Correction: Abelian Varieties defined over their Fields of Moduli, I

Bull London Math. Soc, 4 (1972), 370–372. The proof of the theorem contains an error. Before giving acorrect proof, we state two lemmas. LEMMA 1. Let K/k be a cyclic Galois extension of degree m, let generate Gal (K/k), and let (A, I, ) be defined over K. Supposethat there exists an isomorphism :(A,I,) (A, I, ) over K suchthat v^m–1 ... = 1, where v is the canonical isomorphism(A^m, I^m, ^m) (A, I, ). Then (A, I, ) has a model over k, whichbecomes isomorphic to (A, I, ) over K. Proof. This follows easily from [7], as is essentially explainedon p. 371. LEMMA 2. Let G be an abelian pro-finite group and let : G Q/Z be a continuous character of G whose image has order p.Then either: (a) there exist subgroups G' and H of G such that H is cyclicof order p^m for some m, (G') = 0, and G = G' x H, or (b) for any m > 0 there exists a continuous character m ofG such that p^m _m = . Proof. If (b) is false for a given m, then there exists an element G, of order p^r for some r m, such that () ¦ 0. (Considerthe sequence dual to 0 Ker (p^m) G ^pm G). There exists an opensubgroup G_o of G such that (G₀) = 0 and has order p^r in G/G₀.Choose H to be the subgroup of G generated by , and then aneasy application to G/G₀ of the theory of finite abelian groupsshows the existence of G' (note that () ¦ 0 implies that is not a p-th. power in G). We now prove the theorem. The proof is correct up to the statement(iv) (except that (i) should read: F' k₁ F'ab). To removea minor ambiguity in the proof of (iv), choose to be an elementof Gal (F'_ab/k₂) whose image $$\stackrel{\¯}{\sigma}$$ in Gal (k₁/k₂) generates this last group. The error occursin the statement that the canonical map v : A^P A acts on pointsby sending a^p a; it, of course, sends a a. The proof is correct, however, in the case that it is possibleto choose so that ^p = 1 (in Gal (F'/k₂)). By applying Lemma 2 to G = Gal (F'_ab/k₂) and the map G Gal(k₁/k₂) one sees that only the following two cases have to beconsidered. (a) It is possible to choose so that ^pm = 1, for some m, andG = G' x H where G' acts trivially on k₁ and H is generatedby . (b) For any m > 0 there exists a field K, F'ab K k₁ k₂is a cyclic Galois extension of degree p^m. In the first case, we let K F'_ab be the fixed field of G'.Then (A, I, ), regarded as being defined over K, has a modelover k₂. Indeed, if m = 1, then this was observed above, butwhen m > 1 the same argument applies. In the second case, let : (A, I, ) (A$$\stackrel{\¯}{\sigma}$$, I$$\stackrel{\¯}{\sigma }$$, $$\stackrel{\¯}{\sigma}$$) be an isomorphism defined over k₁ and let v ... ^p–1 = µ(R). If is replaced by for some Aut_k1((A, I, )) then is replacedby ^P. Thus, as µ(R) is finite, we may assume that ^pm–1= 1 for some m. Choose K, as in (b), to be of degree p^m overk₂. Let _m be a generator of Gal (K/k₂) whose restriction tok₁ is $$\stackrel{\¯}{\sigma }$$. Then : (A, I, ) (A^{$$\stackrel{\¯}{\sigma }$$}, I^{$$\stackrel{\¯}{\sigma}$$}, ^{$$\stackrel{\¯}{\sigma }$$} = (A^{$$\stackrel{\¯}{\sigma}$$m}, I^{$$\stackrel{\¯}{\sigma }$$m}, ^{$$\stackrel{\¯}{\sigma}$$m} is an isomorphism defined over K and v ^mpm–1, ... ^m =^pm–1 = 1, and so, by) Lemma 1, (A, I, ) has a model overk₂ which becomes isomorphic to (A, I, over K. The proof may now be completed as before. Addendum: Professor Shimura has pointed out to me that the claimon lines 25 and 26 of p. 371, viz that µ(R) is a puresubgroup of _R^*t, does not hold for all rings R. Thus this condition,which appears to be essential for the validity of the theorem,should be included in the hypotheses. It holds, for example,if µ(R) is a direct summand of µ(F).     

Analytic Discs Attached From the Outside Have Knotted Boundaries

Let be a pseudoconvex domain in C² with smooth boundary, andlet be a smooth embedded analytic disc intersecting transversally along the curve A. Then A isknotted in . 2000 Mathematics Subject Classification 32U99.     

Quasi-Affinity in certain Classes of Operators

The family of operators S + V (, C, Re > 0), where V isan injective S-Volterra operator (that is, [S, V[ = V²) and— A — V^–1 generates a uniformly bounded C₀-semigroup,is studied in the context of similarity and of the weaker quasi-affinityrelation. It is shown that S is similar to S + V for all , C,Re > 1, and is a quasi-affine transform of S + tV for allt 0 and 0 < < 1.     

An Example on Sobolev Space Approximation

For each d2 we construct a connected open set R^d such that = int (clos()), and for each k 1 and each p [1, ), the subsetW^k, () fails to be dense in the Sobolev space W^{k, p}(), in thenorm of W^{k, p}(). 1991 Mathematics Subject Classification 46E35,46F05.     

The Representation of Some Integers as a Subset Sum

Let A N. The cardinality (the sum of the elements) of A willbe denoted by |A| ((A)). Let m N and p be a prime. Let A {1, 2,...,p}. We prove thefollowing results. If |A| [(p+m–2)/m]+m, then for every integer x such that0 x p – 1, there is B A such that |B| = m and (B) x mod p. Moreover, the bound is attained. If |A| [(p+m–2)/m]+m!, then there is B A such that |B| 0 mod m and (B) = (m – 1)!p. If |A| [(p + 1)/3]+29, then for every even integer x such that4p s x p(p + 170)/48, there is S A such that x = (S). In particular,for every even integer a 2 such that p 192a – 170, thereare an integer j 0 and S A such that (S) = a^j+1.     

Correction: on the Intersection Matrices of Curves lying on Irregular Algebraic Surfaces

Professor W. F. Hammond has kindly drawn my attention to a blunderin 4 of the above paper. He referred to the ( – 2r) xß submatrix D of the skew-symmetric matrix displayednear the top of page 181, of which it is asserted that it issquare and non-singular, and pointed out that, from the factthat the matrix of which D forms part is regular, it may onlybe deduced that the columns of D are linearly independent; thatis, it only follows that – 2r ß. The validity of the equation – 2r = ß is essentialto the succeeding argument and, fortunately, may be establishedby alternative means. Using the nomenclature of the paper, wehave on F the set ₁^*, ..., _2r^*, ₁^*, ..., _ß^* of independent3-cycles (independent because they cut independent 1-cycleson the curve C), which may be completed, to form a basis forsuch cycles on F, by a further set ₁', ..., _2q–2r–pof independent 3-cycles, each of which meets C in a cycle homologousto zero on C. The cycles ₁^*, ..., ^* are invariant cycles andare independent on F so that, if > 2r + ß, thereis a non-trivial linear combination ^* of these having zero intersectionon C with each of the cycles ₁^*, ..., _2r^*, ₁^*, ..., _ß^*.Thus we have. (^* ._k^*)c = 0 = (^* ._i^*)c i.e. (^* ._k^*) = 0 = (^* ._i^* on F (1 k 2r; 1 i ß). Furthermore, (_j . C) 0 on C and we have (^* ._j .C)_C = 0 i.e. (^* ._j) = 0 on F (1 j 2q – 2r – ß). It now follows that ^* 0 on F (for it has zero intersectionwith every member of a basic set of 3-cycles on F). But thiscondradicts the assumption that ^* is a non-trivial linear combinationof the independent cycles ₁^*, ...,^*; and hence < 2r + ß.     

Sub- and Superadditive Properties of Fejer's sine Polynomial

Let be Fejér's sine polynomial. We prove the following statements.

1. The inequality holds for all x, y (0, ) with x + y < if and only if 0 and + rß 1.
2. The converse of the above inequality is valid for allx, y (0, ) with x + y < if and only if 0 and + rß 1.
3. For all n N and x, y [0, ] we have . Both bounds are best possible.

2000 Mathematics Subject Classification 42A05, 26D05 (primary),39B62 (secondary).     

Cardinal Invariants and Eventually Different Functions

In this paper we study several kinds of maximal almost disjointfamilies. In the main result of this paper we show that forsuccessor cardinals , there is an unexpected connection betweeninvariants a_e(), b() and a certain cardinal invariant m_d(⁺)on ⁺. As a corollary we get for example the following result.For a successor cardinal , even assuming that ^< = and 2= ⁺, the following is not provable in Zermelo–Fraenkelset theory. There is a ⁺-cc poset which does not collapse andwhich forces a() = ⁺ < a_e() = ⁺⁺ = 2. We also apply the ideasfrom the proofs of these results to study a = a() and non(M).2000 Mathematics Subject Classification 03E17 (primary), 03E05(secondary).     

Heat Kernel Estimates and LP Spectral Independence of Elliptic Operators

Let be an open subset of R^d, and let T_p for p[1, ) be consistentC₀-semigroups given by kernels that satisfy an upper heat kernelestimate. Denoting their generators by A_p, we show that thespectrum (A_p) is independent of p[1, ). We also treat the caseof weighted L^p-spaces for weights that satisfy a subexponentialgrowth condition. An example shows that independence of thespectrum may fail for an exponential weight. 1991 MathematicsSubject Classification 47D06, 47A10, 35P05.     

An infinite Family of 4-Arc-Transitive Cubic Graphs Each with Girth 12

If p is any prime, and is that automorphism of the group SL(3,p) which takes each matrix to the transpose of its inverse,then there exists a connected trivalent graph (p) on vertices with the split extensionSL(3, p) as a group of automorphisms acting regularly on its4-arcs. In fact if p 3 then this group is the full automorphismgroup of (p), while the graph (3) is 5-arc-transitive with fullautomorphism group SL(3,3)0 x C₂. The girth of (p) is 12, exceptin th case p = 2 (where the girth is 6). Furthermore, in allcases (p) is bipartite, with SL(3, p) fixing each part. Alsowhen p 1 mod 3 the graph (p) is a triple cover of another trivalentgraph, which has automorphism group PSL(3, p)0 acting regularlyon its 4-arcs. These claims are proved using elementary theoryof symmetric graphs, together with a suitable choice of threematrices which generate SL(3, Z). They also provide a proofthat the group 4⁺(a¹²) described by Biggs in Computational grouptheor(ed. M. Atkinson) is infinite.     

Uncountable Saturated Structures have the Small Index Property

We prove the following theorem. Let m be an uncountable saturatedstructure of cardinality = ^< and assume that G is a subgroupof Aut (m) whose index is less than or equal to . Then thereexists a subset A of cardinality strictly less than such thatevery automorphism of m leaving A pointwise fixed is in G.     

On the ideals and singularities of secant varieties of Segre varieties

We find minimal generators for the ideals of secant varietiesof Segre varieties in the cases of _k(¹ x ⁿ x ^m) for all k, n,m, ₂(ⁿ x ^m x ^p x ^r) for all n, m, p, r (GSS conjecture for fourfactors), and ₃(ⁿ x ^m x ^p) for all n, m, p and prove they arenormal with rational singularities in the first case and arithmeticallyCohen–Macaulay in the second two cases.     

Smoothness of Holonomies for Codimension 1 Hyperbolic Dynamics

Hyperbolic invariant sets of C¹⁺ diffeomorphisms where eitherthe stable or unstable leaves are 1-dimensional are consideredin this paper. Under the assumption that the has local productstructure, the authors prove that the holonomies between the1-dimensional leaves are C¹⁺ for some 0 < < 1.     

Trivial centralizers for codimension-one attractors

We show that if is a codimension-one hyperbolic attractor fora C^r diffeomorphism f, where 2 r , and f is not Anosov, thenthere is a neighborhood of f in Diff^r(M) and an open and denseset of such that any g has a trivial centralizer on thebasin of attraction for .     

The Dirichlet problem by variational methods

Let ^N be a bounded open set and C( ). Assume that has an extensionC() such that H^–1().Then by the Riesz representation theorem there exists a unique

We show that u+ coincides with the Perron solutionof the Dirichlet problem

This extends recent results by Hildebrandt [Math. Nachr. 278(2005), 141–144] and Simader [Math. Nachr. 279 (2006),415–430], and also gives a possible answer to Hadamard'sobjection against Dirichlet's principle.     

Boundaries of reduced free group C*-algebras

We prove that the crossed product C*-algebra C*_r(, ) of a freegroup with its boundary sits naturally between the reducedgroup C*-algebra C*_r and its injective envelope I(C*_r). In otherwords, we have natural inclusion C*_r C*_r(, ) I(C*_r) of C*-algebras.     

The Polynomials Associated with a Julia Set

We prove that, with two exceptions, the set of polynomials withJulia set J has the form {pⁿ:nN,} where p is one of these polynomialsand is the symmetry group of J. The exceptions occur when Jis a circle or a straight line segment.     

Extremal Properties of Contraction Semigroups on Hilbert and Banach Spaces

Let f be a unit vector and T = {T(t) = e^tA: t 0} be a (C₀)contraction semigroup generated by A on a complex Hilbert spaceX. If |T(t)f,f| 1 as t then f is an eigenvector of A correspondingto a purely imaginary eigenvalue. If one allows X to be a Banachspace, the same situation can be considered by replacing T(t)f,fby (T(t)f) where is a unit vector in X^* dual to f. If |(T(t)f)| 1, as t , is f an eigenvector of A? The answer is sometimesyes and sometimes no.     

Volume of a small Extrinsic Ball in a Submanifold

For a submanifold M^p R, we determine a two-term asymptoticformula for vol (M^p B(x)) for x M^p as 0. The second termis a quadratic curvature invariant of the second fundamentalform of the imbedding. Imbedded spheres are characterized amongcompact hypersurfaces by this term.     

On A Minimization Problem for Vector Fields in L1

This paper treats the problem of minimizing the norm of vectorfields in L¹ with prescribed divergence. The ridge of . playsan important role in the analysis, and in the case where R²is a polygonal domain, the ridge is thoroughly analysed andsome examples are presented. In the case where Rⁿ is a Lipschitzdomain and the divergence is a finite positive Borel measure,the infimum is calculated, and it is shown that if an extremalexists, then it is of the form ₁ = –Fd, where F is a nonnegativefunction and d(x) is the distance from x to the boundary .Finally, if R² is a polygonal domain and the measure is representedby a nonnegative continuous function, then an explicit expressionfor the extremal is given, and it is proven that this extremalis unique.