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Two decades ago, Steven Roman, Daniel E. Loeb and Gian-Carlo Rota introduced a family of harmonic numbers in their study of harmonic logarithms. We propose to refer to those numbers as Roman harmonic numbers . With the purpose of revitalizing the study of these mathematical objects, we recall here their known properties and unveil additional ones. An integral representation, several generating relations, and a collection of sum rules involving those numbers are presented. It is also shown that higher derivatives of the Pochhammer and reciprocal Pochhammer symbols are easily expressed in terms of Roman harmonic numbers. [ABSTRACT FROM AUTHOR]

Suppose ρ 1 and ρ 2 are two pure ℓ -adic degree n representations of the absolute Galois group of a number field K of weights k 1 and k 2 respectively, having equal normalized Frobenius traces T r ( ρ 1 ( σ v ) ) / N v k 1 / 2 and T r ( ρ 2 ( σ v ) ) / N v k 2 / 2 at a set of primes v of K with positive upper density. Assume further that the algebraic monodromy group of ρ 1 is connected and ρ 1 is absolutely irreducible. We prove that ρ 1 and ρ 2 are twists of each other by a power of the ℓ -adic cyclotomic character times a character of finite order. As a corollary, we deduce a theorem of Murty and Pujahari proving a refinement of the strong multiplicity one theorem for normalized eigenvalues of newforms. [ABSTRACT FROM AUTHOR]

In this paper, we present a complete classification of all imaginary n -quadratic fields of class number 1. This classification is done using theoretical methods to narrow down the list of potential number fields of class number 1. The final result is produced through computations in the Sage mathematics software [19] . [ABSTRACT FROM AUTHOR]

For any cubic number field K and any conductor ideal f of K we describe how to find all orders of K with conductor f . The result depends only on the factorization of the rational prime numbers in K , and it also yields an elementary proof for a formula for the zeta function of orders in a cubic number field. [ABSTRACT FROM AUTHOR]

Marc Krasner proposed a theory of limits of local fields in which one relates the extensions of a local field to the extensions of a sequence of related local fields. The key ingredient in his approach was the notion of valued hyperfields, which occur as quotients of local fields. Pierre Deligne developed a different approach to the theory of limits of local fields which replaced the use of hyperfields by the use of what he termed triples, which consist of truncated discrete valuation rings plus some extra data. We study the relationship between Krasner's valued hyperfields and Deligne's triples. [ABSTRACT FROM AUTHOR]

We examine congruence relations satisfied by the fundamental unit of a pure cubic field with a power integral basis and relate those to its class number. Our approach also yields in an elementary way the congruence relations for the fundamental unit of a real quadratic field of odd class number obtained by Z. Zhang and Q. Yue in 2014. [ABSTRACT FROM AUTHOR]

Given a commutative ring R with 1 ∈ R and the multiplicative group R ⁎ of units, an element u ∈ R ⁎ is called an exceptional unit if 1 − u ∈ R ⁎ , i.e., if there is a u ′ ∈ R ⁎ such that u + u ′ = 1 . We study the case R = Z n : = Z / n Z of residue classes mod n and determine the number of representations of an arbitrary element c ∈ Z n as the sum of two exceptional units. As a consequence, we obtain the sumset Z n ⁎ ⁎ + Z n ⁎ ⁎ for all positive integers n , with Z n ⁎ ⁎ denoting the set of exceptional units of Z n . [ABSTRACT FROM AUTHOR]

Abstract: “Most” hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by the appropriate Chow variety, and provides bounds on the probability that a random curve over a finite field is reducible. [Copyright &y& Elsevier]

Abstract: Let R be a complete discrete valuation ring of mixed characteristic with fraction field K. We study stable models of p-cyclic covers of . First, we determine the monodromy extension, the monodromy group, its filtration and the Swan conductor for special covers of arbitrarily high genus with potential good reduction. In the case we consider hyperelliptic curves of genus 2. [Copyright &y& Elsevier]

Abstract: We prove that the Ohno–Zagier relation of multiple zeta values can be deduced from the regularized double shuffle relation. At the same time, we provide an algebraic proof of Ohno–Zagier relation. [Copyright &y& Elsevier]

Abstract: Let K be a number field and an elliptic curve defined over K for . We prove that there exists a number field L containing K such that there are infinitely many such that has positive rank, equivalently all four elliptic curves have growth of the rank over each of quadratic extensions , more strongly, for any , We also prove that if each elliptic curve for can be written in Legendre form over a cubic extension K of a number field k, then there are infinitely many such that for is of positive rank. [Copyright &y& Elsevier]

Abstract: Let be a prime ideal in the ring of integers R of a number field F, with , and assume that has degree of inertia f and ramification index . Let be the smallest positive integer s such that every integer that is expressible as a sum of k-th powers is a sum of s k-th powers . We prove that . We also use known bounds for to obtain bounds for for any positive integer m. [Copyright &y& Elsevier]

Abstract: In this note we present a short and elementary proof of Heckeʼs reciprocity law for Hecke–Gauss sums of number fields. [Copyright &y& Elsevier]

Abstract: Based on algebraic number theory we construct some families of rotated -lattices with full diversity which can be good for signal transmission over both Gaussian and Rayleigh fading channels. Closed-form expressions for the minimum product distance of those lattices are obtained through algebraic properties. [Copyright &y& Elsevier]

Abstract: In the present paper we generalise transference theorems from the classical geometry of numbers to the geometry of numbers over the ring of adeles of a number field. To this end we introduce a notion of polarity for adelic convex bodies. [Copyright &y& Elsevier]

Abstract: Using work of Colmez, we give a quick algorithm for obtaining a clean fundamental domain for the action on of the totally positive units of a totally real cubic field. The fundamental domain consists of two infinite solid cones in , one generated by and , the other by and . Here are certain fundamental totally positive units, included in by the usual geometric embedding, which we show to be easily computable from any set of fundamental units of k. Similar cones were found by Thomas and Vasquez in 1980, and by Halbritter and Pohst in 2000, but their methods did not result in practical algorithms. [Copyright &y& Elsevier]

Abstract: Let K be a real quadratic field with discriminant d, and for a (fractional) ideal a of K, let Na be the norm of a. For a given fractional ideal I of K, and Dirichlet character χ of conductor q, we define where the sum is over all integral ideals a of K which are equivalent to I. We give a short, easily computable formula to evaluate , using familiar objects from considerations of K. We generalize our formula to with , though the result obtained is not quite so satisfactory as that for . We discuss connections between these formulae and small class numbers. [Copyright &y& Elsevier]

Abstract: Let E be an elliptic curve over and let K be a quadratic imaginary field that satisfies the Heegner hypothesis. We study the arithmetic of E over ring class extensions of K, with particular focus on the case when E has analytic rank at least 2 over . We also point out an issue in the literature regarding generalizing the Gross–Zagier formula, and offer a conjecturally correct formula. [Copyright &y& Elsevier]

Abstract: In this paper, we extend the work in [Z. Qi, C. Yang, Moritaʼs theory for the symplectic groups, Int. J. Number Theory 7 (2011) 2115–2137 ] to split reductive groups. We construct and study the holomorphic discrete series representations and the principal series representations of a split reductive group G over a p-adic field F as well as a duality between certain sub-representations of these two representations. [Copyright &y& Elsevier]

Abstract: We study an analog of Serreʼs conjecture over imaginary quadratic fields. In particular, we ask whether the weight recipe of Buzzard, Diamond and Jarvis will hold in this setting. Using a program written by the author, we provide computational evidence that this is in fact the case. In order to justify the method used in the program, we prove that a modular symbols method will work for arbitrary weights over imaginary quadratic fields. [Copyright &y& Elsevier]

Abstract: We show that there exists an upper bound for the number of squares in arithmetic progression over a number field that depends only on the degree of the field. We show that this bound is 5 for quadratic fields, and also that the result generalizes to k-powers for integers . [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and study several norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms on a certain subspace implies Lehmerʼs conjecture and in at least one case that the converse is true as well. We evaluate these norms on a class of algebraic numbers that include Pisot and Salem numbers, and for surds. We prove that the infimum in the construction is achieved in a certain finite dimensional space for all algebraic numbers in one case, and for surds in general, a finiteness result analogous to that of Samuels and Jankauskas for the t-metric Mahler measures. [Copyright &y& Elsevier]

Abstract: It was proved by Cassels and Swinnerton-Dyer that the Littlewood conjecture in simultaneous Diophantine approximation holds for any pair of numbers in a cubic field. Later this result was generalized by Peck to a basis of a real algebraic number field of degree at least 3. By transference, this result provides some solutions for the dual form of Littlewoodʼs conjecture. Here we find another solutions, and using Bakerʼs estimates for linear forms in logarithms of algebraic numbers, we discuss whether the result is best possible. [Copyright &y& Elsevier]

Abstract: The Hadamard product of two power series and is the power series . We define the (Hadamard) grade of a power series A to be the least number (finite or infinite) of algebraic power series, the Hadamard product of which equals A. We study and discuss this notion. [Copyright &y& Elsevier]

Abstract: In this paper, we are interested in the average behavior of the coefficients of Dedekind zeta function over square numbers. In Galois fields of degree d which is odd, when is an integer, we have where , is a polynomial in t of degree , and is an arbitrarily small constant. By using our method, we also rectify the main terms of the k-dimensional divisor problem in some Galois fields over square numbers established by Deza and Varukhina (2008) . [Copyright &y& Elsevier]

Abstract: Conjecturally, the parity of the Mordell–Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers, so the rank modulo 2 is (conjecturally) the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any , so this is purely a parity phenomenon. [Copyright &y& Elsevier]

Abstract: The Euler–Lehmer constants are defined as the limits We show that at most one number in the infinite list is an algebraic number. The methods used to prove this theorem can also be applied to study the following question of Erdös. If is such that and , then Erdös conjectured that If , we show that the Erdös conjecture is true. [Copyright &y& Elsevier]

Abstract: Text: The purpose of this paper is to show that the reflex fields of a given CM-field K are equipped with a certain combinatorial structure that has not been exploited yet. The first theorem is on the abelian extension generated by the moduli and the b-torsion points of abelian varieties of CM-type, for any natural number b. It is a generalization of the result by Wei on the abelian extension obtained by the moduli and all the torsion points. The second theorem gives a character identity of the Artin L-function of a CM-field K and the reflex fields of K. The character identity pointed out by Shimura (1977) in follows from this. The third theorem states that some Pfister form is isomorphic to the orthogonal sum of defined on the reflex fields . This result suggests that the theory of complex multiplication on abelian varieties has a relationship with the multiplicative forms in higher dimension. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=IIwksVYV5YE. [Copyright &y& Elsevier]

Abstract: In this paper, we attempt to prove that the symmetric pairs and are Gelfand pairs where E is a commutative semi-simple algebra over F of dimension 2 and F is a non-archimedean field of characteristic 0. Using Aizenbud and Gourevitch''s generalized Harish-Chandra method and traditional methods, i.e. the Gelfand–Kahzdan theorem, we can prove that these symmetric pairs are Gelfand pairs when E is a quadratic extension field over F for any n, or E is isomorphic to for . Since is a descendant of , we prove that it is a Gelfand pair for both archimedean and non-archimedean fields. [Copyright &y& Elsevier]

Abstract: Text: Let K be a number field, , or the field of rational functions on a smooth projective curve over a perfect field, and let V be a subspace of , . Let be a union of varieties defined over K such that . We prove the existence of a point of small height in , providing an explicit upper bound on the height of such a point in terms of the height of V and the degree of a hypersurface containing , where dependence on both is optimal. This generalizes and improves upon the results of Fukshansky (2006) . As a part of our argument, we provide a basic extension of the function field version of Siegel''s lemma (Thunder, 1995) to an inequality with inhomogeneous heights. As a corollary of the method, we derive an explicit lower bound for the number of algebraic integers of bounded height in a fixed number field. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=R-o6lr8s0Go. [Copyright &y& Elsevier]

Abstract: Because the analytic functional equation holds for Katz p-adic L-function for CM fields, the algebraic functional equation of the Selmer groups for CM fields is expected to hold. In this note we prove it following the specialization principle developed by T. Ochiai (2005) in [Och05]. [Copyright &y& Elsevier]

Abstract: We show that the height of a nonzero algebraic number α that lies in an abelian extension of the rationals and is not a root of unity must satisfy . [Copyright &y& Elsevier]

Abstract: Consider the set of number fields unramified away from 2, i.e., unramified outside . We show that there do not exist any such fields of degrees 9 through 15. As a consequence, the following simple groups are ruled out for being the Galois group of an extension which is unramified away from 2: Mathieu groups and , , and alternating groups for (values were previously known). [Copyright &y& Elsevier]

Abstract: We introduce two types of zeta functions (Ψ-type and ζ-type) of one complex variable associated to an arbitrary number field K. We prove various arithmetic identities which involve both of them. We also study their special values at integral arguments. [Copyright &y& Elsevier]

Abstract: For a fixed number field K, we consider the mean square error in estimating the number of primes with norm congruent to a modulo q by the Chebotarëv Density Theorem when averaging over all and all appropriate a. Using a large sieve inequality, we obtain an upper bound similar to the Barban–Davenport–Halberstam Theorem. [Copyright &y& Elsevier]

Abstract: Let be a local field. In this paper we compute the local coefficients arising from the uniqueness of the Whittaker model for principal series representations of the metaplectic double cover of . Our computations include the relatively hard case of p-adic field of even residual characteristic. [Copyright &y& Elsevier]

Abstract: We will study the torsion of p-ramified Iwasawa modules of the -extension over imaginary quadratic fields k. In this article, we prove that the torsion is trivial except for one case under the assumption that the prime p does not divide the class number of k. We also study generalized Greenberg''s conjecture for the prime 2 and imaginary quadratic fields. [Copyright &y& Elsevier]

Abstract: Let K be an imaginary quadratic field with class number one and ℓ be a rational prime that splits in K. We prove that mod ℓ, a system of Hecke eigenvalues occurring in the first cohomology group of some congruence subgroup Γ of can be realized, up to twist, in the first cohomology with trivial coefficients after increasing the level of Γ by (ℓ). [Copyright &y& Elsevier]

Abstract: Let K be an algebraic number field, and an irreducible, automorphic, cuspidal representation of with analytic conductor . The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant c depending on , and K only, such that π can be decided completely by its local components with norm . [Copyright &y& Elsevier]

Abstract: Text: In this paper we apply Yamamoto''s Theorem [Y. Yamamoto, Dirichlet series with periodic coefficients, in: Proc. Intern. Sympos. “Algebraic Number Theory”, Kyoto, 1976, JSPS, Tokyo, 1977, pp. 275–289] to find the residue modulo a prime power of the linear combination of Dirichlet L-function values at positive integral arguments s such that s and χ are of the same parity, in terms of Euler numbers, whereby we obtain the finite expressions for short interval character sums. The results obtained generalize the previous results pertaining to the congruences modulo a prime power of the class numbers as the special case of . Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=_KAv4FCdVUs. [Copyright &y& Elsevier]

Abstract: Let K be a finite extension of a -adic number field k. By local class field theory there is only a finite number of norm subgroups of the multiplicative group of k that contain the norm group . If X is a subgroup of a group Y, then the interval is the set of subgroups of Y that contain X including X and Y. In the present work we investigate the number of norm groups in the interval for a given finite Galois extension of algebraic number fields. There are finite Galois 2-extensions and Galois extensions of odd degrees such that the corresponding intervals contain only a finite number of norm groups. The main theorem, however, states that for any finite Galois extension of even degree that is not a 2-extension, called -extension, the interval contains infinitely many norm groups. [Copyright &y& Elsevier]

Abstract: Text: By a result of G. Freiman and A. Geroldinger [G. Freiman, A. Geroldinger, An addition theorem and its arithmetical application, J. Number Theory 85 (1) (2000) 59–73] it is known that the set of lengths of factorizations of an algebraic integer (in the ring of integers of an algebraic number field), or more generally of an element of a Krull monoid with finite class group, has a certain structure: it is an almost arithmetical multiprogression for whose difference and bound only finitely many values are possible, and these depend just on the class group. We establish a sort of converse to this result, showing that for each choice of finitely many differences and of a bound there exists some number field such that each almost arithmetical multiprogression with one of these difference and that bound is up to shift the set of lengths of an algebraic integer of that number field. Moreover, we give an explicit sufficient condition on the class group of the number field for this to happen. Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=c61xM-5D6Do. [Copyright &y& Elsevier]

Abstract: In this paper, we will show that there are exactly 3 real quadratic fields of the form with class number 2, where is a square free integer. This completely determines narrow Richaud–Degert type modulo 8 with class number 2. [Copyright &y& Elsevier]

Abstract: We study primitive prime divisors of the terms of , where for K a real quadratic field, and u a unit element of its ring of integers. The methods used allow us to find the terms of the sequence that do not have a primitive prime divisor. [Copyright &y& Elsevier]

Abstract: We study Stickelberger elements associated to -extensions over global function fields of characteristic and show that they are in some sense generically irreducible in the Iwasawa algebras. [Copyright &y& Elsevier]

Abstract: Lower bounds for regulators of algebraic number fields are very important for a variety of applications. Good estimates depend at least on the degree and the discriminant of the considered field. In this paper we present an improved bound which is obtained from more specific field data, e.g. the size of small -values of the integers of the field. This is of considerable interest for computations in practice, for example, of fundamental units. [Copyright &y& Elsevier]

Abstract: Let be a prime. Let with . In the paper we mainly determine by assuming or with . As an application we obtain simple criteria for to be a quadratic residue , where is a squarefree integer such that D is a quadratic residue of p, is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for , where is the Lucas sequence defined by , and . [Copyright &y& Elsevier]

Abstract: For a compositum of quadratic fields , where are square-free odd integers and , we study the group C of circular units of k. We construct a basis of C, compute the index of C in the full group of units of k and derive a lower bound for the divisibility of this index by a power of 2. These results give a lower bound for the divisibility of the class number of the maximal real subfield of k by a power of 2. [Copyright &y& Elsevier]

Abstract: In this note, we look at the composite integers n which divide . [Copyright &y& Elsevier]

Abstract: We first prove Sun''s three conjectures [Z.H. Sun, On the number of incongruent residues of modulo p, J. Number Theory 119 (2006) 210–241; Z.H. Sun, http://sfb.hytc.edu.cn/xsjl/szh/, 2000, June] on the number of rational points of some elliptic curves over finite fields , which are related to the congruence cubic and quartic residue. And we provide some examples and comments concerning these conjectures. [Copyright &y& Elsevier]