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).     

Frame duality properties for projective unitary representations

Let be a projective unitary representation of a countable groupG on a separable Hilbert space H. If the set B of Bessel vectorsfor is dense in H, then for any vector x H the analysis operator_x makes sense as a densely defined operator from B to ²(G)-space.Two vectors x and y are called -orthogonal if the range spacesof _x and _y are orthogonal, and they are -weakly equivalent ifthe closures of the ranges of _x and _y are the same. These propertiesare characterized in terms of the commutant of the representation.It is proved that a natural geometric invariant (the orthogonalityindex) of the representation agrees with the cyclic multiplicityof the commutant of (G). These results are then applied to Gaborsystems. A sample result is an alternate proof of the knowntheorem that a Gabor sequence is complete in L²( ^d) ifand only if the corresponding adjoint Gabor sequence is ²-linearlyindependent. Some other applications are also discussed.     

A Relation between Hausdorff Dimension and a Condition on Time Sets for the Image by the Brownian Sheet to Possess Interior Points

We prove that if W_{N, d} is a Brownian sheet mapping to R^d and E is a set in (0, )^N of Hausdorff dimensiongreater than , then for almost every rotation about a point x and translation _x such that _x(E) (0, )^N, the set _x(E) is such that almost surely W(E) containsinterior points. The techniques are adapted from Kahane andRosen and generalize to higher dimensional time and range.     

The structure of the normalisers of the congruence subgroups of the Hecke group G5

Let = 2cos (/5) and let []. Denote the normaliser ofG₀() of the Hecke group G₅ in PSL₂() by N(G₀()). Then N(G₀())= G₀(/h), where h is the largest divisor of 4 such that h² divides. Further, N(G₀())/G₀() is either 1 (if h = 1), ₂ x ₂ (if h= 2) or ₄ x ₄ (if h = 4).     

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).     

Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let be a bounded connected open set in R^N, N 2, and let –0be the Dirichlet Laplacian defined in L²(). Let > 0 be thesmallest eigenvalue of –, and let > 0 be its correspondingeigenfunction, normalized by ||||₂ = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying Let – 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||₂=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying 1991 Mathematics SubjectClassification 47F05.     

Packing, Tiling, Orthogonality and Completeness

Let R^d be an open set of measure 1. An open set DR^d is calleda ‘tight orthogonal packing region’ for if D–Ddoes not intersect the zeros of the Fourier transform of theindicator function of , and D has measure 1. Suppose that isa discrete subset of R^d. The main contribution of this paperis a new way of proving the following result: D tiles R^d whentranslated at the locations if and only if the set of exponentialsE = {exp 2i, x: } is an orthonormal basis for L²(). (This resulthas been proved by different methods by Lagarias, Reeds andWang [9] and, in the case of being the cube, by Iosevich andPedersen [3]. When is the unit cube in R^d, it is a tight orthogonalpacking region of itself.) In our approach, orthogonality ofE is viewed as a statement about ‘packing’ R^d withtranslates of a certain non-negative function and, additionally,we have completeness of E in L²() if and only if the above-mentionedpacking is in fact a tiling. We then formulate the tiling conditionin Fourier analytic language, and use this to prove our result.2000 Mathematics Subject Classification 52C22, 42B99, 11K70.     

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.     

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.     

Calculs De Facteurs Epsilon De Paires Pour GLn Sur Un Corps Local, I

Soient F un corps commutatif localement compact non archimédienet un caractère additif non trivial de F. Soient unereprésentation du groupe de Weil–Deligne de F,et sa contragrédiente. Nous calculons le facteur (, , ). De manière analogue, nous calculons le facteur (x, , ) pour toute représentationadmissible irréductible de GL_n(F). En conséquence,si F est de caractéristique nulle et si et se correspondentpar la correspondance de Langlands construite par M. Harris,ou celle construite par les auteurs, alors les facteurs (, , s) et (x, , s) sont égaux pour tout nombre complexe s. Let F be a non-Archimedean local field and a non-trivial additivecharacter of F. Let be a representation of the Weil–Delignegroup of F and its contragredient representation. We compute (, , ). Analogously, we compute (x, , ) for all irreducible admissible representations of GL_n(F).Consequently, if F has characteristic zero, and , correspondvia the Langlands correspondence established by M. Harris orthe correspondence constructed by the authors, then we have(, , s) = (x, , s) for all sC. 1991 Mathematics Subject Classification22E50.     

A Variation on a Theorem of Galvin and Hajnal

Let be a singular cardinal of regular uncountable cofinality. Let {(): < } be a continuous increasing sequence withlimit , and let =₍₎₊₍₎, < be regular cardinals. Let I be a normal ideal on , and assume that the reduced product_</I admits a cofinal -scale of ordinal functions. Then ₊, where =||||_I is the I-norm of .     

Singular Cardinal Problem: Shelah's Theorem on 2N{omega}

This is an expository paper giving a complete proof of a theoremof Saharon Shelah: if 2 < for all n < , then 2 < ₄.     

Intersections of Odd Order Hall Subgroups

Given a set of odd primes and a -solvable finite group G, onecan choose three (not necessarily distinct) Hall -subgroupsof G whose intersection is O(G). This statement relies on aresult concerning long orbits in coprime actions of groups ofodd order. 2000 Mathematics Subject Classification 20D20.     

Logarithmic Convexity for Supremum Norms of Harmonic Functions

We prove the following convexity property for supremum normsof harmonic functions. Let be a domain in Rⁿ, ₀ and E a subdomainand a compact sebset of ,respectively. Then there exists a constant = (E, ₀, ) (0, 1) such that for all harmonic functions u on, the inequality is valid.The case of concentric balls E plays a key role in the proof.For positive harmonic funcitons ono osuch balls, we determinethe sharp constant in the inequlity.     

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.     

A reverse Denjoy theorem

Suppose that C₁ and C₂ are two simple curves joining 0 to ,non-intersecting in the finite plane except at 0 and enclosinga domain D which is such that, for all large r, has measure at most 2, where 0 < < .Suppose also that u is a non-constant subharmonic function inthe plane such that u(z) = B(|z|, u) for all large z C₁ C₂.Let A_D(r, u) = inf { u(z):z D and | z | = r }. It is shownthat if A_D(r, u) = O(1) (or A_D(r, u) = o(B(r, u))), then lim_r B(r, u)/r^/2 > 0 (or lim_r log B(r, u)/log r /2).     

The A-Polynomial has Ones in the Corners

1. Definition of the A-polynomial The A-polynomial was introduced in [3] (see also [5]), and wepresent an alternative definition here. Let M be a compact 3-manifoldwith boundary a torus T. Pick a basis , µ of ₁T, whichwe shall refer to as the longitude and meridian. Consider thesubset R_U of the affine algebraic variety R = Hom (₁M, SL₂C)having the property that () and (µ) are upper triangular.This is an algebraic subset of R, since one just adds equationsstating that the bottom-left entries in certain matrices arezero. There is a well-defined eigenvalue map given by taking the top-left entries of () and (µ).1991 Mathematics Subject Classification 57M25, 57M50.     

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let p^h(x) = inf||–₁(Ric_x(,)–2Hess(h_x(,)).Here Ric_x denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if ^h–p^h<0. Here ^h =:+2L_h is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.     

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.     

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.