pinecone/core-2020-05-10-deduplication huggingface dataset:
The Number of k-Sums Modulo k

AbstractLet a1, …, ar be a sequence of elements of Zk, the integers modulo k. Calling the sum of k terms of the sequence a k-sum, how small can the set of k-sums be? Our aim in this paper is to show that if 0 is not a k-sum then there are at least r−k+1 k-sums. This result, which is best possible, extends the Erdo&#x030B;s–Ginzburg–Ziv theorem, which states that if r=2k−1 then 0 is a k-sum. We also show that the same result holds in any abelian group of order k, and make some related conjectures
Remarks on mod-ln Representations, l=3, 5

AbstractLet l=3 or 5. For any integer n>1, we produce an infinite set of triples (L, E1, E2), where L is a number field with degree l3(n−1) over Q and E1 and E2 are elliptic curves over L with distinct j-invariants lying in Q, such that the following conditions hold: (1) the pairs of j-invariants {j(E1), j(E2)} are mutually disjoint, (2) the associated mod-ln representations GL=Gal(L/L)→GL2(Z/ln) are surjective, (3) for almost all primes p of L, we have ln∣ap(E1) if and only if ln∣ap(E2), and (4) the two representations Ei[ln](L) are not related by twisting by a continuous character GL→(Z/ln)×. No such triple satisfying (2)–(4) exists over any number field if we replace l by a prime larger than 5. The proof depends on determining the automorphisms of the group GL2(Z/ln) for l=3, 5 and analyzing ramification in a branched covering of “twisted” modular curves
On Uniformly Distributed Dilates of Finite Integer Sequences

AbstractGiven N nonzero real numbers a1<…<aN, we consider the problem of finding a real number α so that αa1, …, αaN are close to be uniformly distributed modulo one (this question is attributed to Komlos). First, it turns out that it suffices to consider integers a1, …, aN. Given various quantities that measure how close a sequence is to being uniformly distributed, e.g., the size of the largest gap between consecutive points on the circle, discrepancy, or the number of points falling into any interval of size 1/N (“concentration”), we provide upper bounds for the optimal dilate. These bounds depend only on N and they are attained by typical α, i.e., up to α belonging to some set of small measure. We also provide lower bounds for these quantities. Some of our examples are constructed for this purpose by means of probabilistic methods. In case of the discrepancy, the lower and upper bounds match up to logarithms (N/logN vs NlogN). However, in case of the largest gap (logN/N vs N−1/2) and the concentration (exp(clogN/loglog2N) vs N1/3+ε) the lower and upper bounds do not match and the question about the correct asymptotic behavior in terms of N remains open. Finally, we improve on a recent result of Noga Alon and the second author by showing that every set of N integers contains a non-averaging subset of size at least N1/5
The Non-normal Quartic CM-Fields and the Octic Dihedral CM-Fields with Relative Class Number Two

AbstractIn this paper we determine all non-normal quartic CM-fields with relative class number two and all octic dihedral CM-fields with relative class number two: there are exactly 254 non-isomorphic non-normal quartic CM-fields with relative class number two and 95 non-isomorphic octic dihedral CM-fields with relative class number two. This result permits us to find all octic dihedral CM-fields of which ideal class groups are nontrivial and cyclic of 2-power order: there are precisely 16 non-isomorphic such octic dihedral CM-fields
On Old and New Jacobi Forms

AbstractCertain “index shifting operators” for local and global representations of the Jacobi group are introduced. They turn out to be the representation theoretic analogues of the Hecke operators Ud and Vd on classical Jacobi forms, which underlie the theory of Jacobi old- and newforms. Further analogues of these operators on spaces of classical elliptic cusp forms are also investigated. In view of the correspondence between Jacobi forms and elliptic modular forms, this provides some support for a purely local conjecture about the dimension of spaces of spherical vectors in representations of the p-adic Jacobi group
The Distribution of Generalized Sum-of-Digits Functions in Residue Classes

AbstractLet Q={Qj}∞j=0 be a strictly increasing sequence of integers with Q0=1 and such that each Qj is a divisor of Qj+1. The sequence Q is a numeration system in the sense that every positive integer n has a unique “base-Q” representation of the form n=∑j⩾0aj(n)Qj with “digits” aj(n) satisfying 0⩽aj(n)<Qj+1/Qj. A Q-additive function is a function f:N→C of the form f(n)=∑j⩾0fj(aj(n)) where n=∑j⩾0aj(n)Qj is the base-Q representation of n and the component functions fj are defined on {0, 1, …, Qj+1/Qj−1} and satisfy fj(0)=0. We study the distribution of integer-valued Q-additive functions in residue classes. Our main result gives necessary and sufficient conditions for f to be uniformly (resp. non-uniformly) distributed modulo m, for any given prime m. We apply this result to many cases, showing, for example, that the sum-of-digits functions associated with base-Q representations are uniformly distributed modulo any prime m
Swan Modules and Hilbert–Speiser Number Fields

AbstractA number field K is called a Hilbert–Speiser field if for each tamely ramified finite abelian extension N/K the ring of algebraic integers of N, ON, has a normal integral basis over OK, the ring of algebraic integers of K. The classical Hilbert–Speiser theorem proves that the field of rational numbers Q is such a field. It is well known that the class number of a Hilbert–Speiser field must equal 1. We consider tame elementary abelian extensions of a number field K and Swan modules to obtain additional necessary conditions for K to be a Hilbert–Speiser field. We show that among all number fields, the field Q is the only Hilbert–Speiser field
A Factorization Formula for Class Number Two

AbstractLet R be an atomic integral domain. R is a half-factorial domain (HFD) if whenever x1…xn= y1…ym for x1, …, xn, y1, …, ym irreducibles of R, then n=m. A well known result of L. Carlitz (1960, Proc. Amer. Math. Soc.11, 391–392) states that the ring of integers in a finite extension of the rationals is a HFD if and only if the class number of R is less than or equal to 2. If R is such a ring of integers with class number 2, then we use some simple Krull monoids to develop a formula for counting the number of different factorizations of any integer x into products of irreducible elements of R
Asymptotic Behavior of Characteristic Sequences of Integer-Valued Polynomials

AbstractLet D be the ring of integers of a number field K and let E be an infinite subset of D. The D-module Int(E, D) of integer-valued polynomials on E is isomorphic to ⊕∞n=0Ingn where gn is a monic polynomial in D[X] of degree n and In is a fractional ideal of D. For each maximal ideal m of D, let vm be the corresponding valuation of K; we determine here the asymptotic behavior of the characteristic sequences {vm(In)}n∈N in the case where E is a homogeneous subset of D. In order to do this, we first study some properties of ultrametric matrices; then we prove explicit formulas in the case where D is a Dedekind domain with infinite residue fields; finally, we extend these results to the case of number fields
On a Weighted Mean Square Result of the Error Terms of Some Arithmetical Functions

AbstractLet a(n) be an arithmetical function satisfying some conditions and write ∑′n⩽xa(n)=main term+E(x). We obtain an asymptotic formula for a weighted mean square of the error term∫1T|E(y)|2y2σ0+1dy∼clogT for some constants σ0 and c, by modifying a method that seems due to Titchmarsh. This method utilizes the following tools: the Perron formula, the Parseval Theorem, and a Tauberian Theorem. The main difference between the present and the original method is that we include the contribution of the poles of the associated Dirichlet series on a certain line in the main term
On Saturated Distinguished Chains over a Local Field

AbstractFor a separable element α over a local field K, we consider a sequence (α, α1, …, αn) of elements such that αi is of minimal degree over K with v(αi−1−αi)=sup{v(αi−1−β)∣[K(β):K]<[K(αi−1):K]} and that αn belongs to K, where α0=α and v is the unique valuation on the algebraic closure of K with v(K×)=Z. Such a sequence is called a saturated distinguished chain for α over K. We study how these chains are determined from α and see that these chains are closely related to the ramification of the field K(α)
On a Fermat-Type Diophantine Equation

AbstractLet p>3 be an odd prime and ζ a pth root of unity. Let c be an integer divisible only by primes of the form kp−1, (k, p)=1. Let C(i)p be the eigenspace of the ideal class group of Q(ζ) corresponding to ωi, ω being the Teichmuller character. Let B2i denote the 2ith Bernoulli number. In this article we apply the methods (following H. S. Vandiver (1934, Bull. Amer. Math. Soc., 118–123)) which were used by the author (1994, Ph.D. Thesis, California Institute of Technology) to prove a special case Fermat's Last theorem, to study the equation xp+yp=pczp. In particular, we prove the following: Assume p is irregular, and p∣Bp−3. Let q be an odd prime such that q≡1 (modp), and there is a prime ideal Q over q in Q(ζ) whose ideal class generates C(3)p, which is known to be cyclic. If xp+yp=pczp has nontrivial integer solutions, then we show that q∤(pczp/(x+y)). We also give proof of the unsolvability of the above equation for regular primes (p>3), using the results of H. S. Vandiver (1936, Monatsh. Math. Phys.43, 317–320)
Automata and Transcendence in Positive Characteristic

AbstractWe give a generalized and effective version of the Theorem of G. Christol, T. Kamae, M. Mendès France and G. Rauzy. By using this statement, we can prove transcendence results in the completion of the algebraic closure of a function field of positive characteristic. We also obtain weak transcendence statements, i.e., for certain sequences in a function field, the existence of an isomorphism from this function field, which transforms the sequence into a sequence with a transcendental limit
An Elementary Approach to L-Functions modp

AbstractWe give a brief presentation of basic facts concerning the relationship of L-functions modp and the Cartier operator. We take the notion of the Cartier operator for granted but develop all other notions that we need from scratch. An idea taken from Tate's famous “sum of residues equals zero” paper plays a key role and gives the paper a distinctly “functional analytic” flavor. All the ideas in the paper have been “in the air” for some time; our contribution is simply to organize them and to express them succinctly
On a Problem of H. Cohn for Character Sums

AbstractCohn's problem on character sums (see [4], p. 202) asks whether a multiplicative character on a finite field can be characterized by a kind of two level autocorrelation property. Let f be a map from a finite field F to the complex plane such that f(0)=0, f(1)=1, and |f(α)|=1 for all α≠0. In this paper we show that if for all a, b∈F*, we have(q−1)∑α∈Ff(bα)f(α+a)=−∑α∈Ff(bα)f(α), then f is a multiplicative character of F. We also prove that if F is a prime field and f is a real valued function on F with f(0)=0, f(1)=1, and |f(α)|=1 for all α≠0, then ∑α∈Ff(α)f(α+a)=−1 for all a≠0 if and only if f is the Legendre symbol. These results partially answer Cohn's problem
Two Zero-Sum Problems and Multiple Properties

AbstractIn this paper we consider the following open problems: Conjecture0.1.Let S be a sequence of 3n−3 elements in Cn⊕Cn. If S contains no nonempty zero-sum subsequence of length not exceeding n, then S consists of three distinct elements, each appearing n−1 times. Conjecture0.2.Let S be a sequence of 4n−4 elements in Cn⊕Cn. If S contains no zero-sum subsequence of length n, then S consists of four distinct elements, each appearing n−1 times. We show that both Conjecture 0.1 and Conjecture 0.2 are multiplicative, i.e., if Conjecture 0.1 (Conjecture 0.2) holds both for n=k and n=l then it holds also for n=kl
Group Structures of Elementary Supersingular Abelian Varieties over Finite Fields

AbstractLet A be a supersingular abelian variety over a finite field k which is k-isogenous to a power of a simple abelian variety over k. Write the characteristic polynomial of the Frobenius endomorphism of A relative to k as f=ge for a monic irreducible polynomial g and a positive integer e. We show that the group of k-rational points A(k) on A is isomorphic to (Z/g(1)Z)e unless A's simple component is of dimension 1 or 2, in which case we prove that A(k) is isomorphic to (Z/g(1)Z)a×(Z/(g(1)/2)Z×Z/2Z)b for some non-negative integers a, b with a+b=e. In particular, if the characteristic of k is 2 or A is simple of dimension greater than 2, then A(k)≅(Z/g(1)Z)e
On the One-Sided Boundedness of Sums of Fractional Parts ({nα+γ}−12)

AbstractGiven an irrational α in [0, 1), we ask for which values of γ in [0, 1) the sumsC(m, α, γ)≔∑1⩽n⩽m({nα+γ}−12)are bounded from above or from below for all m. When the partial quotients in the continued fraction expansion of α=[0, a1, a2, …] are bounded, say ai⩽A, we give a necessary condition on γ (involving the non-homogeneous continued fraction expansion of γ with respect to α). When the ai⩾2 we give examples of γ that cause one-sided boundedness. In particular, when 2⩽ai⩽A and the a2i−1 (respectively a2i) are all even, we call deduce that C(m, α, γ) is bounded from below (resp. above) if and only if γ={12α+sα} (resp. γ={12α+sα}) for some integer s. The sums |C(m, α, γ)| are always unbounded with |C(m, α, γ)|>clogm for infinitely many m
A Siegel–Weil Identity for G2 and Poles of L-Functions

AbstractRelations among endoscopy liftings of automorphic forms, poles of L-functions, and nonvanishing of certain periods of automorphic forms have long been expected, although they have not been formulated, even conjecturally. We take a first step toward considering the relations by formulating our conjectures for certain types of endoscopy liftings, which generalizes a theorem of Ginzburg et al. (1997, J. Reine Angew. Math.487, 85–114) (n=2 case). By establishing the Siegel–Weil type identity for Eisenstein series of G2, we verify a portion of our conjecture for n=3, among some other results
On Simultaneous Diophantine Approximation in the Vector Space Q+Qα

AbstractBeginning with an improvement to Dirichlet's Theorem on simultaneous approximation, in this paper we introduce an alogrithm, in sympathy with the classical continued fraction algorithm, to generate the sequence of best approximates to the system max{‖α0n‖, ‖α1n‖, …, ‖αLn‖} in the case when α0, α1, …, αL are elements of the vector space Q+Qα. We then produce best possible upper bounds for the associated diophantine inequality. In addition, we consider implications within the geometry of numbers and investigate the converse of the sharpened version of Dirichlet's result. Finally, we close with some remarks on the Littlewood Conjecture in this context
Nonrational Hilbert Modular Threefolds

AbstractLet k be a totally real cubic number field with ring of integers Ok. The Hilbert modular threefold of k is a desingularization of the (natural) compactification of PSL2(Ok)\H3. The goal of this paper is to prove that all rational Hilbert modular threefolds arise from fields with discriminant less than 75125. Specifically, it is shown that if k is a cubic field of discriminant at least 75125, then the arithmetic genus of the Hilbert modular variety of k is negative and hence the variety is not rational. Smaller bounds on the size of the discriminant are obtained for some special classes of cubic fields
Slope of Cusp Forms and Theta Series

AbstractThe aim of this paper is to improve the results of [o] about the theta series associated to the even extremal unimodular quadratic form. We prove that, in degree 3, the associated theta series is still unique for extremal even unimodular lattices of rank 32 and 48. For the ranks 40, 56, 72 we analyze the situation. We discuss also consequences in degree 4. To obtain these results we use some known results about the vanishing of the cusp forms on the hyperelliptic locus Ig. We give also some improvements in this subject, namely we show that the estimate about the slope of cusp forms vanishing on Ig is sharp
Asymptotically Exact Heuristics for (Near) Primitive Roots

AbstractLet g∈Q\{−1, 0, 1}. Let p be a prime. Let ordp(g) denote the exponent of p in the canonical factorization of g. If ordp(g)=0, we define rg(p)=[(Z/pZ)*:〈gmodp〉], that is, rg(p) is the residual index modp of g. For an arbitrary natural number t we consider the set Ng, t of primes p with ordp(g)=0 and rg(p)=t. Write g=±gh0, where g0 is positive and not an exact power of a rational. We introduce a function wg, t(p)∈{0, 1, 2}, for which it is proved, under the Generalized Riemann Hypothesis (GRH), that[formula]where Ng, t(x) is the counting function for Ng, t . This modifies the naive and, under GRH, false heuristic in which one takes wg, t(p)=1
Artin's Conjecture on Average for Composite Moduli

AbstractLet a be an integer ≠−1 and not a square. Let Pa(x) be the number of primes up to x for which a is a primitive root. Goldfeld and Stephens proved that the average value of Pa(x) is about a constant multiple of x/lnx. Carmichael extended the definition of primitive roots to that of primitive λ-roots for composite moduli n, which are integers with the maximal order modulo n. Let Na(x) be the number of natural numbers up to x for which a is a primitive λ-root. In this paper we will prove that the average value of Na(x) oscillates. That is, limx→∞1/x2∑1⩽a⩽xNa(x)>0 and limx→∞1/x2∑1⩽a⩽xNa(x)=0
Gauss Problem for Function Fields

AbstractAccording to a celebrated conjecture of Gauss, there are infinitely many real quadratic fields whose ring of integers is principal. We recall this conjecture in the framework of global fields. If one removes any assumption on the degree, this leads to various related problems for which we give solutions; namely, we prove that there are infinite families of principal rings of algebraic functions in positive characteristic, which are extensions of a given one, and with prescribed Galois, or ramification, properties, at least in some particular cases
Counting Rational Points on K3 Surfaces

AbstractFor any algebraic variety X defined over a number field K, and height function HD on X corresponding to an ample divisor D, one can define the counting function NX, D(B)=#{P∈X(K)∣HD(P)⩽B}. In this paper, we calculate the counting function for hyperelliptic K3 surfaces X which admit a generically two-to-one cover of P1×P1 branched over a singular curve. In particular, we effectively construct a finite union Y=∪Ci of curves Ci⊂X such that NX−Y, D(B)⪡NY, D(B); that is, Y is an accumulating subset of X. In the terminology of Batyrev and Manin [4], this amounts to proving that Y is the first layer of the arithmetic stratification of X. We prove a more precise result in the special case where X is a Kummer surface whose associated Abelian surface is a product of elliptic curves
Computing Power Integral Bases in Quartic Relative Extensions

AbstractWe develop an algorithm for computing all generators of relative power integral bases in quartic extensions K of number fields M. For this purpose we use the main ideas of our previously derived algorithm for solving index form equations in quartic fields (I. Gaál, A. Pethő, and M. Pohst, 1993, J. Symbolic Comput.16, 563–584; 1996, J. Number Theory57, 90–104). In this way we reduce the problem to the resolution of a cubic and several corresponding quartic relative Thue equations over M. These equations determine the generators of power integral bases of K over M up to translation by integers of M and multiplication by unit factors of M. The new method is based on our ability to solve relative Thue equations efficiently by the algorithm in (I. Gaál and M. Pohst, 2000, Math. Comp., to appear). In the case K is an octic field with a quadratic subfield M we can also consider the absolute index of elements of K, having relative index 1 over M. In order to determine all generators of power integral bases of K (over Q) we determine the corresponding translating elements and unit factors properly. This is done by solving an equation similar to an inhomogeneous Thue equation. We illustrate our algorithms with detailed examples
Some Eisenstein Series Identities

AbstractIn this paper we use the theory of elliptic functions to provide different proofs of some Eisenstein series identities of Ramanujan from those given in a recent paper by B. C. Berndt, S. Bhargava, and F. G. Garvan (1995, Trans. Amer. Math. Soc.347, 4136–4244). From one of these identities we derive the inversion formula for the Borweins cubic theta functions via Venkatachanliengar's method. We also derive some striking Eisenstein series identities associated with the Borweins' cubic theta functions
A Two-Variable Artin Conjecture

AbstractLet a, b∈Q* be rational numbers that are multiplicatively independent. We study the natural density δ(a, b) of the set of primes p for which the subgroup of F*p generated by (amodp) contains (bmodp). It is shown that, under assumption of the generalized Riemann hypothesis, the density δ(a, b) exists and equals a positive rational multiple of the universal constant S=∏pprime(1−p/(p3−1)). An explicit value of δ(a, b) is given under mild conditions on a and b. This extends and corrects earlier work of Stephens (1976, J. Number Theory8, 313–332). We also discuss the relevance of the result in the context of second order linear recurrent sequences and some numerical aspects of the determination of δ(a, b)
A Uniqueness Theorem for Rational Points in Projective Space

AbstractThis paper proves a uniqueness result of the following type which has an analogy in Nevanlinna theory. Let K be a number field and S a finite set of places of K containing the infinite places. Let L1, …, L3m+2 be linear forms in m+1 variables with coefficients in Q which are in general position. Let xn, yn be two infinite sequences in Pm(K) such that at least one of them is non-degenerate and such that Lj(xn)≠0, Lj(yn)≠0, and Lj(xn)/Lj(yn) is an S-unit for 1⩽j⩽3m+2. Then there exists an infinite subsequence {nk} with xnk=ynk
On the Total Degree of Certain L-Functions

AbstractAssume a polynomial f∈Fq[x, y] and an additive character ψ of Fq are given. From a set of exponential sums defined by f and ψ one can define an L-function Lf(t), which by results of Dwork and Grothedieck is known to be a rational function. In fact, Lf(t) is the Artin L-function associated to ψ and to an Artin–Schreier covering defined from f. In this note we give bounds for the number of poles of Lf(t) and for its total degree (the number of zeros plus the number of poles). Our bounds are given in terms of the Newton polyhedron of f. The bound for the total degree we give improves, for polynomials in two variables, previous bounds of E. Bombieri (1978, Invent. Math.47, 29–39) and A. Adolphson–S. Sperber (1987, Invent. Math.88, 555–569)
On the Distribution of the Eigenvalues of the Hyperbolic Laplacian for PSL(2, Z), II

AbstractLet H be the upper half plane and X=SL(2, Z)\H the corresponding modular surface. Theory and experiment suggest that the eigenvalues of the hyperbolic Laplacian, Δ, on X, denoted by λj=1/4+t2j, behave in many ways like a random sequence. In particular, for any A>0 the numbers Aλj, j=1, 2, 3, …, should be well distributed modulo 1 (that is to say, there should be square root cancellation in the corresponding Weyl sums). In this paper we show in sharp contrast to the above that the sequence 2tjlog(tj/πe) is not well distributed modulo 1. This reflects a certain structure that the closed geodesics on X carry, precisely that the norms of the hyperbolic conjugacy classes (which correspond to closed geodesics) of Γ are very close to being squares of integers. This phenomenon no doubt occurs for all arithmetic quotients X of H but not for the generic hyperbolic surface