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Let G be an additive finite abelian group. Denote by disc(G) the smallest positive integer t such that every sequence S over G of length | S | ≥ t has two nonempty zero-sum subsequences of distinct lengths. In this paper, we focus on the direct and inverse problems associated with disc(G) for certain groups of rank three. Explicitly, we first determine the exact value of disc(G) for G ≅ C 2 ⊕ C n 1 ⊕ C n 2 with 2 ∣ n 1 ∣ n 2 and G ≅ C 3 ⊕ C 6 n 3 ⊕ C 6 n 3 with n 3 ≥ 1 . Then we investigate the inverse problem. Let L 1 (G) denote the set of all positive integers t satisfying that there is a sequence S over G of length | S | = disc (G) - 1 such that every nonempty zero-sum subsequence of S has the same length t. We determine L 1 (G) completely for certain groups of rank three. [ABSTRACT FROM AUTHOR]

Let p ≡ 1 (mod 8) and q ≡ 7 (mod 8) be two prime numbers. The purpose of this paper is to compute the unit groups of the fields L = Q (2 , p , q) and give their 2-class numbers. [ABSTRACT FROM AUTHOR]

For an infinite family of real biquadratic fields k we give the structure of the Iwasawa module X = X (k ∞) of the Z 2 -extension of k. For these fields, we obtain that λ = μ = 0 and ν = 2 . where λ , μ and ν are the Iwasawa invariants of the cyclotomic Z 2 -extension of k [ABSTRACT FROM AUTHOR]

Let p ≥ 5 be a prime number such that p ≡ 3 (mod 8) and d an odd positive squarefree integer. Put K = Q (2 , - p , d) . In this paper, we compute the 2-rank of the class group of the fields K = Q (2 , - p , d) . Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic and of type (2, 2) or (2, 4). [ABSTRACT FROM AUTHOR]

We state and prove a Gras-type equality relating a certain module generated by Rubin–Stark units and the ray class number. We use for this purpose the notion of generalized index. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to compute the 2-rank of the class group of the fields K = ℚ ( − 2 , q , d) for any odd positive squarefree integer d and any prime integer q ≡ 5 (mod 8). Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic, of type (2, 2), (2, 4) or (2, 2, 2). [ABSTRACT FROM AUTHOR]

In this note, we prove that for every even integer d ≥ 4 , there is a number field K of degree d over Q , such that for each sufficiently large N ∈ N among (2 N + 1) d distinct elements of the form a 1 ω 1 + ⋯ + a d ω d , where ω 1 , ⋯ , ω d is an integral basis of the ring O K and a 1 , ⋯ , a d ∈ Z , | a 1 | , ⋯ , | a d | ≤ N , at least c (log N) d / 2 - 1 elements are unimodular. Here, c is a positive constant that depends on the field and the integral basis only. In particular, for a quartic field K, the lower bound for the number of unimodular elements of such form is at least c log N . We show that that this estimate is best possible for any quartic field K and any ω 1 , ⋯ , ω 4 ∈ O K . [ABSTRACT FROM AUTHOR]

Let G be an additive finite abelian group and Γ ⊂ End (G) be a subset of the endomorphism group of G. A sequence S = g 1 · ... · g ℓ over G is a (Γ -)weighted zero-sum sequence if there are γ 1 , ... , γ ℓ ∈ Γ such that γ 1 (g 1) + ... + γ ℓ (g ℓ) = 0 . We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group Γ ) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic properties of monoids of weighted zero-sum sequences. [ABSTRACT FROM AUTHOR]

The arithmetic nature of the Euler's constant γ is one of the biggest unsolved problems in number theory from almost three centuries. In an attempt to give a partial answer to the arithmetic nature of γ , Murty and Saradha made a conjecture on linear independence of digamma values. In particular, they conjectured that for any positive integer q > 1 and a field K over which the q-th cyclotomic polynomial is irreducible, the digamma values namely ψ (a / q) where 1 ≤ a ≤ q with (a , q) = 1 are linearly independent over K. Further, they established a connection between the arithmetic nature of the Euler's constant γ to the above conjecture. In this article, we first prove that the conjecture is true with at most one exceptional q. Later on we also make some remarks on the linear independence of these digamma values with the arithmetic nature of the Euler's constant γ . [ABSTRACT FROM AUTHOR]

For an imaginary biquadratic number field L = Q (i , d) , where d is an odd square-free integer, let L ∞ be the cyclotomic Z 2 -extension of L. For any integer n ≥ 0 , we denote by Ln the nth layer of L ∞ / L . We study the rank of the 2-primary part of the class group of Ln and then we draw the list of all number fields L where the Galois group of the maximal unramified pro-2-extension of L ∞ is metacyclic. [ABSTRACT FROM AUTHOR]

For a commutative cancellative monoid M, we introduce the notion of the length density of both a nonunit x ∈ M {x\in M} , denoted LD ⁡ (x) {\operatorname{LD}(x)} , and the entire monoid M, denoted LD ⁡ (M) {\operatorname{LD}(M)}. This invariant is related to three widely studied invariants in the theory of nonunit factorizations, L ⁢ (x) {L(x)} , ℓ ⁢ (x) {\ell(x)} , and ρ ⁢ (x) {\rho(x)}. We consider some general properties of LD ⁡ (x) {\operatorname{LD}(x)} and LD ⁡ (M) {\operatorname{LD}(M)} and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid M with irrational length density, we show that if M is finitely generated, then LD ⁡ (M) {\operatorname{LD}(M)} is rational and there is a nonunit element x ∈ M {x\in M} with LD ⁡ (M) = LD ⁡ (x) {\operatorname{LD}(M)=\operatorname{LD}(x)} (such a monoid is said to have accepted length density). While it is well known that the much studied asymptotic versions of L ⁢ (x) {L(x)} , ℓ ⁢ (x) {\ell(x)} , and ρ ⁢ (x) {\rho(x)} (denoted L ¯ ⁢ (x) {\overline{L}(x)} , ℓ ¯ ⁢ (x) {\overline{\ell}(x)} , and ρ ¯ ⁢ (x) {\overline{\rho}(x)}) always exist, we show the somewhat surprising result that LD ¯ ⁢ (x) = lim n → ∞ ⁡ LD ⁡ (x n) {\overline{\operatorname{LD}}(x)=\lim_{n\rightarrow\infty}\operatorname{LD}(x^% {n})} may not exist. We also give some finiteness conditions on M that force the existence of LD ¯ ⁢ (x) {\overline{\operatorname{LD}}(x)}. [ABSTRACT FROM AUTHOR]

The modular trace of the normalized Hauptmodul has been extended to the Galois trace of a class invariant by Kaneko. It is an important issue to search for class invariants for which the Galois traces have modular properties. The crucial point in this paper is that we initiate a new notion, called Siegel resolvents; we define the Siegel resolvents as the quadratic polynomials of Siegel functions of level 3, so that they are modular functions of level 3 as well. We construct real-valued class invariants over imaginary quadratic fields by using the singular values of Siegel resolvents at imaginary quadratic irrationals. We also prove that the generating series of their Galois traces become a weakly holomorphic modular form with weight 3/2. This shows that the work of D. Zagier on traces of singular moduli can be extended to the modular functions of higher level. [ABSTRACT FROM AUTHOR]

Let d be an odd square-free integer and ζ 8 a primitive 8-th root of unity. The purpose of this paper is to investigate the rank of the 2-class group of the fields L d = Q (ζ 8 , d) . [ABSTRACT FROM AUTHOR]

After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

In this paper we show if E is a totally real Galois extension of Q of degree s then there is a cubic extension K of E with signature (s, s) such that in the group of units of K there are s multiplicatively independent positive units with the following property: if α is any of those s units then there are two conjugates of α over Q which are complex conjugate numbers lying on the circle | z | = α - 1 / 2 . In particular, our result combined with a recent result of Otiman implies that the corresponding Oeljeklaus–Toma manifold X(K, U) admits a pluriclosed metric. [ABSTRACT FROM AUTHOR]

Let be an imaginary bicyclic biquadratic number field, where d is an odd negative square-free integer and its second Hilbert 2-class field. Denote by the Galois group of . The purpose of this note is to investigate the Hilbert 2-class field tower of and then deduce the structure of G. [ABSTRACT FROM AUTHOR]

After Zagier's significant work (in: Bogomolov and Katzarkov (eds) Motives, polylogarithms and hodge theory, part I, International Press, Somerville, 2002) on traces of singular moduli, Bruinier and Funke (J Reine Angew Math 594:1–33, 2006) generalized his result to the traces of singular values of modular functions on modular curves of arbitrary genus. In class field theory, the extended ring class field is a generalization of the ray class field over an imaginary quadratic field. By using Shimura's reciprocity law, we construct primitive generators of the extended ring class fields by using Siegel functions of arbitrary level N ≥ 2 and identify their Galois traces with Fourier coefficients of weight 3/2 harmonic weak Maass forms. This would extend the results of Jeon et al. (Math Ann 353:37–63, 2012) and Jung et al. (Modularity of Galois traces of Weber's resolvents, under revision). [ABSTRACT FROM AUTHOR]

Let H be a transfer Krull monoid over a subset G0 of an abelian group G with finite exponent. Then every non-unit a ∈ H can be written as a finite product of atoms, say a = u 1 · ... · u k. The set L (a) of all possible factorization lengths k is called the set of lengths of a, and H is said to be half-factorial if | L (a) | = 1 for all a ∈ H. We show that, if a ∈ H is a non-unit and | L (a ⌊ (3 exp (G) − 3) / 2 ⌋ ) | = 1 , then the smallest divisor-closed submonoid of H containing a is half-factorial. In addition, we prove that, if G0 is finite and | L (∏ g ∈ G 0 g 2 ord (g) ) | = 1 , then H is half-factorial. [ABSTRACT FROM AUTHOR]

Let ε be an algebraic unit of the degree n ⩾ 3. Assume that the extension ℚ(ε)/ℚ is Galois. We would like to determine when the order ℤ[ε] of ℚ(ε) is Gal(ℚ(ε)/ℚ)-invariant, i.e. when the n complex conjugates ε1, ..., εn of ε are in ℤ[ε], which amounts to asking that ℤ[ε1, ..., εn] = ℤ[ε], i.e., that these two orders of ℚ(ε) have the same discriminant. This problem has been solved only for n = 3 by using an explicit formula for the discriminant of the order ℤ[ε1, ε2, ε3]. However, there is no known similar formula for n > 3. In the present paper, we put forward and motivate three conjectures for the solution to this determination for n = 4 (two possible Galois groups) and n = 5 (one possible Galois group). In particular, we conjecture that there are only finitely many cyclic quartic and quintic Galois-invariant orders generated by an algebraic unit. As a consequence of our work, we found a parametrized family of monic quartic polynomials in ℤ[X] whose roots ε generate bicyclic biquadratic extensions ℚ(ε)/ℚ for which the order ℤ[ε] is Gal(ℚ(ε)/ℚ)-invariant and for which a system of fundamental units of ℤ[ε] is known. According to the present work it should be difficult to find other similar families than this one and the family of the simplest cubic fields. [ABSTRACT FROM AUTHOR]

In this article, we show that certain abelian extensions K with unit rank greater than or equal to three have cyclic class group if and only if it has a Euclidean ideal class. This result improves an earlier result of Murty and Graves. One can improve this result up to unit rank 2 if one assumes the Elliott and Halberstam conjecture (see Conjecture 1 in preliminaries). These results are known under generalized Riemann hypothesis by the work of Lenstra (J Lond Math Soc 10:457–465) [see also Weinberger (Proc Symp Pure Math 24:321–332)]. [ABSTRACT FROM AUTHOR]

We collect some statements regarding equivalence of the parities of various class numbers and signature ranks of units in prime power cyclotomic fields. We correct some misstatements in the literature regarding these parities by providing an example of a prime cyclotomic field where the signature rank of the units and the signature rank of the circular units are not equal. [ABSTRACT FROM AUTHOR]

Let d be a non-square positive integer such that the period of the continued fraction expansion of d is even. We give some relations between some properties of partial quotients of the continued fraction expansion of d , which emerge from numerical experiments. [ABSTRACT FROM AUTHOR]

Let K be a number field. In the terminology of Nagell a unit ε of K is called exceptional if 1 − ε is also a unit. The existence of such a unit is equivalent to the fact that the unit equation ε 1 + ε 2 + ε 3 = 0 is solvable in units ε 1 , ε 2 , ε 3 of K . Numerous number fields have exceptional units. They have been investigated by many authors, and they have important applications. In this paper we deal with a generalization of exceptional units. We are interested in the smallest integer k with k ≥ 3 , denoted by ℓ ( K ) , such that the unit equation ε 1 + … + ε k = 0 is solvable in units ε 1 , … , ε k of K . If no such k exists, we set ℓ ( K ) = ∞ . Apart from trivial cases when ℓ ( K ) = ∞ , we give an explicit upper bound for ℓ ( K ) . We obtain several results for ℓ ( K ) in number fields of degree at most 4, cyclotomic fields and general number fields of given degree. We prove various properties of ℓ ( K ) , including its magnitude, parity as well as the cardinality of number fields K with given degree and given odd resp. even value ℓ ( K ) . Finally, as an application, we deal with certain arithmetic graphs, namely we consider the representability of cycles. We conclude the paper by listing some problems and open questions. [ABSTRACT FROM AUTHOR]

Abstract: We investigate in detail a homomorphism which we call the 2‐Selmer signature map from the 2‐Selmer group of a number field K to a nondegenerate symmetric space, in particular proving the image is a maximal totally isotropic subspace. Applications include precise predictions on the density of fields K with given narrow class group 2‐rank and with given unit group signature rank. In addition to theoretical evidence, extensive computations for totally real cubic and quintic fields are presented that match the predictions extremely well. In an Appendix with Richard Foote, we classify the maximal totally isotropic subspaces of orthogonal direct sums of two nondegenerate symmetric spaces over perfect fields of characteristic 2 and derive some consequences, including a mass formula for such subspaces. [ABSTRACT FROM AUTHOR]

For a knot K in a homology 3-sphere Σ, let M be the result of 2 / q -surgery on K , and let X be the universal abelian covering of M . Our first theorem is that if the first homology of X is finite cyclic and M is a Seifert fibered space with N ≥ 3 singular fibers, then N ≥ 4 if and only if the first homology of the universal abelian covering of X is infinite. Our second theorem is that under an appropriate assumption on the Alexander polynomial of K , if M is a Seifert fibered space, then q = ± 1 (i.e. integral surgery). [ABSTRACT FROM AUTHOR]

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Then every non-unit a∈H can be written as a finite product of atoms, say a=u1·…·uk. The set L(a) of all possible factorization lengths k is called the set of lengths of a. There is a constant M∈N such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d∈Δ∗(H), where Δ∗(H) denotes the set of minimal distances of H. We study the structure of Δ∗(H) and establish a characterization when Δ∗(H) is an interval. The system L(H)={L(a)∣a∈H} of all sets of lengths depends only on the class group G, and a standing conjecture states that conversely the system L(H) is characteristic for the class group. We confirm this conjecture (among others) if the class group is isomorphic to Cnr with r,n∈N and Δ∗(H) is not an interval. [ABSTRACT FROM AUTHOR]

For an algebraic number α, the metric Mahler measure $${m_1(\alpha)}$$ was first studied by Dubickas and Smyth [4] and was later generalized to the t-metric Mahler measure $${m_t(\alpha)}$$ by the author [16]. The definition of $${m_t(\alpha)}$$ involves taking an infimum over a certain collection N-tuples of points in $$\overline{\mathbb{Q}}$$ , and from previous work of Jankauskas and the author, the infimum in the definition of $${m_t(\alpha)}$$ is attained by rational points when $${\alpha\in \mathbb{Q}}$$ . As a consequence of our main theorem in this article, we obtain an analog of this result when $${\mathbb{Q}}$$ is replaced with any imaginary quadratic number field of class number equal to 1. Further, we study examples of other number fields to which our methods may be applied, and we establish various partial results in those cases. [ABSTRACT FROM AUTHOR]

Let O K and C K be respectively the ring of integers and the class group of a number field K . For each integer q ≥ 2 , denote by ∏ q ( K ) the product of all the maximal ideals of O K with norm q , if these ideals do not exist we set ∏ q ( K ) = O K . The Pólya group of K is the subgroup of C K generated by the classes of the ideals ∏ q ( K ) , and K is called a Pólya field if the module of integer-valued polynomials over O K has a regular basis. In this paper, we determine Pólya group of any imaginary bicyclic biquadratic number field, and thus we deduce all the imaginary bicyclic biquadratic Pólya fields. [ABSTRACT FROM AUTHOR]

Let G be an additive finite abelian group. For a positive integer k , let s ≤ k ( G ) denote the smallest integer l such that each sequence of length l has a non-empty zero-sum subsequence of length at most k . Among other results, we determine s ≤ k ( G ) for all finite abelian groups of rank two. [ABSTRACT FROM AUTHOR]

Let K be a totally real number field of degree $$r\,=\,[K:\mathbb {Q}]$$ and let p be an odd rational prime. Let $$K_{\infty }$$ denote the cyclotomic $$\mathbb {Z}_{p}$$ -extension of K and let $$L_{\infty }$$ be a finite extension of $$K_{\infty }$$ , abelian over K. In this article, we extend results of Büyükboduk (Compos Math 145(5):1163-1195, 2009) relating characteristic ideal of the $$\chi $$ -quotient of the projective limit of the ideal class groups to the $$\chi $$ -quotient of the projective limit of the r-th exterior power of units modulo Rubin-Stark units, in the non semi-simple case, for some $$\overline{\mathbb {Q}_{p}}$$ -irreducible characters $$\chi $$ of $$\mathrm {Gal}(L_{\infty }/K_{\infty })$$ . [ABSTRACT FROM AUTHOR]

We compute the p -adic regulator of cyclic cubic extensions of Q with discriminant up to 10 16 for 3 < p < 100 , and observe the distribution of the p -adic valuation of the regulators. We find that for almost all primes, the observation matches the model that the entries in the regulator matrix are random elements with respect to the obvious restrictions. Based on this random matrix model, a conjecture on the distribution of the valuations of p -adic regulators of cyclic cubic fields is stated. [ABSTRACT FROM AUTHOR]

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. For $$k\in \mathbb N$$ , let $$ \mathcal U_k(H)$$ denote the set of all $$m\in \mathbb N$$ with the following property: There exist atoms $$u_1, \ldots , {{u}_{k}}, v_1, \ldots , {{v}_{m}}\in H$$ such that $$u_1\cdot \ldots \cdot {{u}_{k}}=v_1\cdot \ldots \cdot v_m$$ . It is well-known that the sets $$\mathcal U_k (H)$$ are finite intervals whose maxima $$\rho _k(H)=\max \mathcal U_k(H) $$ depend only on G. If $$|G|\le 2$$ , then $$\rho _k (H) = k$$ for every $$k \in \mathbb N$$ . Suppose that $$|G| \ge 3$$ . An elementary counting argument shows that $$\rho _{2k}(H)=k\mathsf D(G)$$ and $$k\mathsf D(G)+1\le \rho _{2k+1}(H)\le k\mathsf D(G)+\lfloor \frac{\mathsf D(G)}{2}\rfloor $$ where $$\mathsf D(G)$$ is the Davenport constant. In [11] it was proved that for cyclic groups we have $$k\mathsf D(G)+1 = \rho _{2k+1}(H)$$ for every $$k \in \mathbb N$$ . In the present paper we show that (under a reasonable condition on the Davenport constant) for every noncyclic group there exists a $$k^*\in \mathbb N$$ such that $$\rho _{2k+1}(H)= k\mathsf D(G)+\lfloor \frac{\mathsf D(G)}{2}\rfloor $$ for every $$k\ge k^*$$ . This confirms a conjecture of A. Geroldinger, D. Grynkiewicz, and P. Yuan in [13]. [ABSTRACT FROM AUTHOR]

The set of distances of a monoid or of a domain is the set of all d ∈ N with the following property: there are irreducible elements u 1 , … , u k , v 1 , … , v k + d such that u 1 ⋅ … ⋅ u k = v 1 ⋅ … ⋅ v k + d , but u 1 ⋅ … ⋅ u k cannot be written as a product of l irreducible elements for any l with k < l < k + d . We show that the set of distances is an interval for certain seminormal weakly Krull monoids which include seminormal orders in holomorphy rings of global fields. [ABSTRACT FROM AUTHOR]

Let F q be a finite field with char F q = p and n > 0 an integer with gcd ( n , log p ⁡ q ) = 1 . Let ( ) ⁎ : F q ( x 0 , … , x n − 1 ) → F q ( x 0 , … , x n − 1 ) be the F q -monomorphism defined by x i ⁎ = x i + 1 for 0 ≤ i < n − 1 and x n − 1 ⁎ = x 0 q . For f , g ∈ F q ( x 0 , … , x n − 1 ) ∖ F q , define f ∘ g = f ( g , g ⁎ , … , g ( n − 1 ) ⁎ ) . Then ( F q ( x 0 , … , x n − 1 ) ∖ F q , ∘ ) is a monoid whose invertible elements are called global P -forms. Global P -forms were first introduced by H. Dobbertin in 2001 with q = 2 to study a certain type of permutation polynomials of F 2 m with gcd ( m , n ) = 1 ; global P -forms with q = p for an arbitrary prime p were considered by W. More in 2005. In this paper, we discuss some fundamental questions about global P -forms, some of which are answered and others remain open. [ABSTRACT FROM AUTHOR]

Inspired by the Odlyzko root discriminant and Golod–Shafarevich p -group bounds, Martinet (1978) asked whether an imaginary quadratic number field K / Q must always have an infinite Hilbert 2-class field tower when the class group of K has 2-rank 4, or equivalently when the discriminant of K has 5 prime factors. No negative results are known. Benjamin (2001, 2002) and Sueyoshi (2004, 2009, 2010) systematically established infinite 2-towers for many K in question, by casework on the associated Rédei matrices. Others, notably Mouhib (2010), have also made progress, but still many cases remain open, especially when the class group of K has small 4-rank. Recently, Benjamin (2015) made partial progress on several of these open matrices when the class group of K has 4-rank 1 or 2. In this paper, we partially address many open cases when the 4-rank is 0 or 2, affirmatively answering some questions of Benjamin. We then investigate barriers to our methods and ask an extension question (of independent interest) in this direction. Finally, we suggest places where speculative refinements of Golod–Shafarevich or group classification methods might overcome the “near miss” inadequacies in current methods. [ABSTRACT FROM AUTHOR]