Moments and non-vanishing of L -functions over subgroups of optimal index

We prove the existence of certain rationally rigid triples in${E}_{8}(p)$for good primes$p$(i.e. $p>5$) thereby showing that these groups occur as Galois groups over the field of rational numbers. We show that these triples arise from rigid triples in the algebraic group and prove that they generate an interesting subgroup in characteristic zero. As a byproduct of the proof, we derive a remarkable symmetry between the character table of a finite reductive group and that of its dual group. We also give a short list of possible overgroups of regular unipotent elements in simple exceptional groups.

Let $A$ be an abelian variety defined over a number field $F$. We prove a control theorem for the fine Selmer group of the abelian variety $A$ which essentially says that the kernel and cokernel of the natural restriction maps in a given $\mathbb{Z}_p$-extension $F_\infty/F$ are finite and bounded. We emphasise that our result does not have any constraints on the reduction of $A$ and the ramification of $F_\infty/F$. As a first consequence of the control theorem, we show that the fine Tate-Shafarevich group over an arbitrary $\mathbb{Z}_p$-extension has trivial $\Lambda$-corank. We then derive an asymptotic growth formula for the $p$-torsion subgroup of the dual fine Selmer group in a $\mathbb{Z}_p$-extension. However, as the fine Mordell-Weil group needs not be $p$-divisible in general, the fine Tate-Shafarevich group needs not agree with the $p$-torsion of the dual fine Selmer group, and so the asymptotic growth formula for the dual fine Selmer groups do not carry over to the fine Tate-Shafarevich groups. Nevertheless, we do provide certain sufficient conditions, where one can obtain a precise asymptotic formula.

Let A be an abelian variety defined over a number field F . We prove a control theorem for the fine Selmer group of the abelian variety A which essentially says that the kernel and cokernel of the natural restriction maps in an arbitrarily given \mathbb{Z}_p -extension F_\infty/F are finite and bounded. We emphasise that our result does not have any constraints on the reduction of A and the ramification of F_\infty/F . As a first consequence of the control theorem, we show that the fine Tate-Shafarevich group over an arbitrary \mathbb{Z}_p -extension has trivial \Lambda -corank. We then derive an asymptotic growth formula for the p -torsion subgroup of the dual fine Selmer group in a \mathbb{Z}_p -extension. However, as the fine Mordell-Weil group need not be p -divisible in general, the fine Tate-Shafarevich group need not agree with the p -torsion of the dual fine Selmer group, and so the asymptotic growth formula for the dual fine Selmer groups do not carry over to the fine Tate-Shafarevich groups. Nevertheless, we do provide certain sufficient conditions, where one can obtain a precise asymptotic formula.

Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = ℚp, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = ℤp, the ring of padic integers. Classical wavelet theories, which require a non trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of the discrete quotient Ĝ/H⊥ to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to socalled wavelet sets in the dual group Ĝ. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.

In this paper, we compare the structure of Selmer groups of certain classes of Galois representations over an admissible $p$-adic Lie extension. Namely, we show that the $\pi$-primary submodules of the Pontryagin dual of the Selmer groups of two Galois representations have the same elementary representations when the two Galois representations in question are either Tate dual to each other or are congruent to each other. In the first situation, our result gives a partial answer to the question of Greenberg on whether the Pontryagin dual of the Selmer groups of two Galois representations that are Tate dual to each other are pseudo-isomorphic (up to a twist of the Iwasawa algebra). In the second situation, our result will be applied to study the variation of the $\pi$-primary submodules of the dual Selmer groups of certain specialization of a big Galois representation. One of the important ingredient in our proofs is an asymptotic formula for $\pi$-primary modules over a noncommutative Iwasawa algebra which can be viewed as a generalization of a weak analog of the classical Iwasawa asymptotic formula.

Using the Iwasawa structure theorem for connected locally compact Hausdorff groups we show that every locally compact Hausdorff group G is homeomorphic to Rn × K × D, where n is a non-negative integer, K is a compact group and D is a discrete group. This makes recent results on cardinal numbers associated with the topology of locally compact groups more transparent. For abelian G, we note that the dual group, Ĝ, is homeomorphic to This leads us to the relationship card G = ω0(Ĝ) + 2ω0(G), where ω (respectively, ω0) denotes the weight (respectively local weight) of the topological group. From this classical results such as card G = 2 card Ĝ for compact Hausdorff abelian groups, and ω(G) = ω(Ĝ) for general locally compact Hausdorff abelian groups are easily derived.

Consider a locally compact group G=Q⋉V such that V is abelian and the action of Q on the dual abelian group Vˆ has a free orbit of full measure. We show that such a group G can be quantized in three equivalent ways:(1)by reflecting across the Galois object defined by the canonical irreducible representation of G on L2(V);(2)by twisting the coproduct on the group von Neumann algebra of G by a dual 2-cocycle obtained from the G-equivariant Kohn–Nirenberg quantization of V×Vˆ;(3)by considering the bicrossed product defined by a matched pair of subgroups of Q⋉Vˆ both isomorphic to Q.In the simplest case of the ax+b group over the reals, the dual cocycle in (2) is an analytic analogue of the Jordanian twist. It was first found by Stachura using different ideas. The equivalence of approaches (2) and (3) in this case implies that the quantum ax+b group of Baaj–Skandalis is isomorphic to the quantum group defined by Stachura.Along the way we prove a number of results for arbitrary locally compact groups G. Using recent results of De Commer we show that a class of G-Galois objects is parametrized by certain cohomology classes in H2(G;T). This extends results of Wassermann and Davydov in the finite group case. A new phenomenon is that already the unit class in H2(G;T) can correspond to a nontrivial Galois object. Specifically, we show that any nontrivial locally compact group G with group von Neumann algebra a factor of type I admits a canonical cohomology class of dual 2-cocycles such that the corresponding quantization of G is neither commutative nor cocommutative.

Fourth-grade students' understanding of the order and equivalence of rational numbers was investigated in 11 interviews with each of 12 children during an 18-week teaching experiment. Six children were instructed individually and as a group at each of two sites. The instruction relied heavily on the use of manipulative aids. Children's explanations of their responses to interview tasks were used to identify strategies for comparing fraction pairs of three types: same numerators, same denominators, and different numerators and denominators. After extensive instruction, most children were successful but some continued to demonstrate inadequate understanding. Previous knowledge relating to whole numbers sometimes interfered with learning about rational numbers. Rational number concepts are among the most complex and most important mathematical ideas that children encounter before they reach secondary school. The increased attention being given to research on children's acquisition of such concepts reflects their importance. Recent results from national assessments have shown that children have significant difficulty learning and applying concepts related to rational numbers. In a recent national assessment, 30% of the nation's 13-year-olds added the numerators and the denominators to find the sum of 1/2 and 1/3 (Post, 1981), even though a bit of reflection would have suggested that the sum of two positive quantities should not be less than one of them. Only 24% of the 13-year-olds were able to estimate 12/13 + 7/8 by selecting the correct response, 2, from [1, 2, 19, 21, I don't know]. The most recent Minnesota State Assessment in Mathematics identified fractions as the topic most in need of attention (Minnesota Department of Education & Minnesota Council of Teachers of Mathematics, 1976), a finding that is not unique to Minnesota. State and national assessments suggest that students have often failed to internalize a workable concept of rational number. Students often do not consider numerators and denominators in relation to one another but, rather,

For a given finite dimensional Hopf algebra H we describe the set of all equivalence classes of cocycle deformations of H as an affine variety, using methods of geometric invariant theory. We show how our results specialize to the Universal Coefficients Theorem in the case of a group algebra, and we also give examples from other families of Hopf algebras, including dual group algebras and Bosonizations of Nichols algebras. In particular, we use the methods developed here to classify the cocycle deformations of a dual pointed Hopf algebra associated to the symmetric group on three letters. We also give an example of a cocycle deformation over a dual group algebra, which has only rational invariants, but which is not definable over the rational field. This differs from the case of group algebras, in which every 2-cocycle is equivalent to one which is definable by its invariants.

We generalize a well-known result proved by Filaseta and Trifonov in 2002 that the Bessel polynomials of all degrees are irreducible over the field of rational numbers. The proof given here of our generalization appears to be simpler than the known proofs of Filaseta-Trifonov Theorem. We use some recent results by Lehmer, Luca, Najman and Shorey regarding the largest prime divisor of a product of consecutive integers, which play a significant role in making the proof shorter. The backbone of our proof is a theorem based on an extension introduced by Ore of the concept of Newton polygons. We illustrate the utility of our generalization by showing that it leads to new classes of monic irreducible polynomials with integer coefficients for which the known irreducibility criteria for polynomials do not seem to be applicable.

We establish various new results on a problem proposed by Mahler [Some suggestions for further research. Bull. Aust. Math. Soc.29 (1984), 101–108] concerning rational approximation to fractal sets by rational numbers inside and outside the set in question. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich, and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover, we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle-third set $C$ to the set $C$, in terms of the denominator $q$. We further discuss patterns of rational numbers in fractal sets. We highlight two of them: firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets; and secondly, we find properties of the denominator structure of rational points in ‘missing-digit’ Cantor sets, generalizing claims of Nagy and Bloshchitsyn.

It is shown that the convergence of convolution products of probability measures on certain non-locally compact topological abelian groups can be veriÞed by means of characteristic function- als. Analogous results are obtained also for almost everywhere con- vergence of series of independent random elements in the considered groups. A connection with the Sazonov property of the groups is discussed. 1. Preliminaries. Throughout the paper N denotes the set of natu- ral numbers; Q;R, and C are, respectively, the Þelds of rational, real, and complex numbers with the ordinary (Euclidean) metric, and T denotes the multiplicative group of complex numbers of modulus 1 with the metric in- duced from C. For a topological abelian group X we denote by X 0 the topological dual group which consists of all continuous (unitary) characters h : X !T; the group operation in X 0 is the natural pointwise multiplication. No topology in X 0 is speciÞed. A topological abelian group X is called dually separated or DS-group if X 0 separates the points of X; in other words, X is a DS-group if for any dierent x1;x2 2 X there is a character h 2 X 0 such that h(x1) 6 h(x2). Hausdorlocally compact abelian (LCA-) groups and any additive subgroup of any Hausdorlocally convex space are examples of DS-groups. Below we shall see another type of examples too. Let X be a completely regular Hausdortopological space. Denote by Mt(X) the set of all Radon probability measures n deÞned on the Borel o-algebra of X. In Mt(X) we consider only the weak topology (for all the notions unexplained here the reader is referred to (1) and (2)). For Þxed x 2 X we denote by ex the Dirac measure concentrated at x. The Prokhorov theorem says that a subset M o Mt(X) is relatively compact if

It is shown that the convergence of convolution products of probability measures on certain non-locally compact topological abelian groups can be veriÞed by means of characteristic function- als. Analogous results are obtained also for almost everywhere con- vergence of series of independent random elements in the considered groups. A connection with the Sazonov property of the groups is discussed. 1. Preliminaries. Throughout the paper N denotes the set of natu- ral numbers; Q;R, and C are, respectively, the Þelds of rational, real, and complex numbers with the ordinary (Euclidean) metric, and T denotes the multiplicative group of complex numbers of modulus 1 with the metric in- duced from C. For a topological abelian group X we denote by X 0 the topological dual group which consists of all continuous (unitary) characters h : X !T; the group operation in X 0 is the natural pointwise multiplication. No topology in X 0 is speciÞed. A topological abelian group X is called dually separated or DS-group if X 0 separates the points of X; in other words, X is a DS-group if for any dierent x1;x2 2 X there is a character h 2 X 0 such that h(x1) 6 h(x2). Hausdorlocally compact abelian (LCA-) groups and any additive subgroup of any Hausdorlocally convex space are examples of DS-groups. Below we shall see another type of examples too. Let X be a completely regular Hausdortopological space. Denote by Mt(X) the set of all Radon probability measures n deÞned on the Borel o-algebra of X. In Mt(X) we consider only the weak topology (for all the notions unexplained here the reader is referred to (1) and (2)). For Þxed x 2 X we denote by ex the Dirac measure concentrated at x. The Prokhorov theorem says that a subset M o Mt(X) is relatively compact if

Canonical maps on a two-torus in phase space are quantized under most general conditions. Recent results by Keating et al (1999 Nonlinearity 12 579) are thus fully extended in two directions: (a) The translational component of a general canonical map is included in the quantization. (b) All values of Planck's constant, consistent with the toral boundary conditions (BCs), are considered; generically, these values are rational numbers whose numerator must satisfy a number-theoretical condition. Besides the condition on Planck's constant, the quantization is possible only for particular, 'allowed' BCs on the torus. The general equation determining these BCs is derived. Allowed BCs may not exist in some cases; representative examples are the irrational skew translations and Kronecker maps. Exact versions of Egorov's theorem are shown to hold under some conditions. Composition and representation properties of the quantization scheme are studied.

Letp > q > 1 be two coprime integers. In this paper, we prove several results about subsets of the interval [0, 1) which does or does not contain all the fractional parts {ξ (p /q)n }, n = 0, 1, 2, …, for certain non‐zero real number ξ. We show, for instance, that there are no real ξ for which the union of two intervals [8/39, 18/39] ∪ [21/39, 31/39] contains the set {ξ (3/2)n }, n ∈ N. The most important aspect of this result is that the total length of both intervals 20/39 is greater than 1/2: the same result as above for [0, 1/2) would imply that there are no Mahler's Z ‐numbers which the best known unsolved problem in this area. On the other hand, it is shown that there are infinitely many ξ for which {ξ (3/2)n } ∈ (5/48, 43/48) for each integer n ≥ 0. We also give simpler proofs of few recent results in this area. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)