pinecone/core-2020-05-10-deduplication huggingface dataset:
Supersingular Abelian Varieties over Finite Fields

AbstractLet A be a supersingular abelian variety defined over a finite field k. We give an approximate description of the structure of the group A(k) of k-rational points of A in terms of the characteristic polynomial f of the Frobenius endomorphism of A relative to k. Write f=∏geii for distinct monic irreducible polynomials gi and positive integers ei. We show that there is a group homomorphism ϕ:A(k)→∏(Z/gi(1)Z)ei that is “almost” an isomorphism in the sense that the sizes of the kernel and the cokernel of ϕ are bounded by an explicit function of dimA
Automorphic Forms, Definite Quaternion Algebras, and Atkin–Lehner Theory on Trees

AbstractAssociated to the multiplicative group of a definite quaternion algebra over Q are two notions of automorphic forms. One is a complex-valued function on the idele group. The other, a p-adic-valued function on a p-adic “upper half-plane,” arises from considering differentials on a Mumford curve associated to the quaternion algebra. Explicit connections between these forms are explored. In particular, forms of the second type are seen to arise from certain newforms of the first type. In order to define newforms, the Atkin–Lehner theory is developed for certain spaces of functions on the tree associated to PGL2(Qp). As an application, Schneider's proposed p-adic L-series associated to a differential on a Mumford curve is seen to be better interpreted as an adelic form twisted by a p-adic character
On the Structure of (OK/I)×

AbstractIn this paper we investigate the structure of the unit group of OK/I where K is a global number field, and I is a nonzero ideal in the ring of integers OK. The case I=0 is given by the Dirichlet unit theorem. By the Chinese remainder theorem we may assume that I is a prime power pn. We obtain an explicit decomposition of (OK/pn)× in cyclic groups for almost all primes p, namely those lying above a rational prime p satisfying p>e where e=e(p, Z) is the ramification index. In particular we obtain the structure of (OK/pn)× for all unramified p
Local Newforms with Global Applications in the Jacobi Theory

AbstractThe study of spherical representations of the Jacobi group begun in (R. Schmidt, 1998, Abh. Math. Sem. Univ. Hamburg68, 273–296) is continued. Using certain index shifting operators, the notion of age of such a representation is introduced, as well as the notion of a local newform. The precise structure of the space of spherical vectors, in particular its dimension, is determined in terms of the age. The age of the spherical principal series representations is computed, and the local newforms amongst these are determined. Restricting to the classical situation, it is shown that the local index shifting operators essentially coincide with well-known Hecke operators on classical modular forms. This leads to some global applications of the local results
Reduction, Dynamics, and Julia Sets of Rational Functions

AbstractWe consider a rational function φ(z)∈K(z) in one variable defined over an algebraically closed field K which is complete with respect to a valuation v. We study how the reduction (modulo v) of such functions behaves under composition, and in particular under iteration. We also investigate the relationship between bad reduction and the Julia set of φ. In particular, we prove that under certain conditions, bad reduction is equivalent to having a nonempty Julia set. We also give several examples of maps not satisfying those conditions and having both bad reduction and an empty Julia set
On Classical and ℓ-Adic Modular Forms of Levels Nℓm and N

AbstractLet ℓ be any prime number ⩾2, and let N be any integer ⩾1 with (N, ℓ)=1. We generalize J.-P. Serre's Théorème 5.4 in (1976, Enseign Math. (2)22, 227–260) and Théorème 12 in (1973, Lecture Notes in Mathematics, Vol. 350, pp. 191–268, Springer-Verlag, Berlin/Heidelberg) for classical modular forms on Γ0(ℓm) and (Γ1(ℓ) for ℓ⩾3) and ℓ-adic modular forms on SL2(Z) to arbitrary classical modular forms of nebentypus of level Nℓm and ℓ-adic modular forms of level N. We express any modular form of nebentypus of level Nℓm as an ℓ-adic modular form of level N. We extend Ribet's Theorem (2.1) in [8] on modℓ Galois representations for odd characteristics to the case of characteristic 2. Finally, we give another method using rigid analysis
Siegel Modular Forms, L-Functions, and Satake Parameters

AbstractIn this paper we take a first step towards a multiplicity-one result for Siegel modular forms on Spn(Z). We study two L-functions associated to Siegel modular forms, the spinor zeta function in genus 2 and the standard zeta function for arbitrary genus. Both the spinor and standard zeta function are defined as products over all primes and we show that the factors for almost all primes determine the L-function. The study of these zeta functions naturally leads to the study of an invariant related to Siegel modular forms, Satake p-parameters. Our result equivalently states that for a simultaneous Hecke eigenform, the Satake parameters for almost all primes determine the Satake parameters for all primes up to an occasional variation in sign
On Gekeler's Conjecture for Function Fields

AbstractLet P be a monic irreducible polynomial in Fq[T] such that d=degP is even. We have obtained (B. Anglès, 1999, J. Number Theory79, 258–283), when q is odd, a class number congruence modulo P for the ideal class number of Fq[T, P] which is similar to the famous Ankeny–Artin–Chowla formula. As a consequence, we have that this latter class number is divisible by the characteristic of Fq if and only if the Bernoulli–Carlitz number B((qd−1)/2) is divisible by P. This result is a special case of Gekeler's conjecture. In these notes, we give a class number congruence for the ideal class number of any totally real subfield F of the Pth cyclotomic function field (Theorem 2). As expected, this formula involves the Bernoulli–Carlitz numbers. It also appears in this formula which we call the regulator modulo P of F:RF. In the case where F=Fq(T, P ), then RF≢0 (modP). Unfortunately, this is not the case for general F. In Section 3, we show that if RF≢0 (modP), then Gekeler's conjecture is true for the field F (Theorem 4)
On Uniqueness Polynomials and Bi-URS for p-adic Meromorphic Functions

AbstractLet W be an algebraically closed field of characteristic zero, and let K be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. A(K) will denote the ring of entire functions in K and M(K) will denote the field of meromorphic functions in K. In this paper we give some classes of uniqueness polynomials for M(K) and show the existence of a bi-URS for M(K) of the form ({a1, a2, a3, a4}, {ω}). Also a sufficient condition of uniqueness range sets for M(K) in terms of uniqueness polynomials is given
Interpolation Formulas and Auxiliary Functions

AbstractWe prove an interpolation formula for “semi-cartesian products” and use it to study several constructions of auxiliary functions. We get in this way a criterion for the values of the exponential map of an elliptic curve E defined over Q. It reduces the analogue of Schanuel's conjecture for the elliptic logarithms of E to a statement of the form of a criterion of algebraic independence. We also consider a construction of auxiliary function related to the four exponentials conjecture and show that it is essentially optimal. For analytic functions vanishing on a semi-cartesian product, we get a version of the Schwarz lemma in which the exponent involves a condition of distribution reminiscent of the so-called technical hypotheses in algebraic independence. We show by two examples that such a condition is unavoidable
A Note on Zeta Measures over Function Fields

AbstractLet r be a power of a prime number p, Fr be the finite field of r elements, and Fr[T] be the polynomial ring over Fr. As an analogue to the Riemann zeta function over Z, Goss constructed the zeta function ζFr[T](s) over Fr[T]. In order to study this zeta function, Thakur calculated the divided power series associated to the zeta measure μx on Fr[T]v, where v is a finite place of Fr(T). This paper calculates the divided power series associated to the zeta measure on Fr[T]∞=Fr[[1T]] and expresses ζFr[T](s) by an integral of some locally analytic function
On the Distribution of Small Powers of a Primitive Root

AbstractLet Ng={gn:1⩽n⩽N}, where g is a primitive root modulo an odd prime p, and let fg(m, H) denote the number of elements of Ng that lie in the interval (m, m+H], where 1⩽m⩽p. H. Montgomery calculated the asymptotic size of the second moment of fg(m, H) about its mean for a certain range of the parameters N and H and asked to what extent this range could be increased if one were to average over all the primitive roots (modp). We address this question as well as the related one of averaging over the prime p
Degrees of Parametrizations of Elliptic Curves by Shimura Curves

AbstractIn this paper, we study some properties of parametrizations of elliptic curves by Shimura curves. Fix a square-free positive integer N and an isogeny class E of elliptic curves of conductor N defined over Q. Consider a pair (D, M) such that N=DM and the number of prime factors of D is even. Let J be the Jacobian of Shimura curve XD0(M) associated with an Eichler order of level M in an indefinite quaternion albebra of discriminant D defined over Q. There is a unique E in E and a homomorphism J→E having the connected kernel. For a prime r∣N, we study the map on groups of connected components of Néron fibers at r induced from J→E. We show that if r divides D, then the map is surjective. Moreover, we study some relations among degrees of parametrizations XD0(M)→E when D and M vary. Also, we describe a method of computing the degree of XD0(M)→E when D>1
On Artin's Conjecture for Rank One Drinfeld Modules

AbstractLet k be a global function field with a chosen degree one prime divisor ∞, and O⊂k is the subring consisting of all functions regular away from ∞. Let φ be a sgn-normalized rank one Drinfeld O-module defined over O′, the integral closure of O in the Hilbert class field of O. We prove an analogue of the classical Artin's primitive roots conjecture for φ. Given any a≠0 in O′, we show that the density of the set consisting of all prime ideals P′ in O′ such that a (modP′) is a generator of φ(O′/P′) is always positive, provided the constant field of k has more than two elements
A Converse Theorem for Epsilon Factors

AbstractWe prove the following theorem: Let F be a nonarchimedean local field of characteristic zero and K a quadratic extension of F. Let S be the set of characters of K* trivial on F*. Let χ1 and χ2 be two characters of K* such that χ1∣F*=χ2∣F*≠1. Let ψ be a nontrivial additive character of F and ψK=ψ∘trK/F. If ε(χ1λ, ψK)=ε(χ2λ, ψK) for all λ∈S then χ1 and χ2 agree on all units in the ring of integers in K and on all elements of trace zero. If, in addition, the conductor of χ1∣F* is not zero then χ1=χ2
On the 4-rank of the Tame Kernel K2(O) in Positive Definite Terms

AbstractThe paper is about the structure of the tame kernel K2(O) for certain quadratic number fields. There has been recent progress in making explicit the 4-rank of the tame kernel of quadratic number fields and even in obtaining results about the 8-rank. The emphasis of this paper is to determine the 4-rank of the tame kernel in definite terms. Our characterizations are in terms of positive definite binary quadratic forms X2+32Y2, X2+2pY2, 2X2+pY2 over Z. The results make numerical computations readily available, and the characterizations might generate some interest in “density results” concerning the 4-rank of tame kernels
Class Groups and Modular Lattices

AbstractWe show that in the case of 2-dimensional lattices, Quebbemann's notion of modular and strongly modular lattices has a natural extension to the class group of a given discriminant, in terms of a certain set of translations. In particular, a 2-dimensional lattice has “extra” modularities essentially when it has order 4 in the class group. This allows us to determine the conditions on D under which there exists a strongly modular 2-dimensional lattice of discriminant D, as well as how many such lattices there are. The technique also applies to the question of when a lattice can be similar to its even sublattice
Gap Unification Theory: A Note on the Treatment of Partitions via Their Gaps

AbstractRecently, several authors have been treating partitions by considering the gaps between their consecutive parts. Many of the existing results involving “partitions with parts in the gaps” bear a striking resemblance to certain weighted partition identities that also arise from the gaps of partitions. In this paper, we show that this is not merely a coincidence, and we demonstrate that many of these weighted partition identities are equivalent to identities involving partitions with parts in the gaps. We intend that this brief note will more or less act as a Rosetta Stone for some of the literature on the gap-theoretic study of partitions
On the Distribution of Squares of Hypercomplex Integers

AbstractLet A be a real quadratic algebra of dimension s⩾3 which satisfies the basic relations of hypercomplex systems. For a large positive parameter X, let A(X) denote the number of squares α2 with α∈A, α integral, and all s components of α2 lying in the interval [−X, X]. With particular regard to Cayley's octaves, and generalizing former results concerning Gaussian integers by H. Müller and W. G. Nowak, and Hurwitz integral quaternions by the author, we show thatA(X)=cXs/2−dX(s−1)/2+O(X(logX)−1/2+X(s−2)/2δ(X))(X→∞),where c and d are certain positive constants depending on s, and δ(X) is any upper bound of the error term in the divisor problem, e.g. δ(X)=X23/73+ε
Porosity in Conformal Infinite Iterated Function Systems

AbstractIn this paper we deal with the problem of porosity of limit sets of conformal (infinite) iterated function systems. We provide a necessary and sufficient condition for the limit sets of these systems to be porous. We pay special attention to the systems generated by continued fractions with restricted entries and we give a complete description of the subsets I of positive integers such that the set JI of all numbers whose continued fraction expansion entries are contained in I, is porous. We then study such porous sets in greater detail examining their Hausdorff dimensions, Hausdorff measures, packing measures, and other geometric characteristics. We also show that the limit set generated by the complex continued fraction algorithm is not porous, the limit sets of all plane parabolic iterated function systems are porous, and of all real parabolic iterated function systems are not porous. We provide a very effective necessary and sufficient condition for the limit set of a finite conformal iterated function system to be porous
On Optimal Subset Representations of Integer Sets

AbstractIn this paper, we investigate representations of sets of integers as subset sums of other sets of minimal size, achieving results on the nature of the representing set as well as providing several reformulations of the problem. We apply one of these reformulations to prove a conjecture and extend a theorem of David Moulton concerning the case when the set to be represented consists of a geometric sequence. Finally, we provide a number of interesting questions for possible future research in this relatively new area
On the Parity of Exponents in the Prime Factorization of Factorials

AbstractIn 1997 Berend proved a conjecture of Erdős and Graham by showing that for every positive integer r there are infinitely many positive integers n with the property thatep(1)(n!)≡ep(2)(n!)≡…≡ep(r)(n!)≡0mod2,where p(1)=2, p(2)=3, p(3)=5,… is the sequence of primes in ascending order, and ep(m) denotes the order of the prime p in the prime factorization of the positive integer m. This article presents conjectures and results for the more complicated situation where an arbitrary pattern of residues modulo 2 is given
Minimal Extensions of Algebraic Groups and Linear Independence

AbstractWe introduce the notion of a minimal extension of t-groups. Linear independence of the coordinates of the logarithm of an algebraic point in a minimal extension of t-groups follows naturally from linear independence of the coordinates of the image in the tangent space of the base t-group. We illustrate this principle through a leisurely parade of examples. In particular, we establish a general theorem about divided derivatives for t-modules. Minimal extensions turn out to correspond to Frattini covers for t-groups
Quadratic Diophantine Inequalities

AbstractWe consider systems of quadratic diophantine inequlities. For example, suppose that Q1 and Q2 are real diagonal quadratic forms in s variables, where one has s⩾10. Suppose also that every form αQ1+βQ2 with (α, β)∈R2\{0} has at least 5 nonzero coefficients, one irrational coefficient, at least one negative coefficient, and at least one positive coefficient. Then for any ε>0, there exists a nonzero integral vector x∈Zs such that |Q1(x)|<ε and |Q2(x)|<ε. We also prove a result on systems of R quadratic diophantine inequalities under more complicated restrictions
Deformations of Pseudorepresentations

AbstractThe deformation theory of Galois representations has been the subject of much research in the past several years. This work is a study of deformations of pseudorepresentations, which were first introduced by Wiles. In general, representations might only admit a “versal” deformation and a versal deformation ring. On the other hand, pseudorepresentations always have a universal deformation. Let π be a residual pseudorepresentation and let ρ be an odd, 2-dimensional residual representation such that π comes from ρ. If this representation is absolutely irreducible, then it has a universal deformation, and the universal deformation rings Rπ and Rρ of π and ρ are (canonically) isomorphic. More generally, there is a natural map φ:Rπ→Rρ which can give some relationship between the two rings. Early on we define the kernel of a pseudorepresentation and its factorization through any quotient by a normal subgroup contained in this kernel. Later we use an appropriate factorization to find the universal deformation ring of a specific residual pseudorepresentation, for which the above map is not an isomorphism. This computation also relies on finding the tangent spaces of both the original and the factorized pseudorepresentations, which turn out to have the same dimension. Finally, we consider the deformation theory of pseudorepresentations in more generality, and we give a generalization of the above deformation problem by capturing its relevant properties
Lattice Points and Generalized Diophantine Conditions

AbstractWe prove that (∫2π0(|Nθ(t)−|D|t2|)2dθ)1/2=O(t2/3), where Dθ is a rotation of a convex domain in R2 and Nθ(t)=#{Z2∩tDθ}. It follows that for any δ>0, there exists a set of θ's of measure 2π−δ, such that for t∈Λ, where Λ is any lacunary sequence, |Nθ(t)−|D|t2|⩽CΛt2/3log(t). Moreover, we prove, under some additional assumptions, that for almost every θ,Nθ(t)−|D|t2=O(t2/3), (*)up to a small logarithmic transgression. We also prove that if D is convex and finite type, and also in some infinite type situations, Nτ(t)=#{Z2∩tD+τ}, τ∈T2, the two-dimensional torus, and‖Nτ(t)−t2|D|‖L2(T2)⩽Ct1/2, (**)the optimal bound, then N(0, 0)(t)=t2|D|+O(t2/3). We conclude that (**) cannot in general hold if the boundary of D has order of contact ⩾4 with one of its tangent lines
Hilbert Modular Forms Modulo pm: The Unramified Case

AbstractThis paper is about Hilbert modular forms on certain Hilbert modular varieties associated with a totally real field L. Let p be unramified in L. We reduce to the inert case and consider modular forms modulo pm. We study the ideal of modular forms with q-expansion equal to zero modulo pm, find canonical elements in it, and obtain as a corollary the congruences for the values of the zeta function of L at negative integers. Our methods are geometric and also have applications to lifting of Hilbert modular forms and compactification of certain modular varieties
On Representations of Integers by Indefinite Ternary Quadratic Forms

AbstractLet f be an indefinite ternary integral quadratic form and let q be a nonzero integer such that −qdet(f) is not a square. Let N(T, f, q) denote the number of integral solutions of the equation f(x)=q where x lies in the ball of radius T centered at the origin. We are interested in the asymptotic behavior of N(T, f, q) as T→∞. We deduce from the results of our joint paper with Z. Rudnick that N(T, f, q)∼cEHL(T, f, q) as T→∞, where EHL(T, f, q) is the Hardy–Littlewood expectation (the product of local densities) and 0⩽c⩽2. We give examples of f and q such that c takes the values 0, 1, 2
Transcendence of Sturmian or Morphic Continued Fractions

AbstractWe prove, using a theorem of W. Schmidt, that if the sequence of partial quotients of the continued fraction expansion of a positive irrational real number takes only two values, and begins with arbitrary long blocks which are “almost squares,” then this number is either quadratic or transcendental. This result applies in particular to real numbers whose partial quotients form a Sturmian (or quasi-Sturmian) sequence, or are given by the sequence (1+(⌊nα⌋mod2))n⩾0, or are a “repetitive” fixed point of a binary morphism satisfying some technical conditions
An Asymptotic Formula for a Trigonometric Sum of Vinogradov

AbstractWe obtain a representation formula for the trigonometric sum f(m, n)≔ ∑m−1a=1|sin(πan/m)|sin(πa/m) and deduce from it the upper bound f(m, n)<(4/π2)mlogm+ (4/π2)(γ−log(π/2)+2CG)m+O(m/logm), where CG is the supremum of the function G(t)≔∑∞k=1log|2sinπkt|/(4k2−1), over the set of irrationals. The coefficients on both the main term and the second term are shown to be best possible. This improves earlier bounds for f(m, n). It is conjectured that CG=G(2)≈ 0.236. We also obtain the following asymptotic formula: If α is a real algebraic integer of degree 2 with 0<α<1, then for any rational approximation n/m of α with 0<n<m we have f(m, n)=(4/π2)mlogm+(4/π2)(γ−log(π/2)+2G(α))m+ Oα (|α−nm|1/2mlogm)
Large Sieve Estimates on Arcs of a Circle

AbstractLet 0⩽α<β⩽2π and let Δ=def{eiθ:θ∈[α, β]}. We show that for generalized (non–negative) polynomials P of degree r and p>0, we have∑j=1m|P(aj)|p|aj−eiα||aj−eiβ|+β−αpr+121/2⩽cτ(pr+1)∫βα|P(eiθ)|pdθ, where a1, a2, …, am∈Δ, c is an absolute constant (and, thus, it is independent of α, β, p, m, r, P, {aj}) and τ is an explicitly determined constant which measures the number of points {aj} in a small interval. This implies large sieve inequalities for generalized (non–negative) trigonometric polynomials of degree r on subintervals of [0, 2π]. The essential feature is the uniformity of the estimate in α and β
Level Zero Types and Hecke Algebras for Local Central Simple Algebras

AbstractLet D be a central division algebra and A×=GLm(D) the unit group of a central simple algebra over a p-adic field F. The purpose of this paper is to give types (in the sense of Bushnell and Kutzko) for all level zero Bernstein components of A× and to establish that the Hecke algebras associated to these types are isomorphic to tensor products of Iwahori Hecke algebras. The types which we consider are lifted from cuspidal representations τ of M(kD), where M is a standard Levi subgroup of GLm and kD is the residual field of D. Two types are equivalent if and only if the corresponding pairs (M(kD), τ) are conjugate with respect to A×. The results are basically the same as in the split case A×=GLn(F) due to Bushnell and Kutzko. In the non-split case there are more equivalent types and the proofs are technically more complicated