is a paper by S. Fermigier "Exemples des courbes elliptiques de
grand rang sur Q(T) et sur Q poss\'edant des points d'ordre 2.
Comptes Rendus, Paris, 322I (1996), 949-952. It produces examples with
rank 8, 14 respectively. Details of how it was found to appear elsewhere
(unspecified).
The following paper amounts to finding all rational points
on a genus 2 curve with reducible jacobian (although it's
not phrased that way):
G.C. Young. On the solution of a pair of simultaenous equations connected
with the nuptial numbers of Plato.
Proc London Math Soc (2) 23, 27-44 (1924).
A torsion reference:
E.W. Howe, F. Le\'prevost et B. Poonen. Sous-groupes de torsion
d'ordres \'elev\'es de jacobiennes d\'ecomposables de courbes
de genre 2.
Comptes Rendus, Paris 323 I (1996), 1031-1034.
Another reference: R.A. Rankin. Burnside's uniformization. Acta
Arithmetica 79 (1997), 53-57. This gives a uniformization of Y^2=X(X^4-1)
due to Burnside and claimed to be the first explicit uniformization of a
curve of genus greater than 1.
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