pinecone/core-2020-05-10-deduplication huggingface dataset:
A Certain Power Series and the Inhomogeneous Continued Fraction Expansions

AbstractA recent paper (J. Number Theory42(1992), 61–87) announced various arithmetical properties of the Mahler functionf(θ, φ;x,  y)=∑∞k=1∑1≤m≤kθ+φxkym. Unfortunately the arguments of that paper are marred by an error whereby the arguments hold only forφ=0 (or whenbn=1 for all positive integersn). We show how to correct the arguments so that they do hold for generalφ. Moreover, another paper (J. Number Theory43(1993), 293–318) soon afterwards happened again to discuss the subject using different expressions for the power series forf. We show that the corrected results here do coincide with those later results notwithstanding our alternative presentatio
Contributions to a General Theory of View-Obstruction Problems, II

AbstractIn view-obstruction problems, congruent copies of a closed, centrally symmetric, convex bodyC, centred at the points of the shifted lattice (12, 12, ..., 12)+Znin Rn, are expanded uniformly. The expansion factor required to touch a given subspaceLis denoted byν(C,L) and for each dimensiond, 1≤d≤n−1, the relevant expansion factors are used to determine a supremumν(C,d)=sup {ν(C,L): dimL=d,Lnot contained in a coordinate hyperplane}.Here a method for obtaining upper bounds onν(C,L) for “rational” subspacesLis given. This leads to many interesting results, e.g. it follows that the supremaν(C,d) are always attained and a general isolation result always holds. The method also applies to give simple proofs of known results for three dimensional spheres. These proofs are generalized to obtainν(B,n−2) and a Markoff type chain of related isolations for spheresBin Rnwithn≥4. In another part of the paper, the subspaces occurring in view-obstruction problems are generalized to arbitrary flats. This generalization is related to Schoenberg's problem of billiard ball motion. Several results analogous to those forν(C,L) andν(C,d) are obtained
A Family of Étale Coverings of the Affine Line

AbstractIt this note we prove the following theorem. Letπalg1(A1C) be the algebraic fundamental group of the affine line overC, whereCis the completion of the algebraic closure ofFq((1/T)), andFqis a field withqelements. IfFqhas at least four elements, then we show that there is a continuous surjectionπalg1(A1C)→limSL2?(A/I)/{+ ±1}, whereA=Fq[T] and the inverse limit is over the family of non-zero, proper ideals ofA. This result is proved by using the moduli of Drinfel'dA-modules of rank two overCwithI-level structures; these curves give (tamely) ramified covers of the line and the tame ramification is removed using a variant of Abhyankar's lemma
Spectra of Hypergraphs and Applications

AbstractTo a regular hypergraph we attach an operator, called its adjacency matrix, and study the second largest eigenvalue as well as the overall distribution of the spectrum of this operator. Our definition and results extend naturally what is known for graphs, including the analogous threshold bound[formula]fork-regular graphs. As an application of our results, we obtain asymptotic behavior, asNtends to infinity, of the dimension of the space generated by classical cusp forms of weight 2 levelNand trivial character which are eigenfunctions of a fixed Hecke operatorTpwith integral eigenvalues
Algebraic Numbers Close to 1 and Variants of Mahler's Measure

AbstractGiven a rational functionRand a real numberp⩾1, we definehp(R) as theLpnorm of max{log|R|, 0} on the unit circle. In this paper we study the behaviour ofhp(R) providing various bounds for it. Our results lead to an explicit construction of algebraic numbers close to 1 having small Mahler's measure and small degree, which shows that a lower bound for the distance |α−1| recently given by M Mignotte and M. Waldschmidt is also sharp. From our bounds also follows a statement on polynomials equivalent to the Riemann hypothesis
Another Look at the Index Formulas of Cyclotomic Number Theory

AbstractTo the cyclotomic number fieldKgenerated by the roots of unity of orderfwe attach a Galois module which is a hybrid of the Stickelberger ideal and the group of circular units; this Galois module also admits interpretation as the universal punctured distribution of conductorf. We embed our Galois module in another naturally occurring Galois module, and prove that the index is exactly the class number ofK. By avoiding even-odd splittings and the analytic class number formula, we are able to avoid the consideration of {±1}-cohomology groups as well. Cohomological considerations become necessary only when making comparisons to the classical index formulas of Kummer, Hasse, Iwasawa, and Sinnott
Log-Algebraicity of TwistedA-Harmonic Series and Special Values ofL-Series in Characteristicp

AbstractWe find special points in the Carlitz module related, on the one hand, to the values ats=1 of characteristicpDirichletL-function analogues, and on the other hand, to the values at negative integral values ofsof a characteristicpRiemann zeta-function analogue. The special points are constructed with the help of a general theorem asserting the “log-algebraicity” of the “twistedA-harmonic series” associated to a rank one sign-normalized ellipticA-module. Concerning the “special point index” we prove a Kummer-type criterion and raise some Vandiver-type questions
K-Admissibility of Wreath Products of Cyclicp-Groups

AbstractLetKbe a number field and letGbe a wreath product of cyclicp-groups. We show that ifpis odd, thenGisK-admissible if and only ifGis cyclic orphas at least two divisors inK. Ifp=2 we obtain a similar partial result. This work relies on a determination of the Galois structure of the group of l-units in certain local fields. The main theorem is used to prove our conjecture, which was open forp=2, that a metacyclicp-groupGisK-admissible if and only if it occurs as a Galois group over two completions ofK. In addition, we prove that the property ofK-admissibility is inherited by metacyclicp-subgroups
Fermat–Euler Theorem in Algebraic Number Fields

AbstractIn this paper a (maximal) generalization of the classical Fermat–Euler theorem for finite commutative rings with identity is proved. Maximal means that we show how to extend the original Fermat–Euler theorem to all of the elements of such rings with the best possible choice of exponents. The proofs are based on an idempotent technique of Schwarz. The results are then applied to Dedekind's ringsRsatisfying the following finiteness condition:[formula]Further specialization of proved results to some special cases of Dedekind's ringsRdepends then upon a good detailed knowledge of the structure of the group of units of the corresponding residue class ringR/I. The most known prototypes of such rings are besides Znthe algebraic number fields. Amongst these the simplest cases represent the quadratic fields, including one of their oldest representatives, the ring of the Gaussian integers
Quadratic Forms onFq[T]

AbstractIn this paper, we study the number of representations of polynomials of the ringFq[T] by diagonal quadratic forms[formula]whereA1, …, Asare given polynomials andY1, …, Ysare polynomials subject to satisfying the most restrictive degree conditions. WhenA1, …, Asare pairwise coprime, ands⩾5, we use the ordinary circle method; whenA1, …, A4are pairwise coprime we adapt Kloosterman's method to the polynomial case and we get an asymptotic estimate for the number R(A1, …, As; M) of representations of a polynomialMas a sum(Q). We also deal with the particular cases=4,A1=A2=D,A3=A4=1, whereDis a square-free polynomial. In this particular case, the number R(A1, …, A4; M) is the number of representations ofMas a sum of two norms of elements of the quadratic extension[formula]satisfying the most restrictive degree conditions
A Converse Theorem for Jacobi Forms

AbstractLetf(qτ, qz)=∑n, rc(n, r)qnτqrzbe a power series whose coefficients satisfy a particular periodicity condition depending on the integerrmodulo 2m. We first associate tof(qτ, qz) a 2m-vector-valued functionΛ(f, s) via a generalized Mellin transform. Then we show that the functionΛ(f, s) is entire, bounded on vertical strips and satisfies certain matrix functional equation if, and only if,f(qτ, qz) is the Fourier expansion of a Jacobi cusp form of indexminvariant under the group SL(2, Z)⋉Z2. This is the direct analogue of Hecke's converse theorem for elliptic cusp forms in the context of Jacobi cusp forms on SL(2, Z)⋉Z2
Uniformly Approximable Numbers and the Uniform Approximation Spectrum

AbstractWe say a real numberαis uniformly approximable if the upper bound in Dirichlet's theorem, from diophantine approximation, of 1/(Q+1)qmay be sharpened toc(α)/(Q+1)2for all sufficiently largeQ. Here we begin by showing that the set of uniformly approximable numbers is precisely the set of badly approximable numbers. In additition, the optimal lower bound ofc(α), referred to as the uniform approximation constant, is explicitly given. This allows us to introduce the notion of a uniform approximation spectrum. We conclude with a determination of the smallest values of this new spectrum and a comparison of this spectrum with other spectra
Approximation of Real Numbers by Rationals: Some Metric Theorems

AbstractLetxbe a real number in [0, 1], Fnbe the Farey sequence of ordernandρn(x) be the distance betweenxand Fn. The first result concerns the average rate of approximation:[formula]The second result states that any badly approximable number is better approximable by rationals than all numbers in average. Namely, we show that ifx∈[0, 1] is a badly approximable number thenc1⩽n2ρn(x)⩽c2for all integersn⩾1 and some constantsc1>0,c2>0. The last two theorems can be considered as analogues of Khinchin's metric theorem regarding the behaviour of inferior and superior limits ofn2ρn(x)f(logn), whenn→∞, for almost allx∈[0, 1] and suitable functionsf(·)
On Schur Properties of Random Subsets of Integers

AbstractA classic result of I. Schur [9] asserts that for everyr⩾2 and fornsufficiently large, if the set [n]={1, 2, …, n} is partitioned intorclasses, then at least one of the classes contains a solution to the equationx+y=z. Any such solution withx≠ywill be called aSchur triple. Let us say thatA⊆[n] has theSchur propertyif for every partition (or 2-coloring) ofA=R∪B(forredandblue), there must always be formed somemonochromaticSchur triple, i.e., belonging entirely to eitherRorB. Our goal here is to show thatn1/2is a threshold for the Schur property in the following sense: for anyω=ω(n)→∞, almost all setsA⊆[n] with |A|<n1/2/ωdo not possess the Schur property, while almost allA⊆[n] with |A|>ωn1/2do
Optimal Representations by Sumsets and Subset Sums

AbstractLetAbe a set of non-negative integers. In our previous paper forh∈Z,h⩾2 we estimated from below the cardinality ofhAand this allowed us to obtain sharp estimates for suchG, that every integerg⩾Gmay be represented by a sum of elements ofA(the linear diophantine problem of Frobenius). However, “not only the number of elements ofhAis important, but also the way they are situated” (G. Freiman). In this paper we show thathAalways contains long continuous chains of integers and derive an estimation for the number of summands required for representation ofgof the above mentioned type
Torsion Subgroups of Elliptic Curves over Quadratic Extensions

AbstractLetEbe an elliptic curve over Q. Observing certain relations between the torsion subgroups ofEandED, theD-quadratic twist ofE, we prove that the torsion subgroups ofEis stable for all but finitely many quadratic extensions. Moreover, using the result ofK. Ono, we classify the torsion subgroup ofEover all quadratic extensions whenEis of the formE:y2=x(x+M)(x+N), whereMandNare integers. In the special case when torsion subgroup ofEover Q is isomorphic to Z/2Z⊕Z/8Z, we prove that the torsion subgroup ofEis always stable under quadratic extensions
Some Consequences of a Result of Jean Coquet

AbstractLetqbe a positive integer,q⩾2. In 1979, Coquet proved that aq-multiplicative function of modulus 1 is pseudo-random if and only if its Fourier–Bohr spectrum is empty. In this article, we show that a consequence of this result is that the existence of a Fourier–Bohr spectrum for the image by a character of a R/Z-valuedq-additive functionfis equivalent to the existence of some constantαand a sequenceAnin R/Z such that the sequence of measures 1/[x]∑n⩽xδ(f(n)−αn−ALogq(x))converges vaguely to a probability measureνon R/Z; we extend this result toq-additive functions with values in a locally compact abelian group
On Sums of Consecutive Squares

AbstractIn this paper we consider the problem of characterizing those perfect squares that can be expressed as the sum of consecutive squares where the initial term in thus sum isk2. This problem is intimately related to that of finding all integral points on elliptic curves belonging to a certain family which can be represented by a Weierstraß equation with parameterk. All curves in this family have positive rank, and for those of rank 1 a most likely candidate generator of infinite order can be explicitly given in terms ofk. We conjecture that this point indeed generates the free part of the Mordell–Weil group and give some heuristics to back this up. We also show that a point which is modulo torsion equal to a nontrivial multiple of this conjectured generator cannot be integral. Forkin the range 1⩽k⩽100 the corresponding curves are closely examined, all integral points are determined and all solutions to the original problem are listed. It is worth mentioning that all curves of equal rank in this family can be treated more or less uniformly in terms of the parameterk. The reason for this lies in the fact that in Sinnou David's lower bound of linear forms in elliptic logarithms—which is an essential ingredient of our approach—the rank is the dominant factor. Also the extra computational effort that is needed for some values ofkin order to determine the rank unconditionally and construct a set of generators for the Mordell–Weil group deserves special attention, as there are some unusual features
On the Construction of Indecomposable Positive Definite Hermitian Forms over Imaginary Quadratic Fields

AbstractLetDmbe the ring of integers of an imgainary quadratic fieldQ(−m)withm≡3 (mod4). Then there are indecomposable positive definite hermitianDm-lattices of given ranknand given discriminantdwith exactly eight exceptions ifn≠2 and six exceptions ifn=2 and assume the generalized Riemann hypothesis. In these exceptional cases there are no lattices with the desired properties. In particular, this result holds without assuming the generalized Riemann hypothesis, if the square-freem≡−1 (mod8) orm≡−1 (mod12) or the class number ofQ(−m)is unity
Computing Associated Orders and Galois Generating Elements of Unit Lattices

AbstractLetMbe either the field of rationals Q or a quadratic imaginary number field. We denote byNan abelian number field extension ofM, which is supposed to be real ifM=Q. We putΓ=Gal(N/M) and letUNbe the unit lattice ofN. We describe an algorithm that first computes the associated order AN/M={λ∈Q[Γ]/(∑γ∈Γγ)∣λ(UN)⊆UN} ofUNand then answers the question of whether or notUNis a free, rank one module over AN/M. If the answer is positive a generator also is computed. In particular, we can computationally decide whether a Minkowski unit exists
Estimating the 2-rank of Cubic Fields by Selmer Groups of Elliptic Curves

AbstractFrey and his coauthors have established a relationship between the 2-torsion of the Selmer group of an elliptic curve of the special formE:y2=x3±k2and the 2-class number of pure cubic fieldK=Q((∓k2)1/3)=Q((∓k)1/3).In the present paper we prove a far-reaching generalization of an analogous relationship between the 2-rank of any non-Galois cubic number fieldKand the 2-torsion of the Selmer group of a corresponding elliptic curve. We implemented the resulting algorithm and used it, e.g., to produce four cubic number fields of exact 2-rank 7. The 2-rank of number fields is of special interest because if it is sufficiently large the number field has an infinite class field tower. In particular, the four fields of 2-rank 7 turn out to have infinite class field towers
Criterion for the Equality of Norm Groups of Idele Groups of Algebraic Number Fields

AbstractOne of the fundamental theorems of global class field theory states that there is a one-to-one correspondence between finite Abelian extensions of an algebraic number fieldkand the norm groups of the idele class groupCk=Jk/k* ofk. More generally, for finite extensionsKandLofkthere is the following group theoretic interpretation ofNK/kCK⊆NL/kCL. LetEbe a finite Galois extension ofkcontainingKandL, and letG=G(E/k),H=G(E/K), andN=G(E/L) be the corresponding Galois groups. It follows by global class field theory thatNK/kCK⊆NL/kCLiffG′H⊆G′N, whereG′ is the commutator subgroup ofG. In the present work we prove thatNK/kJK⊆NL/kJLiff every element ofHof prime power order is conjugate inGto an element ofN. We also show that the same group theoretic condition is equivalent toN(K/k)⊆N(L/k), whereN(K/k) is the group of elements ofk* that are local norms everywhere fromKtok. We then use this group theoretic criterion to investigate the equality of norm groups as subgroups ofk*
Number Theoretic Correlation Inequalities for Dirichlet Densities

AbstractOur main discovery is the inequalityD(A, BD[A, B]⩾DADB,whereA,Bare arbitrary sets of positive integers, (A, B)={(a, b):a∈A, b∈B} is the set of largest common divisors, [A, B]={[a, ^;b]:a∈A, b∈B} is the set of least common multiples, andDdenotes the lower Dirichlet density. It is much more general than our recent inequality for multiples of sets, which in turn is sharper than Behrend's well-known inequality. We also extend another recently discovered inequality, which does not seem to have number theoretic predecessors
On Sums of Fractional Parts {nα+γ}

AbstractWe use the continued fraction expansion ofαto obtain a simple, explicit formula for the sumCm(α, γ)=∑1⩽k⩽m({kα+γ}−12)whenαis irrational. From this we deduce a number of elementary bounds on the growth and behaviour ofCm(α, γ). In particular, we show that as m varies the extent of the fluctuations in size can be determined almost entirely from the non-homogeneous continued fraction expansion ofγwith respect toα. These sums are closely related to the discrepancy of the sequence ({nα}); we state a related explicit formula that yields similar bounds for the discrepancy. Sums of this form also occur in a lattice point problem of Hardy and Littlewood
Norm and Trace of thej-invariants of Drinfeld Modules Associated to Hyperelliptic Curves

AbstractLetk/Fq(x) be a quadratic extension that is ramified over the unique pole ofx, and letAbe the integral closure of Fq[x] ink. Thenkis the function field analogue of an imaginary quadratic number field. A rank-one DrinfeldA-moduleφof generic characteristic is the analogue of an elliptic curve with complex multiplications by the full ring of integers of an imaginary quadratic number field. In this paper, we compute the degrees of the trace and norm down to Fq[x] ofj(φ), thej-invariant ofφ. Our results generalize previous ones wherekwas assumed to have genusg=1
Explicit Representation of Fundamental Units of Some Real Quadratic Fields, II

AbstractIn the preceding paper, (K. Tomita,Proc. Japan Acad. Ser. A Sci. Math.71(1995), 41–43), for all real quadratic fields Q(d) such that the periodkdof the continued fraction expansion ofωd=(1+d)/2 is equal to 3, we describedTd,Udexplicitly in the fundamental unitεd=(Td+Udd)/2 (>1) of Q(d) andditself by using two parameters appearing in the continued fraction expansion ofωd; In this paper, we provide analogous results for all real quadratic fields Q(d) with period 4 and 5 by using at most four parameters
A Note on Sinnott's Definition of Circular Units of an Abelian Field

AbstractThis note deals with a slight change of Sinnott's definition of circular units of an abelian field. The reason for this new definition giving a bigger group of units is as follows. Sinnott's formula for the index of circular units in the full group of units contains two factors which are difficult to handle: the index of the Sinnott module and (in the case of an imaginary field) an unknown power of 2. If we use the mentioned definition, the power of 2 in the corresponding formula can be easily described
Transcendence and the Carlitz–Goss Gamma Function

AbstractIn this article, we show that a value of the Carlitz–Goss gamma function for the ringFq[X] is transcendental over the fieldFq(X) if and only if the argument is not an element of N={0, 1, 2, …}. We also answer a question of J.-P. Allouche, and we show the transcendence of monomials built on the values of the Carlitz–Goss gamma function. This concludes the transcendence study of the values of this function initiated and developed by D. Thakur and pursued by A. Thiery, J. Yu, and L. Denis and J.-P. Allouche. The proof of our result, as that of J.-P. Allouche, uses derivation of formal power series and the theorem of G. Christol, T. Kamae, M. Mendès France, and G. Rauzy
One-Dimensional Crystals and Quadratic Residues

AbstractThe main problem in crystallography is recovering the electronic density from the diffraction peak intensities. The one-dimensional model leads to recover a discrete Fourier series in Znwith integral coefficients from its absolute value, which has arithmetical implications. In this paper we prove that the constant absolute value of Gaussian sums determines them among a class of exponential sums. This implies that if diffraction peak intensities are constant except for one of them, then, modulo translations, we obtain a quadratic residue molecule
Ramification Groups of Nonabelian Kummer Extensions

AbstractThe reciprocity law of Coleman for the Hilbert norm residue symbol has allowed the computation of the conductors of the abelian Kummer extensionsQp(a,ζpn)/Qp(ζpn)witha∈Qpandζpna primitive (pn)th root of unity for a fixed primepand all positive integersn. From these conductors, we compute the ramification groups of the nonabelian Kummer extensionQp(Q×p)/Qpobtained from adjoining toQpallp-power roots of its elements. More generally, given a similar nonabelian Kummer extension of complete discrete valuation fields, we have a method of computing its ramification groups from the conductors of the abelian Kummer extensions and knowledge of the ramification groups of the cyclotomic extensions
An Upper Bound on Conductors for Pairs

AbstractLetFbe a non-Archimedean local field. Letπbe a smooth irreducible complex representation ofGLm(F), andρbe a smooth irreducible complex representation ofGLn(F). Denote bya,b, andcthe exponents in the conductors ofπ,ρ, and the pair (π,ρ), respectively. IfFhas positive characteristic, the following upper bound is a consequence of the Local Langlands correspondence with Galois representations:c⩽na+mb−inf(a,b).We prove this bound directly, regardless of the characteristic ofF, using results of Jacquet, Piatetski–Shapiro, and Shalika on the essential (“new”) vector for smooth irreducible generic representations ofGLn(F)
Genus Characters and the Arithmetic of Commutative Monoids

AbstractIn this paper we develop an ideal theory for certain submonoids of the nonzero integers. We associate one of these monoids to each quadratic number field and show that the genus theory of ideals and genus characters of the number field are virtually the same as the ideal theory and the characters of the ideal class group of the monoid. We define L-functions associated to characters of the class group of a finite intersection of these monoids and prove an analogue of Dirichlet's theorem on primes in an arithmetic progression. We further prove a curious relationship between the nature of the singularity of the zeta function and the 2-rank of the class group