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Abstract: As part of his work to develop an explicit trace formula for Hecke operators on congruence subgroups of , Hijikata (1974) defines and characterizes the notion of a split order in , where k is a local field. In this paper, we generalize the notion of a split order to for and give a natural geometric characterization in terms of the affine building for . In particular, we show that there is a one-to-one correspondence between split orders in and a collection of convex polytopes in apartments of the building such that the split order is the intersection of all the maximal orders representing the vertices in the polytope. This generalizes the geometric interpretation in the case in which split orders correspond to geodesics in the tree for with the split order given as the intersection of the endpoints of the geodesic. [Copyright &y& Elsevier]

Abstract: Text: We prove five of Sun''s conjectures on the index of the subgroup of fourth powers in the class group of binary quadratic forms. Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=jMuO4RAp1Xw. [Copyright &y& Elsevier]

Abstract: The focus of this paper will be the extension of the Waring–Goldbach problem to all sufficiently large integers, without congruence restrictions. By reintroducing the effect of small primes, we are able to consider questions which more naturally resemble Waring''s problem and the Goldbach conjecture. We extend the results of S.S. Pillai by considering the problem without the use of zero as an addend and we give a small improvement on the number of additional terms required. [Copyright &y& Elsevier]

Abstract: In Schweiger (2003) , Fritz Schweiger introduced the algorithm of the generalized continued fraction (GCF), and in Zhong (2008) , T. Zhong studied some basic metric properties of the GCF. In this paper, under the restriction of , the “0–1” law and the central limit theorem of quotients in the GCF expansions are studied. [Copyright &y& Elsevier]

Abstract: Knowing the number of solutions for a Diophantine equation is an important step to study various arithmetic problems. Igusa originated the study of Igusa zeta functions associated to local Diophantine problems. Multiplying all these local Igusa zeta functions we obtain the global version in the natural way. Unfortunately, investigations on global Igusa zeta functions are rare up to now. Reformulating the global Igusa zeta function via the number of morphisms between algebraic systems we discover a new aspect: the multivariable Euler product of Igusa type and its applications. A purpose of this paper is to encourage experts for further studies on global Igusa zeta functions by treating a simple interesting example. [Copyright &y& Elsevier]

Abstract: Text: Let be relatively prime positive integers. We prove a conjecture of Aukerman, Kane and Sze regarding the largest size of a partition that is simultaneously s-core and t-core by solving an equivalent problem concerning sets S of positive integers with the property that for , whenever and whenever . Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=o1OEug8LryU. [Copyright &y& Elsevier]

Abstract: Text: We show that the algebra and the endomotive of the quantum statistical mechanical system of Bost–Connes naturally arises by extension of scalars from the “field with one element” to rational numbers. The inductive structure of the abelian part of the endomotive corresponds to the tower of finite extensions of that “field,” while the endomorphisms reflect the Frobenius correspondences. This gives in particular an explicit model over the integers for this endomotive, which is related to the original Hecke algebra description. We study the reduction at a prime of the endomotive and of the corresponding noncommutative crossed product algebra. Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=az_0pxm1jrI. [Copyright &y& Elsevier]

Abstract: Let be a prime and with . Suppose for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where is the Lucas sequence given by , and . We also pose many conjectures concerning , or . [Copyright &y& Elsevier]

Abstract: Serre and Stark found a basis for the space of modular forms of weight 1/2 in terms of theta series. In this paper, we generalize their result—under certain mild restrictions on the level and character—to the case of weight 1/2 Hilbert modular forms over a totally real field of narrow class number 1. The methods broadly follow those of Serre–Stark; however we are forced to overcome technical difficulties which arise when we move out of . [Copyright &y& Elsevier]

Abstract: Let A be a quaternion algebra over a number field k and assume that A satisfies the Eichler condition so that some infinite place of k is unramified in A. Let L be a quadratic extension of k which embeds in A. Let denote the ring of integers of k and let B be an -order in L. Suppose that is an Eichler order of A of square-free level S. In this paper, we determine when there exists an embedding over k which gives an optimal embedding of B into in the sense that . This generalises previous work of Eichler [M. Eichler, Zur Zahlentheorie der Quaternionenalgebren, J. Reine Angew. Math. 195 (1955) 127–155] and Chinburg and Friedman [T. Chinburg, E. Friedman, An embedding theorem for quaternion algebras, J. London Math. Soc. 60 (1999) 33–44]. [Copyright &y& Elsevier]

Abstract: This paper continues the study of beta-expansions of 1 where β is a Pisot or Salem number. Special attention is given to regular Pisot numbers, and the Salem numbers that approach these Pisot numbers. [Copyright &y& Elsevier]

Abstract: Let p be an odd prime and be the smallest positive integer s such that every integer is a sum of s kth powers . We establish and provided that k is not divisible by . Next, let , and q be any positive integer. We show that if then for some constant . These results generalize results known for the case of prime moduli. Video abstract: For a video summary of this paper, please visit http://www.youtube.com/watch?v=zpHYhwL1kD0. [Copyright &y& Elsevier]

Abstract: In this paper, we give some explicit evaluations of multiple zeta-star values which are rational multiple of powers of . [Copyright &y& Elsevier]

Abstract: We call a quadratic extension of a cyclotomic field a quasi-cyclotomic field if it is non-abelian Galois over the rational number field. In this paper, we study the arithmetic of any quasi-cyclotomic field, including to determine the ring of integers of it, the decomposition nature of prime numbers in it, and the structure of the Galois group of it over the rational number field. We also describe explicitly all real quasi-cyclotomic fields, namely, the maximal real subfields of quasi-cyclotomic fields which are Galois over the rational number field. It gives a series of totally real fields and CM fields which are non-abelian Galois over the rational number field. [Copyright &y& Elsevier]

Abstract: In 2002, Hartono, Kraaikamp and Schweiger introduced the Engel continued fractions (ECF), whose partial quotients are increasing. Later, Schweiger generalized it into a class of continued fractions with increasing digits and a parameter ϵ, called generalized continued fractions (GCF). In this paper, we will give some arithmetic properties of such an expansion, and show that the GCF holds similar metric properties with ECF under the condition that . But when it comes to the condition that , it does not. [Copyright &y& Elsevier]

Abstract: In this paper we consider arithmetic progressions on Pell equations, i.e. integral solutions whose X-coordinates or Y-coordinates are in arithmetic progression. [Copyright &y& Elsevier]

Abstract: A set of m distinct positive integers is called a Diophantine m-tuple if is a square for each . In this paper, we show that for each integer the Diophantine pair cannot be extended to a Diophantine quintuple. [Copyright &y& Elsevier]

Abstract: For let denote the number of solutions of the equation with . For a prime p let χ denote a multiplicative character . In this paper, we obtain for some constant C and any , which improved the corresponding error terms obtained by A. Ayyad, T. Cochrane and Z.Y. Zheng. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper is to prove a conjecture of the Euler numbers and its generalization by using the analytic methods. That is, for any prime and integer we proved , where are the Euler numbers and the Euler function. [Copyright &y& Elsevier]

Abstract: Let K be a quadratic imaginary number field, and let be the order of conductor f in K. We consider the singular values of the Kleinian normalization φ of the Weierstrass σ-function belonging to an arbitrary proper ideal of . The factorization of these singular values goes back to K. Ramachandra, R. Schertz and W. Bley. But the factorization formula in [W. Bley, Konstruktion von Ganzheitsbasen in abelschen Körpererweiterungen von imaginär-quadratischen Zahlkörpern, Dissertation, Universität Augsburg, 1991] is very implicit and not easy to handle in view of many practical applications. In this paper we provide an explicit factorization formula and give different tools to control this factorization. As an immediate application we prove the generalized principal ideal theorem in the ring class field situation. [Copyright &y& Elsevier]

Abstract: In this paper we develop a method for determining the number of integers without large prime factors lying in a given set S. We will apply it to give an easy proof that certain sufficiently dense sets A and B always produce the expected number of “smooth” sums , , . The proof of this result is completely combinatorial and elementary. [Copyright &y& Elsevier]

Abstract: In this paper we show how one can obtain simultaneous rational approximants for and with a common denominator by means of Hermite–Padé approximation using multiple little q-Jacobi polynomials and we show that properties of these rational approximants prove that 1, , are linearly independent over . In particular this implies that and are irrational. Furthermore we give an upper bound for the measure of irrationality. [Copyright &y& Elsevier]

Abstract: In this paper we give an alternative review over the subject of strata and simple characters as more generally already considered in the references [P. Broussous, M. Grabitz, Pure elements and intertwining classes of simple strata in local central simple algebras, Comm. Algebra 28 (11) (2000) 5405–5442] and [V. Sécherre, Représentations lisses de , I: charactères simples, Bull. Soc. Math. France 132 (3) (2004) 327–396]. Finally, we prove the intertwining implies conjugacy property of simple characters as known in the split case [C.J. Bushnell, P.C. Kutzko, The Admissible Dual of via Compact Open Subgroups, Ann. of Math. Stud., vol. 129, Princeton Univ. Press, 1993]. [Copyright &y& Elsevier]

Abstract: In this paper, we study the function , which associates to every pair of positive integers a and b the number of positive integers c such that the triangle of sides a, b and c is Heron, i.e., it has integral area. In particular, we prove that if p and q are primes, and that for a random choice of positive integers a and b. [Copyright &y& Elsevier]

Abstract: Let n be a positive integer. In this paper we estimate the size of the set of linear forms , where and are integers, as . [Copyright &y& Elsevier]

Abstract: In this paper we consider the integers of the forms and , which are ever focused by F. Cohen, P. Erdős, J.L. Selfridge, W. Sierpiński, etc. We establish a general theorem. As corollaries, we prove that (i) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of four integers , , and has at least two distinct odd prime factors; (ii) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of ten integers , , , , , , , , and has at least two distinct odd prime factors; (iii) there exists an infinite arithmetic progression of positive odd numbers for each term k of which and any nonnegative integer n, each of ten integers , , , , , , , , and has at least two distinct odd prime factors. Furthermore, we pose several related open problems in the introduction and three conjectures in the last section. [Copyright &y& Elsevier]

Abstract: The general case of the Nagell–Ljunggren equation is and odd primes. In this paper we derive a new bound for which there are no solutions. For small q the problem is harder and we achieve a conditional result: for and some additional condition on must hold. Both functions are quadratic and they leave a small gap for , on which we have no general result. The picture is completed by some explicit, unconditional upper bounds for the absolute values , . [Copyright &y& Elsevier]

Abstract: Let be a prime, , , and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for to be a cubic residue of p, where is the fundamental unit of the quadratic field . We also give a general criterion for , where is the Lucas sequence defined by , and (). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms. [Copyright &y& Elsevier]

Abstract: J.-M. Kim, S. Bae and I.-S. Lee showed that there exists an isomorphism between the p-primary part of the ideal class group and p-primary part of the unit group modulo cyclotomic unit group in for all sufficiently large n under some conditions. In the present paper, we shall give an analogue of their result for modular units. [Copyright &y& Elsevier]

Abstract: Lubin conjectures that for an invertible series to commute with a noninvertible series with only simple roots of iterates, two such commuting power series must be endomorphisms of a single formal group. In this paper, we show that if the reduction of these two commuting power series are endomorphisms of a formal group, then themselves are endomorphisms of a formal group. [Copyright &y& Elsevier]

Abstract: In this paper we obtain an improved asymptotic formula on the frequency of k-free numbers with a given difference. We also give a new upper bound of Barban–Davenport–Halberstam type for the k-free numbers in arithmetic progressions. [Copyright &y& Elsevier]

Abstract: In this paper we prove the best possible upper bounds for the number of elements in a set of polynomials with integer coefficients all having the same degree, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients. Moreover, we prove that there does not exist a set of more than 12 polynomials with integer coefficients and with the property from above. This significantly improves a recent result of the first two authors with Tichy [A. Dujella, C. Fuchs, R.F. Tichy, Diophantine m-tuples for linear polynomials, Period. Math. Hungar. 45 (2002) 21–33]. [Copyright &y& Elsevier]

Abstract: In this paper, we characterize the irreducible generic representations of where F is a p-adic field in terms of their γ-factors and central character. This is known as the local converse theorem. This result refines the one proved by Henniart in 1993 [G. Henniart, Caracterisation de la correspondance de Langlands locale par les facteurs ε de paires, Invent. Math. 113 (1993) 339–350]. [Copyright &y& Elsevier]

Abstract: It is conjectured by Erdős, Graham and Spencer that if with , then this sum can be decomposed into n parts so that all partial sums are ⩽1. This is not true for as shown by , , . In 1997, Sándor proved that Erdős–Graham–Spencer conjecture is true for . In this paper, we reduce Erdős–Graham–Spencer conjecture to finite calculations and prove that Erdős–Graham–Spencer conjecture is true for . Furthermore, it is proved that Erdős–Graham–Spencer conjecture is true if and no partial sum (certainly not a single term) is the inverse of an positive integer. [Copyright &y& Elsevier]

Abstract: The arithmetic Kodaira–Spencer class of the universal elliptic curve was introduced in [A. Buium, Differential modular forms, J. Reine Angew. Math. 520 (2000) 95–167]; its reduction mod p was explicitly computed by Hurlburt [C. Hurlburt, Isogeny covariant differential modular forms modulo p, Compos. Math. 128 (1) (2001) 17–34]. In this paper the complicated expression of Hurlburt is shown to be the unique solution of a simple partial differential equation subject to a certain initial condition and weight condition. [Copyright &y& Elsevier]

Abstract: This paper is concerned with modular lattices over cyclotomic fields. In particular, the notion of Arakelov modular ideal lattice is introduced. All the cyclotomic fields over which there exists an Arakelov modular lattice of given level are characterised. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the reciprocity map of unramified class field theory for smooth projective surfaces over non-archimedean local fields which do not have potentially good reduction. We will construct two types of smooth projective surfaces whose reciprocity maps modulo positive integers are not injective. The first type is the case where the kernel of the reciprocity map is not divisible. The second is the case where the kernel of the reciprocity map is divisible, but where nevertheless the reciprocity map modulo some integer is not injective. [Copyright &y& Elsevier]

Abstract: For finite subsets of a field, their sumset is given by . In this paper, we study various restricted sumsets of with restrictions of the following forms:Furthermore, we gain an insight into relations among recent results on this area obtained in quite different ways. [Copyright &y& Elsevier]

Abstract: In this paper we prove that for a prime and , if , then is not a subgroup of which partially verifies a conjecture of J. Browkin. [Copyright &y& Elsevier]

Abstract: A rational cuboid is a rectangular parallelepiped whose edges and face diagonals all have rational lengths. In this paper, we consider the problem: are there rational cuboids with a given face? In a sense, we reduce the problem to a finite calculation. [Copyright &y& Elsevier]

Abstract: This paper discusses tetrahedra with rational edges forming an arithmetic progression, focussing specifically on whether they can have rational volume or rational face areas. Several infinite families are found which have rational volume, a face can have rational area only if its edges are themselves in arithmetic progression, and a tetrahedron can have at most one such rational face area. [Copyright &y& Elsevier]

Abstract: Let be a -sheaf. Building on previous work of Drinfeld, Anderson, Taguchi, and Wan, Böckle and Pink (A cohomological theory of crystals over function fields, in preparation), develop a cohomology theory for . Böckle (Math. Ann. 323 (2002) 737) uses this theory to establish the analytic continuation of the L-series associated to (which is a characteristic p-valued “Dirichlet series”) and the logarithmic growth of the degrees of its special polynomials. In this paper, we shall show that this logarithmic growth is all that is needed to analytically continue the original L-series as well as all associated partial L-series. Moreover, we show that the degrees of the special polynomials attached to the partial L-series also grow logarithmically. Our tools are Böckle''s original results, non-Archimedean integration, and the very strong estimates of Amice (Bull. Soc. Math. France 92 (1964) 117). Along the way, we define certain natural modules associated with non-Archimedean measures (in the characteristic 0 case as well as in characteristic p). [Copyright &y& Elsevier]

Abstract: Let be a Galois extension of number fields and let be an abelian variety defined over . In this paper we establish the relation between the irreducible characters of the Galois group and the simple factors of the restriction of scalars of from to . Then we derive some equivalences of Birch and Swinnerton–Dyer conjectures. [Copyright &y& Elsevier]

Abstract: We have previously proved Kummer congruences mod primes p such that for the universal divided Bernoulli numbers . In this paper we strengthen these congruences to hold mod powers of p. [Copyright &y& Elsevier]

In his 1964 paper, de Bruijn (Math. Comp. 18 (1964) 537) called a pair (a,b) of positive odd integers good, if Z=aS⊖2bS, where S is the set of nonnegative integers whose 4-adic expansion has only 0''s and 1''s, otherwise he called the pair (a,b) bad. Using the 2-adic integers we obtain a characterization of all bad pairs. A positive odd integer u is universally bad if (ua,b) is bad for all pairs of positive odd integers a and b. De Bruijn showed that all positive integers of the form u=2k+1 are universally bad. We apply our characterization of bad pairs to give another proof of this result of de Bruijn, and to show that all integers of the form u=φpk(4) are universally bad, where p is prime and φn(x) is the nth cyclotomic polynomial. We consider a new class of integers we call de Bruijn universally bad integers and obtain a characterization of such positive integers. We apply this characterization to show that the universally bad integers u=φpk(4) are in fact de Bruijn universally bad for all primes p>2. Furthermore, we show that the universally bad integers φ2k(4), and more generally, those of the form 4k+1, are not de Bruijn universally bad. [Copyright &y& Elsevier]

In the study of substitutative dynamical systems and Pisot number systems, an algebraic condition, which we call ‘weak finiteness’, plays a fundamental role. It is expected that all Pisot numbers would have this property. In this paper, we prove some basic facts about ‘weak finiteness’. We show that this property is valid for cubic Pisot units and for Pisot numbers of higher degree under a dominant condition. [Copyright &y& Elsevier]

In 1934, Romanoff proved that there are a positive proportion natural numbers which can be expressed as the sum of a prime and a power of 2. In this paper, a quantitative version of this theorem is given. We show that the proportion is larger than 0.0868 and for a positive proportion of odd integers the number of such representations is between 1 and 16. [Copyright &y& Elsevier]

In this paper, we define L-series generalizing the Herglotz–Zagier function (Ber. Verhandl. Sächsischen Akad. Wiss. Leipzig 75 (3–14) (1923) 31, Math. Ann. 213 (1975) 153), and the double zeta function (First European Congress of Mathematics, Vol. II, Paris, 1992, pp. 497–512; Progress in Mathematics, Vol. 120, Birkhäuser, Basel, 1994) and evaluate them after meromorphic continuation at integer points in their extended domains. This is accomplished in three steps. First, when α:Z→C is a periodic function and h(n)=∑j=1n j-1 are the harmonic numbers, we establish identities relating these series to the L-seriesH(α,s)=∑lower limit n=1, upper limit ∞ α(n)h(n)n-s, Re(s)>1,and the Dirichlet L-function. Second, we prove that H(α,s) has a meromorphic continuation to C and evaluate H(α,s) at s=-2l for each integer l⩾0 in terms of Hurwitz zeta functions, generalized Euler''s constants, a finite sum, and zeta functions closely resembling the Hurwitz zeta function. Third, we combine steps one and two with well-known facts concerning the Dirichlet L-function to obtain the desired evaluations. Our results for H(α,s) generalize previous work of Apostol and Vu (J. Number Theory 19 (1) (1984) 85) in the case α is identically one with period one. [Copyright &y& Elsevier]

Let F be a finite field with q elements, and T a transcendental element over F. In this paper, we construct infinitely many real function fields of any fixed degree over F(T) with ideal class numbers divisible by any given positive integer greater than 1. For imaginary function fields, we obtain a stronger result which shows that for any relatively prime integers m and n with m,n>1 and relatively prime to the characteristic of F, there are infinitely many imaginary fields of fixed degree m such that the class group contains a subgroup isomorphic to (Z/nZ)m−1. [Copyright &y& Elsevier]

The formal group of an elliptic curve at a finite prime of the field of definition has proven to be a useful tool in studying the elliptic curve. Moreover, these formal groups are interesting in themselves. In this paper we define and study formal Drinfeld modules in a general setting. We also define the formal Drinfeld module associated to a Drinfeld module at a finite prime. The results are applied to the uniform boundedness conjecture for Drinfeld modules. [Copyright &y& Elsevier]