REMARKS AND ERRATA

Maximally complete fields

• A remark: Irving Kaplansky tells me that the residue field part of his “Hypothesis A,” namely the condition that every polynomial of the form

a_{0}x^{p^{n}} + a_{1}x^{p^{n−1}} + ⋯ + a_{n−1}x^{p} + a_{n}x + a_{n+1}

with each a_{i} in the residue field k have a root in k, was shown by Whaples to be equivalent to the condition that k have no extensions of degree divisible by p. See the “Afterthought” to “Maximal Fields with Valuations” in [Kap95].

Union-closed families

• p. 256, Theorem 1, in condition 2: Change F ⊎ G = G to F ⊎ G ⊆ G. A similar change should be made to the beginning of lines −5 and −3 on p. 257, and to the beginning of line 6 on p. 260. (Thanks to Theresa Vaughan.)

Computational aspects of curves of genus ≥ 2

• In the printed version, “positive integer g” should be changed to “g ≥ 2” in the statement of the Shafarevich conjecture in Section 11. The statement “For each number field K and set of places S, there are at most finitely many genus 1 curves over K with good reduction outside S” implies that the Shafarevich–Tate group of every elliptic curve over a number field is finite. The latter is not yet proved.

The number of intersection points made by the diagonals of a regular polygon

• The published version contains a typo introduced while texing the manuscript: in Theorem 1, the 232 in the formula for I(n) should be 262, as the routines in ngon.m give. (Thanks to Steve Sommars for noticing this.)

The classification of preperiodic points of quadratic polynomials over ℚ: a refined conjecture

• The proof of Proposition 1 in Section 4 contains an error: the point (2, \sqrt{33}) is not on C! Instead (−2, \sqrt{33}) is; apparently a sign got dropped halfway through the computation. To complete the 2-descent correctly one must use also the 2-adic information. The end result of the computation is the same as before, so the main results of the paper still hold. (Thanks to Ken Kramer for noticing the error.)

Date: February 28, 2016.

The Cassels–Tate pairing on polarized abelian varieties

• In the printed version, Section 2 suggests that the maximal divisible subgroup M_{div} of an abelian group M equals the set of m ∈ M such that for all n ≥ 1 there exists x ∈ M such that nx = m. This is false in general (the latter set can be larger), but it is true if the p-torsion subgroup M[p] is finite for each prime p. The latter condition holds for each group in the paper for which the notation M_{div} is used, so the rest of the paper is unaffected. (Thanks to Hendrik Lenstra for noticing the error.)

• Let X be a smooth projective geometrically integral surface over a finite field of characteristic p, and let ℓ be a prime not equal to p. The question of Tate in Section 11, whether a certain antisymmetric pairing on Br(X)_{nd}(ℓ) is always alternating, is still open. But [LLR05] proves in any case that Br(X) is of square order (if it is finite, or even if Br(X)(ℓ) is finite for any prime ℓ).

Mordell-Lang plus Bogomolov

• In the printed version, Remark 1 following Proposition 5 should be replaced by the following, because heights associated to effective divisors are not guaranteed to be bounded below for points on the divisor itself. (Thanks to Najmuddin Fakhruddin for noticing this.)

“Condition (∗) is satisfied for (U, f) if there exists an integral projective variety V containing U as an open dense subset, and an ample line bundle ℒ on V such that f extends to a morphism f̄: V → V and a height associated to 𝒩 := f̄^{∗}ℒ ⊗ ℒ^{⊗−q} in (Pic V) ⊗ ℚ is bounded below for some 1 < q ∈ ℚ. The condition on 𝒩 is satisfied, for instance, if 𝒩 is the pullback of an ample sheaf under some morphism of varieties.”

In the application to semiabelian varieties, one can then take f̄ = [m] for some m ≥ 2, q = m, and 𝒩 = ℒ_{1}^{⊗(m^{2}−m)}. The results of the paper still hold.

Genus-two curves with 22 torsion points

The displayed definition of the function f is not the right one for its intended purpose. To test whether the point (u, v) is torsion, one needs to express the vector of integrals along γ_{u} as a linear combination of both the vector of integrals along X_{t}(ℝ) and the vector of integrals along an 1-cycle representing an independent complex-conjugation-invariant class in H_{1}(X(ℂ), ℚ), and to define f(t, u) as that pair of coefficients. Probably this new f still has a 2-dimensional image, but this has not yet been checked, so the proof as it stands is incomplete.

Algebraic families of nonzero elements of Shafarevich-Tate groups

• Section 2.6 implicitly assumes that A is principally polarized, which is the case in the application. If A is a general abelian variety, Y should be a torsor of Â, and it is Â that should be identified with Pic_{X/k}^{0}. (Thanks to my co-author for noticing this.)

Squarefree values of multivariable polynomials

The following changes should be made to the printed version:

• In Theorem 3.2 and Lemma 6.2 the condition “x_{n} appears in f(x)” should be strengthened to “x_{n} appears in each irreducible factor of f(x)”.

• The statement of Theorem 8.1 is OK, but some changes are needed in the proof, since one cannot ensure that t will be among the t_{i_{α}} at the end. One should remark that in the generalization of Lemma 7.2 it suffices to have t_{i}/t_{j} ∉ K^{p} for some i, j, and then only allow subsets {i_{α_{1}}, …, i_{α_{r}}} for which the corresponding t_{i_{α}} satisfy this condition on ratios: this can be done provided deg D is sufficiently large.

The William Lowell Putnam Mathematical Competition 1986–2000: problems, solutions, and commentary

The “Related question” on page 68 is wrong. Condition (b) should be replaced by the hypothesis that all rows and columns of M have the same sum.

Orbits of automorphism groups of fields

In the printed version:

• In the proof of Lemma 1.6, the statement M_{d} = cM_{d} holds but does not follow from the previous lines of the proof. It was used in the last sentence to show that multiplication-by-c maps M_{d} isomorphically to cM. Luckily, the latter also follows from M_{d} = cM and cM = c^{2}M and the fact that cM is torsion-free. (Thanks to E. Mehmet Kiral for noticing the gap.)

• The first paragraph of the proof of Lemma 2.8 should read as follows:

Since M − N generates M as a module, the sequence (f^{m}M)_{m≥1} also contains only finitely many sets. But this sequence is decreasing, so f^{m}M = f^{m+1}M for some m.

Thanks to P. K. Sharma for noticing the error.

Twists of X(7) and primitive solutions to x^{2} + y^{3} = z^{7}

In the printed version, in the proof of Lemma 4.6, “twist by 1/3” should be “twist by −1/3”.

Unramified covers of Galois covers of low genus curves

In Remark 1.2 of the printed version, it should be assumed that Y has genus at least 2. (Thanks to Amador Martin-Pizarro for noticing this.)

Isomorphism types of commutative algebras of finite rank over an algebraically closed field

Marco Pellegrini and Chiara Tamburini pointed out some redundant entries in Table 1 of the printed version. These arose from an incorrect classification of symmetric bilinear forms in characteristic 2. This has been corrected in the online version.

Smooth hypersurface sections containing a given subscheme over a finite field

The variable b should be c in a few places in the printed version: in “For d ≥ b” in the proof of Lemma 2.1, and in the statement and proof of Lemma 3.2.

The set of nonsquares in a number field is diophantine

In the printed version, the equation at the end of the proof of Corollary 1.2 should be A_{n+1} = A_{1} ∪{t^{2} : t ∈ A_{n} and −t ∈ A_{n}}. (Thanks to Jean-Louis Colliot-Thélène for noticing this.)

Random maximal isotropic subspaces and Selmer groups

• In the printed version, in the proof of Proposition 2.6(a), “codimension 1 subspaces of W ” should be “codimension 1 subspaces of W not containing v”. The same extra condition should be imposed on W_{1}.

• The observation that a Selmer group could be an intersection of maximal isotropic subspaces in a finite dimensional space (Remark 4.15) appeared earlier in a more limited context, but with a similar proof. Namely, for elliptic curves E over a number field k with E[2] ⊂ E(k), the 2-Selmer group Sel_{2} E was shown to be an intersection of two subspaces of a finite-dimensional 𝔽_{2}-vector space that were maximal isotropic with respect to a symmetric bilinear pairing (slightly weaker than being maximal isotropic with respect to a quadratic form): in [CTSSD98], see Proposition 1.2.1 in conjunction with Proposition 1.1.1 and the remark following it.

• Warning: The references to “PR11” are to the arXiv version http://arxiv.org/pdf/1104.2105v1.pdf, not to the published version [PR11].

Average rank of elliptic curves

The following corrections should be made in the printed version:

• The construction in [BS15, second half of §4.1] of a positive-density family of elliptic curves in which the root number is equidistributed is actually taken from [Won01, p. 25 and §9], so the latter should have been credited.

• In the first paragraph of §4.2, S(ℚ_{p})^{min} should be S(ℤ_{p})^{min}. (Thanks to Ruthi Hortsch for noticing this.)

• In Lemma 4.3, it is necessary to add the hypothesis that f is locally solvable. (Thanks to Jack Thorne for noticing this.)

Néron–Severi groups under specialization

Here is a more detailed explanation of why the homomorphism Pic X → Pic X_{K} in (3.4) is an isomorphism. (This came out of a discussion with Kęstutis Česnavičius.)

First, X is smooth over a regular local ring R, so X and X_{K} are regular. This means that Pic X and Pic X_{K} can be understood as Weil divisor class groups.

Let X → Y → Spec R be the Stein factorization of X → Spec R. Then Y is finite over Spec R, and Y is the normalization of Spec R in X [SP, Tag 03H0], so Y is a semilocal Dedekind scheme.

Since X → Spec R is smooth, the special fiber X_{k} is a disjoint union of irreducible divisors D of X. Any such D maps to some point y of Y lying above the closed point of Spec R. Since Y is a semilocal Dedekind ring, y is a principal divisor on Y. Let F be the fiber of X → Y above y, so F is a principal divisor on X. Now F is contained in X_{k}, and F is connected (by definition of Stein factorization), and F contains a connected component D of X_{k} (even scheme-theoretically, since X_{k} is reduced), so F = D. Thus D is principal. The kernel of Pic X → Pic X_{K} is spanned by the classes of such divisors D, so Pic X → Pic X_{K} is injective.

It is also surjective, since if E is an irreducible divisor on X_{K}, its Zariski closure in X is an irreducible divisor of X whose class maps to the class of E.

References

[BS15] Manjul Bhargava and Arul Shankar, Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0, Ann. of Math. (2) 181 (2015), no. 2, 587–621, DOI 10.4007/annals.2015.181.2.4. MR3275847

[CTSSD98] J.-L. Colliot-Thélène, A. N. Skorobogatov, and Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math. 134 (1998), no. 3, 579–650, DOI 10.1007/s002220050274. MR1660925 (99k:11095)

[Kap95] Irving Kaplansky, Selected papers and other writings, Springer-Verlag, New York, 1995. With an introduction by Hyman Bass. MR1340874 (97a:01074)

[LLR05] Qing Liu, Dino Lorenzini, and Michel Raynaud, On the Brauer group of a surface, Invent. Math. 159 (2005), no. 3, 673–676. MR2125738

[PR11] Bjorn Poonen and Eric Rains, Self cup products and the theta characteristic torsor, Math. Res.

Lett. 18 (2011), no. 6, 1305–1318, DOI 10.4310/MRL.2011.v18.n6.a18. MR2915483

[SP] The  Stacks  Project  authors,  Stacks  Project,  2015.  Available  at

http://stacks.math.columbia.edu .

[Won01] Siman Wong, On the density of elliptic curves, Compositio Math. 127 (2001), no. 1, 23–54, DOI 10.1023/A:1017514507447. MR1832985 (2002d:11066)