On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅¹⁰⁹. We determine all these K's with dK?¹⁰⁶. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅¹⁰¹¹.     

A theorem on analytic strong multiplicity one

Let K be an algebraic number field, and π=⊗πv an irreducible, automorphic, cuspidal representation of GLm(AK) with analytic conductor C(π). The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant c depending on ε>0,m, and K only, such that π can be decided completely by its local components πv with norm N(v)<c⋅C(π)^2m+ε.     

On geometric SDPS-sets of elliptic dual polar spaces

Let n∈N?{0,1} and let K and K^′ be fields such that K^′ is a quadratic Galois extension of K. Let Q⁻(2n+1,K) be a nonsingular quadric of Witt index n in PG(2n+1,K) whose associated quadratic form defines a nonsingular quadric Q⁺(2n+1,K^′) of Witt index n+1 in PG(2n+1,K^′). For even n, we define a class of SDPS-sets of the dual polar space DQ⁻(2n+1,K) associated to Q⁻(2n+1,K), and call its members geometric SDPS-sets. We show that geometric SDPS-sets of DQ⁻(2n+1,K) are unique up to isomorphism and that they all arise from the spin embedding of DQ⁻(2n+1,K). We will use geometric SDPS-sets to describe the structure of the natural embedding of DQ⁻(2n+1,K) into one of the half-spin geometries for Q⁺(2n+1,K^′).     

Pillai's conjecture revisited

We prove a generalization of an old conjecture of Pillai (now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3^x−2^y=c has, for |c|>13, at most one solution in positive integers x and y. In fact, we show that if N and c are positive integers with N?2, then the equation |(N+1)^x−N^y|=c has at most one solution in positive integers x and y, unless (N,c)∈{(2,1),(2,5),(2,7),(2,13),(2,23),(3,13)}. Our proof uses the hypergeometric method of Thue and Siegel and avoids application of lower bounds for linear forms in logarithms of algebraic numbers.     

Revisiting queueing output processes: a point process viewpoint

After some historical notes concerning queueing output processes N _dep??, the paper discusses methods for establishing asymptotic linear relations for var??N _dep??(0,t], whether in the crude form B ₁ t or the more detailed form B ₁ t+B ₀+o(1) for t→∞. The crude form holds whenever the process N _adm of customers admitted to service has a linear asymptote, and then (var??N _dep??(0,t])/t and (var??N _adm(0,t])/t share a common limit (that may be infinite) in stationary G/G/k/K systems. A standard integral formula for the variance of a stationary orderly point process shows that, if N _dep?? is a renewal process whose generic lifetime X has finite second moment, then B ₁=(var??X)/([E(X)]²), and the more detailed linear asymptote holds when E(X ³) is finite. Geometric ergodicity of the queue size process Q(?) in stationary M/M/k/K systems establishes that the more detailed linear asymptote is true for them. It is conjectured that var??N(0,t]～B ₁ t for any stationary point process N possessing an embedded regenerative structure.     

Siegel's lemma with additional conditions

Let K be a number field, and let W be a subspace of KN, N?1. Let V₁,…,VM be subspaces of KN of dimension less than dimension of W. We prove the existence of a point of small height in , providing an explicit upper bound on the height of such a point in terms of heights of W and V₁,…,VM. Our main tool is a counting estimate we prove for the number of points of a subspace of KN inside of an adelic cube. As corollaries to our main result we derive an explicit bound on the height of a nonvanishing point for a decomposable form and an effective subspace extension lemma.     

Reflection theorem for divisor class groups of relative quadratic function fields

We present the reflection theorem for divisor class groups of relative quadratic function fields. Let K be a global function field with constant field Fq. Let L₁ be a quadratic geometric extension of K and let L₂ be its twist by the quadratic constant field extension of K. We show that for every odd integer m that divides q+1 the divisor class groups of L₁ and L₂ have the same m-rank.     

Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces

Let K be a field of characteristic 0 and let (K^*)ⁿ denote the n-fold Cartesian product of K^*, endowed with coordinatewise multiplication. Let Γ be a subgroup of (K^*)ⁿ of finite rank. We consider equations (*) a₁x₁ + … + a_nx_n = 1 in x = (x₁x_n)Γ, where a = (a₁,a_n)(K^*)ⁿ. Two tuples a, b(K^*)ⁿ are called Γ-equivalent if there is a uΓ such that b = u · a. Gy?ry and the author [Compositio Math. 66 (1988) 329-354] showed that for all but finitely many Γ-equivalence classes of tuples a(K^*)ⁿ, the set of solutions of (*) is contained in the union of not more than 2⁽ⁿ⁺¹! proper linear subspaces of Kⁿ. Later, this was improved by the author [J. reine angew. Math. 432 (1992) 177-217] to (n!)²ⁿ⁺². In the present paper we will show that for all but finitely many Γ-equivalence classes of tuples of coefficients, the set of non-degenerate solutions of (*) (i.e., with non-vanishing subsums) is contained in the union of not more than 2ⁿ proper linear subspaces of Kⁿ. Further we give an example showing that 2ⁿ cannot be replaced by a quantity smaller than n.     

Nontrivial lower bounds for the least common multiple of some finite sequences of integers

We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form un=an(n+t)+b with (a,t,b)∈Z³, a?5, t?0, gcd(a,b)=1. From this, we deduce for instance the lower bound: lcm{¹²+1,²²+1,…,n²+1}?0,32n(1,442) (for all n?1). In the last part of this article, we study the integer lcm(n,n+1,…,n+k) (k∈N, n∈N^∗). We show that it has a divisor dn,k simple in its dependence on n and k, and a multiple mn,k also simple in its dependence on n. In addition, we prove that both equalities: lcm(n,n+1,…,n+k)=dn,k and lcm(n,n+1,…,n+k)=mn,k hold for an infinitely many pairs (n,k).     

Representations by quaternary quadratic forms whose coefficients are 1, 4, 9 and 36

Explicit formulae are determined for the number of representations of a positive integer by the quadratic forms ax²+by²+cz²+dt² with a,b,c,d∈{1,4,9,36}, gcd(a,b,c,d)=1 and a?b?c?d.     

Commutative Algebras In Which Polynomials Have Infinitely Many Roots

Let K be a field. Then there exists a commutative K-algebra A such that each polynomial in K[X] of degree at least 2 has infinitely many roots in A. If B is a finite-dimensional commutative K-algebra and char(K) ≠ 3 (resp., char(K ) = 3), then X ² + X + 1 (resp., X ² + X-1) has only finitely many roots in B. Relevant examples are also given, especially of K-algebras of the form K + N, where N is the nilradical.     

Instances of the Conjecture of Chang

We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that if ρ is a cardinal such that ρ ≧ ω₁, and (ρ⁺⁺,ρ⁺↠(ρ⁺, ρ), then 0⁺ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries, we get: Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧ K ⁺ ≧ω₂ and (λ⁺, λ)↠(K ⁺,K. Then 0⁺ exists. Theorem B.Assume that there exists a singularcardinal κ such that(K ⁺,K↠(ω₁, ω₀. Then 0⁺ exists. Theorem C.Assume that (λ ⁺⁺, λ). Then 0⁺ exists (also ifK=ω ₀. Remark. Here, as in the paper of Donder and Koepke, “O⁺ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model. Theorem D.Assume that (λ ⁺⁺, λ)↠(K ⁺, K) and thatK≧ω ₁. Then 0⁺ exists. Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω ₂, or that κ is inaccessible. We shall recall this in due place. Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K ⁺, K↠(ω_n +1, ω_n), for κ singular andn≧0. 0266 0152 V 3 The consistency of several statements of this sort—a prototype of which is (N _ω+1,N _ω)↠(ω₁, ω₀) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the author of the present paper.     

Wild tame-by-cyclic extensions

Suppose G is a semi-direct product of the form Z/pⁿ?Z/m where p is prime and m is relatively prime to p. Suppose K is a complete discrete valuation field of characteristic p>0 with algebraically closed residue field. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration, and we calculate its dimension in terms of the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p³.     

Additive Hilbert's Theorem 90 in the ring of algebraic integers

For a number field K, we give a complete characterization of algebraic numbers which can be expressed by a difference of two K-conjugate algebraic integers. These turn out to be the algebraic integers whose Galois group contains an element, acting as a cycle on some collection of conjugates which sum to zero. Hence there are no algebraic integers which can be written as a difference of two conjugate algebraic numbers but cannot be written as a difference of two conjugate algebraic integers. A generalization of the construction to a commutative ring is also given. Furthermore, we show that for n ?_ 3 there exist algebraic integers which can be written as a linear form in n K-conjugate algebraic numbers but cannot be written by the same linear form in K-conjugate algebraic integers.     

Limit sets of automatic sequences

We show that for every k-automatic sequence there exists a natural number p>0 such that the sequences of the form (k^pn+j)_n?0 with j=0,…,p−1 are scaling sequences for f. Moreover, we demonstrate that every limit set is the union of certain basic limit sets.     

Explicit constructions of infinite families of MSTD sets

Text

We explicitly construct infinite families of MSTD (more sums than differences) sets, i.e., sets where |A+A|>|A−A|. There are enough of these sets to prove that there exists a constant C such that at least C/r⁴ of the r² subsets of {1,…,r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are at most f(r)/²r/2 for some polynomial f(r)). We conclude by generalizing our method to compare linear forms ?₁A+?+?nA with ?i∈{−1,1}.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=vIDDa1R2.     

The degree of an integral point in Ps minus a hypersurface

Let K be a number field of degree m with ring of integers R and absolute discriminant d_K. Given a hypersurface Z_K of degree d in the projective space P_K^us over K with Zariski closure Z in P_R^s, we give an explicit function of m, d_K, s,d, a Hermitian metric on R^s+1 ⊗_z C, and a projective height of Z defined in [1], 4.1, such that there exists an integral point in P_R^s Z of degree bounded by this function.