pinecone/core-2020-05-10-deduplication huggingface dataset:
Some Remarks on Representations of a Division Algebra and of the Galois Group of a Local Field

AbstractIn this paper we prove a division algebra analogue of a theorem of Jacquet and Rallis about uniqueness ofGLn(k)×GLn(k) invariant linear form on an irreducible admissible representation ofGL2n(k). We propose a conjecture about when this invariant form exists. We prove some results about self-dual representations of the invertible elements of a division algebra and of Galois groups of local fields. The Shalika model has been studied for principal series representations ofGL2(D) forDa division algebra and a conjecture made regarding its existence in general
Cubic Forms as Sum of Cubes of Linear Forms

AbstractThe topic of investigation is cubic formsFover Z innvariables that are representable as a sumL31+L32of two cubes of linear forms with algebraic coefficients. IfZ2(n,X) denotes the number of such formsF, the main result, stated as Theorem 1.3, gives its order of magnitude asZ2(n,X)≍X2n/3, with the implied constants depending onnonly. The gist of our method consists of the analysis of thep-adic conditions for the coefficients of the linear formsL1andL2which stem from the fact thatFis defined over Z. This leads to results concerning local lattices and their connection to global lattices that seem of interest even beyond the treated problem, and which are therefore stated with some more generality in Theorems 4.1 and 4.8. The combination of a fact from the Geometry of Numbers with the above then leads to the main theorem. Even though these steps are applied to changes of variables leading to the special diagonal formX3+Y3, they may be applicable to more general situations, the final goal being the treatment of forms that can be transformed into an arbitrary given formf(X,Y) by a suitable linear, algebraic change of variables. Another, probably difficult generalisation consists in increasing the number of variables to deal with formsX31+…+X3kfork⩾3
A New Light on Minkowski's ?(x) Function

AbstractThe well-known Minkowski's ?(x) function is presented as the asymptotic distribution function of an enumeration of the rationals in (0,1] based on their continued fraction representation. The singularity of ?(x) is proved in two ways: by exhibiting a set of measure one in which ?′(x)=0; and again by actually finding a set of measure one which is mapped onto a set of measure zero and vice versa. These sets are described by means of metrical properties of different systems for real number representation
Class Number Problem for Imaginary Cyclic Number Fields

AbstractLetNbe an imaginary cyclic number field of degree 2n. Whenn=3 orn=2m⩾2, the fieldsNwith class numbers equal to their genus class numbers and the fieldsNwith relative class numbers less than or equal to 4 are completely determined [10, 13, 26, 27]. Now assume thatn⩾5 andnis not a 2-power. In this paper, first we determine all imaginary cyclic number fields of degree 2nwith relative class numbers less than or equal to 4. Second, we determine all imaginary cyclic number fields of degree 2nwith class numbers equal to their genus class numbers
Kloosterman Sets in Reductive Groups

AbstractWe find an explicit formula for the generating function for the sizes of Kloosterman sets (or equivalently, the local Kloosterman zeta function for trivial unipotent characters) in the context of a split reductive connected algebraic group over an nonarchimedean local fieldK. We provide two proofs of the formula: One is based on a representation theoretical interpretation of the generating function, the other uses an explicit parametrization of the Kloosterman sets. The formula implies that in the case of simply connected Chevalley groups over Q, the global Kloosterman zeta function corresponding to the pair of trivial unipotent characters is a product of Riemann zeta functions
Real Polynomials with All Roots on the Unit Circle and Abelian Varieties over Finite Fields

AbstractIn this paper we prove several theorems about abelian varieties over finite fields by studying the set of monic real polynomials of degree 2nall of whose roots lie on the unit circle. In particular, we consider a setVnof vectors inRnthat give the coefficients of such polynomials. We calculate the volume ofVnand we find a large easily-described subset ofVn. Using these results, we find an asymptotic formula—with explicit error terms—for the number of isogeny classes ofn-dimensional abelian varieties overFq. We also show that ifn>1, the set of group orders ofn-dimensional abelian varieties overFqcontains every integer in an interval of length roughlyqn−(1/2)centered atqn+1. Our calculation of the volume ofVninvolves the evaluation of the integral over the simplex {(x1,…,xn)| 0⩽x1⩽…⩽xn⩽1} of the determinant of then×nmatrix [xei−1j, where theeiare positive real numbers
The Shadow Theory of Modular and Unimodular Lattices

AbstractIt is shown that ann-dimensional unimodular lattice has minimal norm at most 2[n/24]+2, unlessn=23 when the bound must be increased by 1. This result was previously known only for even unimodular lattices. Quebbemann had extended the bound for even unimodular lattices to stronglyN-modular even lattices forNin{1, 2, 3, 5, 6, 7, 11, 14, 15, 23}, (*)and analogous bounds are established here for odd lattices satisfying certain technical conditions (which are trivial forN=1 and 2). ForN>1 in (*), lattices meeting the new bound are constructed that are analogous to the “shorter” and “odd” Leech lattices. These include an odd associate of the 16-dimensional Barnes–Wall lattice and shorter and odd associates of the Coxeter–Todd lattice. A uniform construction is given for the (even) analogues of the Leech lattice, inspired by the fact that (*) is also the set of square-free orders of elements of the Mathieu groupM23
Locally Analytic Functions over Completions of Fr[U]

AbstractIt is well known that a continuous functionf:Zp→Qpcan be expanded by Mahler's basis: [formula] withan→0. Amice (Bull. Soc. Math. France92(1964), 117–180) has established conditions on the coefficientsanfor the functionfto be locally analytic, as well as more general results whenZpis replaced by some compact subset of a local field. We study the function field case in this paper. The function field analogue of Mahler's basis is the Carlitz polynomials, and the corre- sponding result for continuous functions has already been established by Wagner (Acta Arith.17(1971), 389–406). We show that the conditions for a continuous function to be locally analytic in the function field case are completely similar to theQpcase. An application to using integral calculus to analytically continue characteristicp-valuedL-series is briefly mentioned at the end of the paper
Igusa Local Zeta Functions and Parabolic Castling Transformation of Prehomogeneous Vector Spaces

AbstractThe castling transformation is a standard procedure of constructing new prehomogeneous vector spaces from a given one. Professor J.-I. Igusa gives a relation between the Igusa local zeta function of a prehomogeneous vector space and the Igusa local zeta function of its castling transform. Y. Teranishi gives one generalization of the castling transformation. This generalized castling transformation is related to parabolic subgroups, hence we call it the parabolic castling transformation. In this paper, we give a relation between the Igusa local zeta function of a prehomogeneous vector space and the Igusa local zeta function of its parabolic castling transform
Palindromy and Ambiguous Ideals Revisited

AbstractThe purpose of this article is to revisit the relationship between ambiguous ideals and palindromy in the simple continued fraction expansions of quadratic irrationals begun by the first author and A. J. van der Poorten (1995,Bull. Austral. Math. Soc.51, 215–233). We present simpler proofs of known results, new interrelationships, and correct some misinterpretations. We do this via the infrastructure of real quadratic fields. The conclusion is that palindromy is ambiguity, when properly viewed
The Sum of Digits Function in Number Fields: Distribution in Residue Classes

AbstractCanonical number systems are the natural generalization ofq-adic number systems to number fields. Such number systems admit a certain representation of each algebraic integer of a given number field with respect to the powers of a given base numberb. The aim of this paper is to study the sum of digits functionνb(γ) of these number systems. In particular, we prove, thatνb(γ) is equidistributed in residue classes modulomin any ideal s. Furthermore, we investigate the sum of digits function over sumsets
Polynomial Distribution and Sequences of Irreducible Polynomials over Finite Fields

AbstractLetk=GF(q) be the finite field of orderq. Letf1(x),f2(x)∈k[x] be monic relatively prime polynomials satisfyingn=degf1>degf2⩾0 andf1(x)/f2(x)≠g1(xp)/g2(xp) for anyg1(x),g2(x)∈k[x]. WriteQ(x)=f1(x)+tf2(x) and letKbe the splitting field ofQ(x) overk(t). LetGbe the Galois group ofKoverk(t).Gcan be regarded as a subgroup ofSn. For any cycle patternλofSn, letπλ(f1,f2,q) be the number of square-free polynomials of the formf1(x)−αf2(x) (α∈k) with factor patternλ(corresponding in the natural way to cycle pattern). We give general and precise bounds forπλ(f1,f2,q), thus providing an explicit version of the estimates for the distribution of polynomials with prescribed factorisation established by S. D. Cohen in 1970. For an application of this result, we show that, ifq⩾4, there is a (finite or infinite) sequencea0,a1,…∈k, whose length exceeds 0.5logq/loglogq, such that for eachn⩾1, the polynomialfn(x)=a0+a1x+…+anxn∈k[x] is an irreducible polynomial of degreen. This resolves in one direction a problem of Mullen and Shparlinski that is an analogue of an unanswered number-theoretical question of A. van der Poorten
On the Quartic Diophantine Equationf(x,y)=f(u,v)

AbstractNo non-trivial solutions are known of the diophantine equationf(x,y)=f(u,v), wheref(x,y) is the general quartic form given byf(x,y)=ax4+bx3y+cx2y2+dxy3+ey4. This paper provides a necessary and sufficient condition for the existence of non-trivial solutions of this diophantine equation. It has also been shown that, using this condition, integer solutions of the equationf(x,y)=f(u,v) can be obtained in specific cases. As an example, integer solutions have been obtained for the equationx4+x3y+x2y2+xy3+y4=u4+u3v+u2v2+uv3+v4,which had not been solved earlier
Urs, Ursim, and Non-Urs for p-adic Functions and Polynomials

AbstractLetWbe an algebraically closed field of characteristic zero, and letKbe an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. LetQnbe the polynomialxn−xn−1+kfor any constantk≠0, (n−1)n−1/nn. LetTn(k) be the set ofndistinct zeros ofQn. For everyn⩾9, we show thatTn(k) is ann-point unique range set (ignoring multiplicities) for bothW[x] and the set A(K) of entire functions inK. However, for everyn>0, we also show thatTn(k) is not a unique range set (counting or ignoring multiplicities) forW(x) and therefore, is also not a unique range set for the set of p-adic meromorphic functions (this was also separately found by Chung-Chun Yang and Xin-Hou Hua). In the same way, we show that there exist no bi-urs for p-adic meromorphic functions of the form ({a,b,c}, {ω}). Moreover, for everyn⩾5, we show that the only linear fractional functions preserving a setTn(k) is the identity, something which was asked (in particular) in Boutabaa and Escassut, “On Uniqueness of p-adic Meromorphic Functions,”Proc. Amer. Math. Soc.(1988)
Random Sidon Sequences

AbstractA subsetAof the set [n]={1,2,…,n}, |A|=k, is said to form aSidon(orBh) sequence,h⩾2, if each of the sumsa1+a2+…+ah,a1⩽a2⩽…⩽ah;ai∈A, are distinct. We investigate threshold phenomena for the Sidon property, showing that ifAnis a random subset of [n], then the probability thatAnis aBhsequence tends to unity asn→∞ ifkn=|An|⪡n1/2h, and thatP(Anis Sidon)→0 provided thatkn⪢n1/2h. The main tool employed is the Janson exponential inequality. The validity of the Sidon propertyatthe threshold is studied as well. We prove, using the Stein–Chen method of Poisson approximation, thatP(Anis Sidon) →exp{−λ} (n→∞) ifkn∼Lambda;·n1/2h(Λ∈R+), whereλis a constant that depends in a well-specified way onΛ. Multivariate generalizations are presented
Onp-adicL-functions and Zp-extensions of Certain Real Abelian Number Fields

AbstractLetkbe a real abelian number field with Galois groupΔandpan odd prime number. Denote byk∞the cyclotomic Zp-extension ofkwith Galois groupΓand byknthenth layer ofk∞/k. Assume that the order ofΔis prime topand thatpsplits completely ink/Q. In this article, we describe the order of theΓ-invariant part of theΨ-component of thep-Sylow subgroup of the ideal class group ofknfor sufficiently largen, in terms of a special value of thep-adicL-function associated toΨ, whereΨis an irreducible Qp-character ofΔ. This allows us to obtain an alternative formulation of Greenberg's criterion for the vanishing of theΨ-components of the cyclotomic Iwasawaλ- andμ-invariants ofkforp. We also compute some examples for cyclic cubic fields andp=5,7
On an Obstruction to the Hasse Norm Principle and the Equality of Norm Groups of Algebraic Number Fields

AbstractLetL/kandT/kbe finite extensions of algebraic number fields. In the present work we introduce the factor group ofk*∩NL/kJLNT/kJTby (k*∩NT/kJT)NL/kL*, whereJLandJTare the idele groups ofLandT, respectively. The main theorem shows that the computation of this factor group can be reduced to the computation in finite group theory, and the computation with Galois groups of local extensions at a finite number of primes of the base fieldk. We then apply the main theorem to establish a number of interesting results on the equality of norm groups as subgroups of the multiplicative group ofk. In particular, we obtain new results on solitary non-Galois extensions of algebraic number fields
Modular Symbols for Q-Rank One Groups and Voronı Reduction

AbstractLetGbe a reductive algebraic group of Q-rank one associated to a self-adjoint homogeneous cone defined over Q, and letΓ⊂Gbe a torsion-free arithmetic subgroup. Letdbe the cohomological dimension ofΓ. We present an algorithm to compute the action of the Hecke operators onHd(Γ;Z). This generalizes the classical modular symbol algorithm, whenΓ⊂SL2(Z), to a setting including Bianchi groups and Hilbert modular groups. In addition, we generalize some results of Voronoı for real positive-definite quadratic forms to self-adjoint homogeneous cones of arbitrary Q-rank
On the Uniform Distribution in Residue Classes of Dense Sets of Integers with Distinct Sums

AbstractA set A⊆{1,…,N} is of the typeB2if all sumsa+b, witha⩾b,a, b∈A, are distinct. It is well known that the largest such set is of size asymptotic toN1/2. For aB2set A of this size we show that, under mild assumptions on the size of the modulusmand on the differenceN1/2−| A| (these quantities should not be too large), the elements of A are uniformly distributed in the residue classes modm. Quantitative estimates on how uniform the distribution is are also provided. This generalizes recent results of Lindström whose approach was combinatorial. Our main tool is an upper bound on the minimum of a cosine sum ofkterms, ∑k1cosλjx, all of whose positive integer frequenciesλjare at most (2−ε)kin size
Haar Bases forL2(Rn) and Algebraic Number Theory

AbstractWe correct an error in the proof of Theorem 1.5 in Lagarias and Wang (J. Number Theory57, 1996, 181–197). We also give a strengthened necessary condition for the existence of a Haar basis of the specified kind for every integer matrixAthat has a given irreducible characteristic polynomialf(x) with |f(0)|=2. A. Potiopa (Master's thesis, Siedlce University, 1997) found that the expanding polynomialg(x)=x4+x2+2 violates this necessary condition. Thus there exists a 4×4 expanding integral matrixAof determinant 2 and characteristic polynomialg(x) which has no Haar-type wavelet basis using an integer digit set D⊆Z4
An Alternate Approach to Solitons for Fq[t]

AbstractWe describe an alternate approach to the solitons for Fq[t] introduced by Anderson. The product of the monic polynomials in Fq[t] of degreeiin a given congruence class can be expressed in a compact form by the solitons “for alliat once.” They also link the arithmetic of Fq[t] gamma values at fractions to the analogues of Jacobians of Fermat curves in the setting of higher dimensional generalizations of Drinfeld modules. We will calculate explicit equations, formulae and bound the support of the divisors of the solitons. We also show how the generalization of this approach differs from the one suggested by Anderson's approach
8-Ranks ofK2of Rings of Integers in Quadratic Number Fields

AbstractLetFbe a quadratic extension of Q and OFthe ring of integers inF. A result of Tate enables one to compute the 2-rank ofK2OFin terms of the 2-rank of the class group. Formulas for the 4-rank ofK2OFexist, but are more involved. We give upper and lower bounds on the 8-rank ofK2OFin terms of the narrow class group. In certain cases the upper and lower bounds agree, and the 8-rank ofK2OFis exactly the 8-rank of the narrow class group. We then give a family of fields for which this equality holds
Fq-Linear Calculus over Function Fields

AbstractWe define analogues of higher derivatives forFq-linear functions over the field of formal Laurent series with coefficients inFq. This results in a formula for Taylor coefficients of aFq-linear holomorphic function, a definition of classes ofFq-linear smooth functions which are characterized in terms of coefficients of their Fourier–Carlitz expansions. A Volkenborn-type integration theory forFq-linear functions is developed; in particular, an integral representation of the Carlitz logarithm is obtained
Quaternary Even Positive Definite Quadratic Forms of Discriminant 4p

AbstractLetp>13 be a prime congruent to 1 modulo 4. Let G be the genus of a quaternary even positive definite Z-lattice of discriminant 4pwhose 2-adic localization has a proper 2-modular Jordan component. We show that the orthogonal group of any lattice from G is generated by −1 and the symmetries with respect to the roots of the lattice. The class number of G is computed. Furthermore, we show that the theta series of degree two coming from the classes in G with non-trivial automorphism groups are linearly independent
Green's Functions for Drinfeld Modular Curves

AbstractLetK=Fq(T) be a rational function field and ∞ the place given by the degree inT. LetL∞/K∞be a finite extension with ramification index not bigger than 2. We show in this paper how the local Néron–Tate height pairing at ∞ on Drinfeld modular curves overKof divisors whose points are defined overL∞can be described through analytic functions onΩ×ΩwhereΩis the Drinfeld upper half plane. The Green's function is locally constant around the cusps. ForX0(N) the Green's function for cusps is then described by Eisenstein series
Irrationality Results for Values of Generalized Tschakaloff Series

AbstractArithmetical properties of values of the entire functionTq(x)=∑∞n=0xn/q(1/2)n(n+1), whereqis a parameter, |q|>1, were first studied by L. Tschakaloff (1921,Math. Ann.80, 62–74;84, 100–114). In this paper we introduce a generalization ofTq(x), given by (1.3), and prove the irrationality results for the values of (1.3) at rational points (see Theorem and Corollaries at the end of Section 1). One of the essential tools in the proof is a variant of Mahler's transcendence method, due to J. H. Loxton and A. J. van der Poorten (1977,in“Transcendence Theory: Advances and Applications,” pp. 211–226, Academic Press, San Diego)
Notes on a Problem of H. Cohn

AbstractWe prove some partial results concerning the following problem:Assume that F is a finite field, aiis a complex number for each i∈F such that a0=0,a1=1, |ai|=1for all i∈F\{0},and∑i∈Fai+jāi=−1for all i∈F\{0}.Does it follow that the function i→aiis a multiplicative character of F? We prove (in the case |F|=p,pis a prime) on the one hand that there is only a finite number of complex solutions; on the other hand we solve completely a modpversion of the problem. The proofs are mainly elementary, except for applying a theorem of Chevalley from algebraic geometr
Number of Points on the Projective Curves aYl=bXl+cZl and aY2l=bX2l+cZ2l Defined over Finite Fields, l an Odd Prime

AbstractThe number of points on the curve aYe=bXe+c (abc≠0) defined over a finite field Fq, q≡1 (mode), is known to be obtainable in terms of Jacobi sums and cyclotomic numbers of order e with respect to this field. In this paper, we obtain explicitly the Jacobi sums and cyclotomic numbers of order e=l and e=2l over finite fields Fq, q=pα≡1 (mode), for odd primes l and any prime p such that the order of p modulo l is even. Contrary to the case p≡1 (mode) considered in the literature, we have obtained these results solely in terms of q and l. We apply these results to evaluate the number of Fqn-rational points on the non-singular projective curves aYl=bXl+cZl and aY2l=bX2l+cZ2l (abc≠0) defined over finite fields Fq, with conditions on q, p, and l as above. Using these evaluations, we obtain explicitly the ζ-function of the former curve aYl=bXl+cZl defined over Fq as a rational function in the variable t. Thereby we corroborate the Weil conjectures (now theorems) for this concrete class of curves
The Prime-to-Adjoint Principle and Unobstructed Galois Deformations in the Borel Case

AbstractFor a given odd two-dimensional representation ρ over Fp of the absolute Galois group GE of a totally real field E which is unramified outside a finite set of places S, Mazur defined a universal deformation ring RGS(ρ). By obstruction theory, the group -- EQUATION OMITTED -- measures to what extent RGS(ρ) is determined by local relations. Using devissage on adρ, we give criteria for the vanishing of -- EQUATION OMITTED -- in terms of vanishing of S-class groups, in terms of Iwasawa invariants, and in terms of special values of p-adic L-functions. If S is the set of places above p and ∞, the condition -- EQUATION OMITTED -- implies that RGS(ρ) is free of dimension 2[E:Q]+1. In this case, we obtain a reformulation of Vandiver's conjecture and asymptotic connections between Greenberg's conjecture and the freeness of RGS(ρ). For larger S, we relate the freeness of the universal deformation ring for minimal deformations to the vanishing of a modified obstruction group -- EQUATION OMITTED -- . Based on this, we can calculate non-free rings RGS(ρ) for some explicit reducible ρ coming from the action of GQ on p-torsion points of elliptic curves
Integral Galois Module Structure of Some Lubin–Tate Extensions

AbstractLetKbe a finite extension of Qp, and suppose thatK/Qpis ramified and that the residue field ofKhas cardinality at least 3. LetK(2)be the second division field ofKwith respect to a Lubin–Tate formal group, and letΓ=Gal(K(2)/K). We determine the associated order inKΓof the valuation ring O(2)ofK(2), and show that O(2)is not free over this order. The integral Galois module structure of certain intermediate fieldsEofK(2)/Kis also considered. In particular, ifp≠2 andKhas residue field of cardinalityporp2, we show that the valuation ring ofEis free over its associated order if and only ifE/Kis either tamely ramified or ap-extension. We also prove that the valuation ring of any weakly ramified abelian extension ofKis free over its associated order
Complements to Li's Criterion for the Riemann Hypothesis

AbstractIn a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only ifλn=∑ρ[1−(1−1/ρ)n] hasλn>0 forn=1, 2, 3, … whereρruns over the complex zeros of the Riemann zeta function. We show that Li's criterion follows as a consequence of a general set of inequalities for an arbitrary multiset of complex numbersρand therefore is not specific to zeta functions. We also give an arithmetic formula for the numbersλnin Li's paper, via the Guinand–Weil explicit formula, and relate the conjectural positivity ofλnto Weil's criterion for the Riemann Hypothesis