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Text Let m be an integer with m ≥ 2 . For A ⊆ Z m and n ∈ Z m , let R 1 ( A , n ) , R 2 ( A , n ) , R 3 ( A , n ) denote the number of solutions of the equation a + a ′ = n with ordered pairs ( a , a ′ ) ∈ A × A , unordered pairs ( a , a ′ ) ∈ A × A ( a ≠ a ′ ) and unordered pairs ( a , a ′ ) ∈ A × A , respectively. In this paper, for i ∈ { 1 , 2 , 3 } , we determine all sets A , B ⊆ Z m such that R i ( A , n ) = R i ( B , n ) for all n ∈ Z m when the cardinality of the symmetric difference of A and B is small or large. These extend some previous results. We also pose some problems for further research. Video For a video summary of this paper, please visit https://youtu.be/stBa9Uy5U0I . [ABSTRACT FROM AUTHOR]

Text Let K be a complete, algebraically closed, non-Archimedean valued field, and let ϕ ∈ K ( z ) with deg ⁡ ( ϕ ) ≥ 2 . In this paper we consider the family of functions ord Res ϕ n ( x ) , which measure the resultant of ϕ n at points x in P K 1 , the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov–Green's function g μ ϕ ( x , x ) attached to the canonical measure of ϕ . Following this, we are able to prove an equidistribution result for Rumely's crucial measures ν ϕ n , each of which is a probability measure supported at finitely many points whose weights are determined by dynamical properties of ϕ . Video For a video summary of this paper, please visit https://youtu.be/YCCZD1iwe00 . [ABSTRACT FROM AUTHOR]

Abstract: A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field. The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald–Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to characterize QM-abelian surfaces which has a model of -type over . The “special” involutions on a corresponding indefinite quaternion algebra (which is defined in Section 2) play an essential role in this paper. [Copyright &y& Elsevier]

Abstract: The aim of the paper is to determine all free separable quadratic algebras over the rings of integers of quadratic fields in terms of the properties of the fundamental unit in the real case. The paper corrects some earlier published results on the subject. [Copyright &y& Elsevier]

In this paper, we prove two results. The first theorem uses a paper of Kim (J. Number Theory 74 (1999) 307) to show that for fixed primes p1,…,pk, and for fixed integers m1,…,mk, with pi∤mi, the numbers (ep1(n),…,epk(n)) are uniformly distributed modulo (m1,…,mk), where ep(n) is the order of the prime p in the factorization of n!. That implies one of Sander''s conjectures from Sander (J. Number Theory 90 (2001) 316) for any set of odd primes. Berend (J. Number Theory 64 (1997) 13) asks to find the fastest growing function f(x) so that for large x and any given finite sequence ϵi∈{0,1}, i⩽f(x), there exists n such that the congruences epi(n)≡ϵi (mod 2) hold for all i⩽f(x). Here, pi is the ith prime number. In our second result, we are able to show that f(x) can be taken to be at least c1(log x/(log log x)6)1/9, with some absolute constant c1, provided that only the first odd prime numbers are involved. [Copyright &y& Elsevier]

In this paper, we prove that the expected number of points in F q 2 of multiplicity m , for 0 ≤ m ≤ q + 1 , with respect to a randomly chosen arrangement of q + 1 lines with different slopes, is ( 1 / ( m ! e ) ) q 2 + O ( q ) , as q → ∞ . We further state that the distance between the number of such points in a randomly chosen arrangement and ( 1 / ( m ! e ) ) q 2 is lower than q ln ⁡ q with probability close to 1 for large q . [ABSTRACT FROM AUTHOR]

In this paper, we study the polynomial Gauss sums over finite fields, and present an analogue of Davenport–Hasse's theorem for the polynomial Gauss sums, which is a generalization of the previous result obtained by Hayes. [ABSTRACT FROM AUTHOR]

Let B be a finite set of natural numbers or complex numbers. Product set corresponding to B is defined by B . B : = { a b : a , b ∈ B } . In this paper we give an upper bound for longest length of consecutive terms of a polynomial sequence present in a product set accurate up to a positive constant. We give a sharp bound on the maximum number of Fibonacci numbers present in a product set when B is a set of natural numbers and a bound which is accurate up to a positive constant when B is a set of complex numbers. [ABSTRACT FROM AUTHOR]

In this paper we focus on finding all the factorials expressible as a product of a fixed number of 2 k -nacci numbers with k ≥ 2 . We derive the 2-adic valuation of the 2 k -nacci sequence and use it to establish bounds on the solutions of the initial equation. In addition, we specify a more general family of sequences, for which we can perform a similar procedure. We also investigate a possible connection of these results with p -regular sequences. [ABSTRACT FROM AUTHOR]

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: 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: 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: 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: 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: 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: 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]

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]