algebraic number fields Research Articles (Page 1)

Let R m be endowed with the Euclidean metric. The covering radius of a lattice Λ ⊂ R m is the least distance r such that, given any point of R m , the distance from that point to Λ is not more than r . Lattices can occur via the unit group of the ring of integers in an algebraic number field K , by applying a logarithmic embedding K ⁎ → R m . In this paper, we examine those lattices which arise from the cyclotomic number field Q ( ζ n ) , for a given positive integer n ≥ 5 such that n ≢ 2 ( mod 4 ) . We then provide improvements to a result of de Araujo in [3] , and conclude with an upper bound on the covering radius for this lattice in terms of n and the number of its distinct prime factors. In particular, we improve [3, Lemma 2] , and show that, asymptotically, it can be improved no further.

The workshop focused on various directions of arithmetic statistics in algebra and number theory. These include statistical problems for random polynomials and varieties, probabilistic Galois theory, and counting and distribution problems for algebraic functions, algebraic number fields, elliptic curves, L -functions, as well as arithmetic problems in non-abelian settings (eg, arithmetic statistics for algebraic groups).

The Pólya group [Formula: see text] of an algebraic number field [Formula: see text] is the subgroup of the ideal class group [Formula: see text] generated by the ideal classes of the products of prime ideals of the same norm. If [Formula: see text] is trivial, then the number field [Formula: see text] is said to be a Pólya field. In the first part of this paper, we extend a recent result of ours and prove the existence of infinitely many number fields having Pólya groups as large as possible and those number fields being the compositum of a simplest cubic field and a quintic field. Using a result of Zylinski, we also prove that those fields are of index [Formula: see text]. Next, we produce three new families [Formula: see text], [Formula: see text] and [Formula: see text] of totally real bi-quadratic Pólya fields [Formula: see text] involving prime numbers [Formula: see text] and [Formula: see text] that satisfy certain quadratic residue conditions among themselves. It is worthwhile to note that in each of these fields, exactly five primes ramify in [Formula: see text] and this is the maximum possible number of ramified primes in a bi-quadratic Pólya field over [Formula: see text].

The Franke filtration is a finite filtration of the space of automorphic forms on a connected reductive linear algebraic group defined over an algebraic number field. The main feature of the filtration is that its quotients can be described in terms of parabolically induced representations using the main values of derivatives of Eisenstein series and the residues of these. The goal of this paper is to provide a complete explicit description of the Franke filtration of the space of automorphic forms on the symplectic group of rank two. The approach is different from the original approach of Franke, and takes into account the full cuspidal support of automorphic forms, that is, the cuspidal automorphic representation from which the Eisenstein series is built and the evaluation point at which it is evaluated. This considerably simplifies the exposition and makes it possible to obtain very explicit results and reveal the phenomena present in the filtration. The considered low rank case exhibits many of the properties and phenomena present in the cases of arbitrary rank. The explicit description of the Franke filtration in this case has important implications and applications in cohomology of congruence subgroups related to the Hilbert–Siegel modular forms of degree two, the Hilbert–Siegel modular variety of degree two and the corresponding Shimura variety.

We consider two algorithmic problems concerning sub-semigroups of Heisenberg groups and, more generally, two-step nilpotent groups. The first problem is Intersection Emptiness, which asks whether a finite number of given finitely generated semigroups have empty intersection. This problem was first studied by Markov in the 1940s. We show that Intersection Emptiness is PTIME decidable in the Heisenberg groups H_n(𝕂) over any algebraic number field 𝕂, as well as in direct products of Heisenberg groups. We also extend our decidability result to arbitrary finitely generated 2-step nilpotent groups. The second problem is Orbit Intersection, which asks whether the orbits of two matrices under multiplication by two semigroups intersect with each other. This problem was first studied by Babai et al. (1996), who showed its decidability within commutative matrix groups. We show that Orbit Intersection is decidable within the Heisenberg group H₃(ℚ).

The paper investigates upper bounds on the period length of functional continued fractions for key elements of hyperelliptic fields over number fields. In the case when the hyperelliptic field is given by a polynomial of odd degree, the finite period length is trivially estimated from above twice the power of the fundamental 𝑆-unit. A more interesting and complicated case is when the hyperelliptic field is given by a polynomial of even degree. In 2019 V.P. Platonov andG.V. Fedorov for hyperelliptic fields ℒ = Q(𝑥)(√𝑓), deg 𝑓 = 2𝑔 +2, over the field Q of rational numbers the exact interval of values 𝑠 ∈ Z is found such that the continued fractions of elements of the form√𝑓/𝑥𝑠 ∈ ℒ∖Q(𝑥) are periodic. In this article, we find a generalization of this result for an arbitrary field as a field of constants. Based on this result, sharp upper estimates for the lengths of the periods are found functional continued fractions of elements of hyperelliptic fields over number fields 𝐾, depending only on the genus 𝑔 of the hyperelliptic field, the degree of extension 𝑘 = [𝐾 : Q] and order 𝑚 of the Jacobian torsion subgroup of the corresponding hyperelliptic curve.

Abstract In this paper, we prove a weak form of the conjecture generalised to algebraic number fields. Given integers satisfying , Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. 291 (1991), 225–230], Stewart and Yu [Duke Math. J. 108 (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height in terms of the radical for algebraic numbers (Győry [Acta Arith. 133 (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field K. Given algebraic integers in a number field K satisfying , we give an upper bound for the logarithm of the projective height in terms of norms of prime ideals dividing , where L is the Hilbert Class Field of K. In many cases, this allows us to give a bound in terms of the modified radical as given by Masser (Proc. Amer. Math. Soc. 130 (2002), no. 11, 3141–3150). Furthermore, by employing a recent bound of Győry (Publ. Math. Debrecen 94 (2019), 507–526) on the solutions of S‐unit equations, our estimates imply the upper bound where is an effectively computable constant. Further, given conditions on the largest prime ideal dividing , we obtain a sub‐exponential bound for in terms of the radical. Independently, as a direct application of his bounds on the solutions of S‐unit equations(Győry ([Publ. Math. Debrecen 94 (2019), 507–526]), Győry (Publ. Math. Debrecen 100 (2022), 499–511) also attains results mentioned above, including the above inequality, but over the base field K, as discussed in Section 6. As a consequence of our results, we will give an application to the effective Skolem–Mahler–Lech problem and give an improvement to a result by Lagarias and Soundararajan (J. Théor. Nombres Bordeaux 23 (2011), no. 1, 209–234) on the XYZ conjecture.

For a smooth projective curve \(\mathcal{C}\) defined over algebraic number field k, we investigate the question of finiteness of the set of generalized Jacobians \({{J}_{\mathfrak{m}}}\) of a curve \(\mathcal{C}\) associated with modules \(\mathfrak{m}\) defined over k such that a fixed divisor representing a class of finite order in the Jacobian J of the curve \(\mathcal{C}\) provides the torsion class in the generalized Jacobian \({{J}_{\mathfrak{m}}}\). Various results on the finiteness and infiniteness of the set of generalized Jacobians with the above property are obtained depending on the geometric conditions on the support of \(\mathfrak{m}\), as well as on the conditions on the field \(k\). These results were applied to the problem of the periodicity of a continuous fraction decomposition constructed in the field of formal power series \(k((1{\text{/}}x))\), for the special elements of the field of functions \(k(\tilde {\mathcal{C}})\) of the hyperelliptic curve \(\tilde {\mathcal{C}}:{{y}^{2}} = f(x)\).

Let 𝒪 K be the ring of integers of an algebraic number field K embedded into ℂ. Let X be a subset of the Euclidean space ℝ d , and D(X) be the set of the squared distances of two distinct points in X. In this paper, we prove that if D(X)⊂𝒪 K and there exist s values a 1 ,...,a s ∈𝒪 K distinct modulo a prime ideal 𝔭 of 𝒪 K such that each a i is not zero modulo 𝔭 and each element of D(X) is congruent to some a i , then |X|≤d+s s+d+s-1 s-1.

Let [Formula: see text] be an algebraic number field. With [Formula: see text] we associate the ring of finite adeles [Formula: see text] In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group [Formula: see text] Specifically, we consider matrix-valued Hamiltonian operators [Formula: see text] where [Formula: see text] is the Vladimirov operator and [Formula: see text] is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman–Kac formula for the Schrödinger semigroup generated by [Formula: see text] This formula is given in terms of the ordered time exponentials.

Much is known about the adele ring of an algebraic number field from the perspective of harmonic analysis and class field theory. However, its ring-theoretical aspects are often ignored. Here, we present a description of the prime spectrum of this ring and study some of the algebraic and topological properties of these prime ideals. We also study how they behave under separable extensions of the base field and give an indication of how this study can be applied in adele rings not of number fields.

Let be an algebraic number field with θ a root of an irreducible trinomial belonging to In this paper, for each prime number p we compute the highest power of p dividing the discriminant of K in terms of the prime powers dividing and discriminant of f(x). An explicit p-integral basis of K is also given for each prime p and a method is described to obtain an integral basis of K from these p-integral bases which is illustrated with examples.

Abstract We obtain a complete description of the fields that are extensions of of degree at most and the cubic polynomials such that the expansion of into a continued fraction in the field of formal power series is periodic. We prove a finiteness theorem for cubic polynomials with a periodic expansion of for extensions of of degree at most . We obtain a description of the periodic elements for the cubic polynomials defining elliptic curves with points of order , . Bibliography: 19 titles.

Explicit integral basis of [formula omitted

We prove a formula, which, given a principally polarized abelian variety $(A,\lambda )$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron–Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,\lambda )$.

We prove that these Cantor sets are made up of transcendental numbers, apart\nfrom their endpoints $0$ and $1$, under some arithmetical assumptions on the\ndata. To that purpose, we establish a criterion of linear independence over the\nfield of algebraic numbers for the three numbers $1$, a characteristic Sturmian\nnumber, and an arbitrary Sturmian number with the same slope.\n

In this paper, we find some inequalities which involve Euler’s function, extended Euler’s function, the function τ, and the generalized function τ in algebraic number fields.

On the modular forms of weight 1/2 over algebraic number fields

Let K=ℚ(𝜃) be an algebraic number field with 𝜃 satisfying an irreducible polynomial x6−m having integer coefficient m. We explicitly provide an integral basis of K along with some examples. As an application, we show that when m is a squarefree integer such that m≡2,3 mod4 and m≢±1 mod9, then K has a power integral basis.

For an algebraic number field [Formula: see text] and a prime number [Formula: see text], let [Formula: see text] be the maximal multiple [Formula: see text]-extension. Greenberg’s generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-[Formula: see text] extension of [Formula: see text] is pseudo-null over the completed group ring [Formula: see text]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.