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We call an abelian variety over a finite field F q super-isolated if its (F q -rational) isogeny class contains a single isomorphism class. In this paper, we use the Honda-Tate theorem to characterize super-isolated ordinary simple abelian varieties by certain algebraic integers. Our main result is that for a fixed dimension g ≥ 3 , there are finitely many such varieties over all finite fields. [ABSTRACT FROM AUTHOR]

Abstract In this paper we describe how to use the algorithmic methods provided by Hunter and Pohst in order to give a complete classification of number fields of degree 8 and signature (2 , 3) with absolute discriminant less than a certain bound. The choice of this bound comes from the local corrections given by prime ideals to the lower estimates for discriminants obtained with the Odlyzko–Poitou–Serre method. [ABSTRACT FROM AUTHOR]

Abstract: Text: For each positive integer m, let , , where and A is a subset of . Recently Chen proved that for each positive integer m, there exists a set such that and for any . In this paper, the following results are proved: (i) for each positive integer m, there exists a set such that and for all with at most 3 exceptions; (ii) for each positive integer m, there exists a set with and such that and for all with at most 3 exceptions. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=GfZ07Fg5qXE. [Copyright &y& Elsevier]

Abstract: This paper explores a simple yet powerful relationship between the problem of counting lattice points and the computation of Dedekind sums. We begin by constructing and proving a sharp upper estimate for the number of lattice points in tetrahedra with some irrational coordinates for the vertices. Besides providing a sharper estimate, this upper bound (Theorem 1.1) becomes an equality (i.e. gives the exact number of lattice points) in a tetrahedron where the lengths of the edges divide each other. This equality condition can then be applied to the explicit computation of the classical Dedekind sums, a topic that is the central focus in the second half of our paper. In this half of the paper, we come up with a number of interesting results related to Dedekind sums, based on our upper estimate (Theorem 1.1). Among these findings, Theorem 1.9 and Theorem 1.10 deserve special attention, for they successfully generalize two of Apostol''s formulas in [T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1997], and also directly imply the famous Reciprocity Law of Dedekind sums. [Copyright &y& Elsevier]

Abstract: We give an alternative computation of the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field (Theorem 2), a particular case of results of Movahhedi–Nguyen Quang Do. This short Note is motivated by the paper [J. Jossey, Galois 2-extensions unramified outside 2, J. Number Theory 124 (2007) 42–76] and, at this occasion, we bring into focus some classical technics of abelian ℓ-ramification which, unfortunately, are often ignored, especially those developed by J.-F. Jaulent with the ℓ-adic class field theory, and by G. Gras in his book on class field theory, and which considerably simplify the study of such subjects; for instance, our proof of Theorem 2 generalizes the purpose of Jossey''s paper in such a way using a result of Herfort–Zalesskii. This Note is mainly an attempt of clarification about ℓ-rationality. [Copyright &y& Elsevier]

Abstract: In this paper, we study the zeta function, named non-abelian zeta function, defined by Lin Weng. We can represent Weng''s rank r zeta function of an algebraic number field F as the integration of the Eisenstein series over the moduli space of the semi-stable -lattices with rank r. For , in the case of , Weng proved that it can be written by the Riemann zeta function, and Lagarias and Suzuki proved that it satisfies the Riemann hypothesis. These results were generalized by the author to imaginary quadratic fields and by Lin Weng to general number fields. This paper presents proofs of both these results. It derives a formula (first found by Weng) for Weng''s rank 2 zeta functions for general number fields, and then proves the Riemann hypothesis holds for such zeta functions. [Copyright &y& Elsevier]

Abstract: Let be a prime, and . In this paper we obtain a general criterion for to be a quartic residue in terms of appropriate binary quadratic forms. Let be a squarefree integer such that , where is the Legendre symbol, and let be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes so that is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where is an odd prime, and . As an application we also obtain a general criterion for , where is the Lucas sequence defined by and . [Copyright &y& Elsevier]

Define D n ( x ) = ∑ k = 0 n ( n k ) 2 x k ( x + 1 ) n − k for n = 0 , 1 , 2 , … and s n ( x ) = ∑ k = 1 n 1 n ( n k ) ( n k − 1 ) x k − 1 ( x + 1 ) n − k for n = 1 , 2 , 3 , … . Then D n ( 1 ) is the n -th central Delannoy number D n , and s n ( 1 ) is the n -th little Schröder number s n . In this paper we obtain some surprising arithmetic properties of D n ( x ) and s n ( x ) . We show that 1 n ∑ k = 0 n − 1 D k ( x ) s k + 1 ( x ) ∈ Z [ x ( x + 1 ) ] for all n = 1 , 2 , 3 , … . Moreover, for any odd prime p and p -adic integer x ≢ 0 , − 1 ( mod p ) , we establish the supercongruence ∑ k = 0 p − 1 D k ( x ) s k + 1 ( x ) ≡ 0 ( mod p 2 ) . As an application we confirm Conjecture 5.5 in [S14a] , in particular we prove that 1 n ∑ k = 0 n − 1 T k M k ( − 3 ) n − 1 − k ∈ Z for all n = 1 , 2 , 3 , … , where T k is the k -th central trinomial coefficient and M k is the k -th Motzkin number. [ABSTRACT FROM AUTHOR]

There is a large literature on the asymptotic distribution of numbers free of large prime factors, so-called smooth or friable numbers. But there is very little known about this distribution that is numerically explicit. In this paper we follow the general plan for the saddle point argument of Hildebrand and Tenenbaum, giving explicit and fairly tight intervals in which the true count lies. We give two numerical examples of our method, and with the larger one, our interval is so tight we can exclude the famous Dickman–de Bruijn asymptotic estimate as too small and the Hildebrand–Tenenbaum main term as too large. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve defined over a number field K . Then its Mordell–Weil group E ( K ) is finitely generated: E ( K ) ≅ E ( K ) t o r × Z r . In this paper, we discuss the cyclic torsion subgroup of elliptic curves over cubic number fields. For N = 169 , 143 , 91 , 65 , 77 or 55, we show that Z / N Z is not a subgroup of E ( K ) t o r for any elliptic curve E over a cubic number field K . [ABSTRACT FROM AUTHOR]

Let F be a totally real algebraic number field, with O F the ring of algebraic integers in F , let L be an O F -lattice on a d -dimensional ( d ≥ 2 ) positive definite quadratic space V over F and let u 0 be a primitive vector in L . The main objective of this paper is to study when a is a fixed (non-unit) algebraic integer in O F and n is a positive rational integer, how the class numbers of lattices translation L + u 0 a n grow as n tends to infinity. [ABSTRACT FROM AUTHOR]

Let a , b , c be fixed coprime positive integers with min ⁡ { a , b , c } > 1 . In this paper, combining the Gel'fond–Baker method with an elementary approach, we prove that if max ⁡ { a , b , c } > 5 × 10 27 , then the equation a x + b y = c z has at most three positive integer solutions ( x , y , z ) . [ABSTRACT FROM AUTHOR]

Let χ range over the non-trivial primitive characters associated with the abelian extensions L / K of a given number field K , i.e. over the non-trivial primitive characters on ray class groups of K . Let f χ be the norm of the finite part of the conductor of such a character. It is known that | L ( 1 , χ ) | ≤ 1 2 Res s = 1 ( ζ K ( s ) ) log ⁡ f χ + O ( 1 ) , where the implied constants in this O ( 1 ) are effective and depend on K only. The proof of this result suggests that one can expect better upper bounds by taking into account prime ideals of K dividing the conductor of χ , i.e. ramified prime ideals. This has already been done only in the case that K = Q . This paper is devoted to giving for the first time such improvements for any K . As a non-trivial example, we give fully explicit bounds when K is an imaginary quadratic number field. [ABSTRACT FROM AUTHOR]

In this paper we first of all determine all possible genera of (odd and even) definite unimodular lattices over an imaginary-quadratic field. The main questions are whether the partial class numbers of lattices with given Steinitz class within one genus are equal for all occurring Steinitz classes and whether the partial masses of those partial genera are equal. We show that the answer to the first question in general is “no” by giving a counter example, while the answer to the second question is “yes” by proving a mass formula for partial masses. Finally, we determine a list of all single-class partial genera and show that a partial genus consists of only one class if and only if the whole genus consists of only one class. [ABSTRACT FROM AUTHOR]

In this paper, we study multizeta values over function fields in characteristic p . For each d ≥ 2 , we show that when the constant field has cardinality >2, the field generated by all multizeta values of depth d is of infinite transcendence degree over the field generated by all single zeta values. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the 3-rank of the ideal class group of the imaginary quadratic field Q ( 4 − 3 18 n + 3 ) is at least 3 for every positive integer n . [ABSTRACT FROM AUTHOR]

Possible transcendental nature of Euler's constant γ has been the focus of study for sometime now. One possible approach is to consider γ not in isolation, but as an element of the infinite family of generalized Euler–Briggs constants. In a recent work [6] , it is shown that the infinite list of generalized Euler–Briggs constants can have at most one algebraic number. In this paper, we study the dimension of spaces generated by these generalized Euler–Briggs constants over number fields. More precisely, we obtain non-trivial lower bounds (see Theorem 5 and Theorem 6 ) on the dimension of these spaces and consequently establish the infinite dimensionality of the space spanned. Further, we study linear and algebraic independence of these constants over the field of all algebraic numbers. [ABSTRACT FROM AUTHOR]

Text In 2014, Herzog, Kaplan and Lev introduced the critical numbers cr ( r , n ) of natural intervals [ r , n ] and determined the values of cr ( 1 , n ) and cr ( 2 , n ) for all n , and for r ≥ 3 , they determined the values of cr ( r , n ) for n ≤ ( 3 r 2 + 3 r ) / 2 and n ≥ ( 5 r 2 + r ) / 2 . In this note, we determine the remaining values of cr ( r , n ) for r ≥ 3 . Video For a video summary of this paper, please visit https://youtu.be/hZSI-Ul3K94 . [ABSTRACT FROM AUTHOR]

Let A K denote the ring of algebraic integers of an algebraic number field K = Q ( θ ) where the algebraic integer θ has minimal polynomial F ( x ) = x n + a x m + b over the field Q of rational numbers with n = m t + u , t ∈ N , 0 ≤ u ≤ m − 1 . In this paper, we characterize those primes which divide the discriminant of F ( x ) but do not divide [ A K : Z [ θ ] ] when u = 0 or u divides m ; such primes p are important for explicitly determining the decomposition of p A K into a product of prime ideals of A K in view of the well known Dedekind theorem. As a consequence, we obtain some necessary and sufficient conditions involving only a , b , m , n for A K to be equal to Z [ θ ] . [ABSTRACT FROM AUTHOR]

In this paper we deal with Diophantine equations involving products of consecutive integers, inspired by a question of Erdős and Graham. [ABSTRACT FROM AUTHOR]

In this paper, we sharpen and simplify our earlier results based on Thue's Fundamentaltheorem and use it to obtain effective irrationality measures for certain roots of particular polynomials of the form ( x − t ) n + ( x + t ) n , where n ≥ 4 is a positive integer and t is a negative integer. For n = 4 and n = 5 , we find infinitely many such algebraic numbers. [ABSTRACT FROM AUTHOR]

Let K be an algebraic number field and B ≥ 1 . For an endomorphism φ : P n → P n defined over K of degree d let R φ ⊂ O K denote its minimal resultant ideal. For a fixed height function h M d n on the moduli space of dynamical systems this paper shows that all such morphisms φ of bounded resultant N K / Q ( R φ ) ≤ B and bounded height h M d n ( 〈 φ 〉 ) ≤ B are contained in finitely many PGL n + 1 ( K ) -equivalence classes. This answers a question of Silverman in the affirmative. [ABSTRACT FROM AUTHOR]

Abstract: Text: Let K be an algebraic number field and the ring of integers in K. Let be the set of all elements which are sums of squares in and the minimal number of squares necessary to represent −1 in . Let be the smallest positive integer t such that every element in is a sum of t squares in . In this note, for , where are two distinct positive square-free integers, we show that and if , then . Video: For a video summary of this paper, please click here or visit http://youtu.be/F6UHHgzfdzA. [Copyright &y& Elsevier]

Abstract: In the paper, the author establishes several integral representations and properties of Stirling numbers of the first kind. [Copyright &y& Elsevier]

Abstract: Text: We prove two congruence results concerning Kloosterman sums over finite fields. The first result concerns the coefficients of the characteristic polynomial over of a Kloosterman sum, and the second result gives a characterisation of ternary Kloosterman sums modulo 27. We use methods from algebraic number theory such as Stickelbergerʼs theorem and the Gross–Koblitz formula, as well as Fourier analysis. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=VJcB6W_PQ0s. [Copyright &y& Elsevier]

Abstract: Text: For any integer , we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order . The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class to be a square. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=H49W8Wyl47o. [Copyright &y& Elsevier]

Abstract: Text: The isomorphism number of an F-crystal over an algebraically closed field of positive characteristic is the smallest non-negative integer such that the -th level truncation of determines the isomorphism class of . When is isoclinic, namely it has a unique Newton slope λ, we provide an efficiently computable upper bound for in terms of λ and the Hodge slopes of . This is achieved by providing an upper bound for the level torsion of introduced by Vasiu. We also check that this upper bound is optimal for many families of isoclinic F-crystals that are of special interest (such as isoclinic F-crystals of K3 type). Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=gVObAZZ1DKE. [Copyright &y& Elsevier]

Abstract: In this paper we give two proofs of the mass formula for definite central division algebras over global function fields, due to Denert and Van Geel. The first proof is based on a calculation of Tamagawa measures. The second proof is based on analytic methods, in which we establish the relationship directly between the mass and the value of the associated zeta function at zero. [Copyright &y& Elsevier]

Abstract: Let p be an odd prime number. For the cyclotomic -extension of a finite algebraic number field F, we denote by the maximal unramified pro-p-extension of . In this paper, using Iwasawa theory and the theory of central class fields, we give necessary conditions for to be abelian, and give sufficient conditions in certain special cases. [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: In this paper we show that, for a K3 surface within a certain class of surfaces and over a number field, the orbit of a point under the group of automorphisms is either finite or its exponent of growth is exactly the Hausdorff dimension of a fractal associated to the ample cone. In particular, the exponent depends on the geometry of the surface and not its arithmetic. For surfaces in this class, the exponent is . [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give irrationality results for numbers of the form where the numbers behave like a geometric progression for a while. The method is elementary, not using differentiation or integration. In particular, we derive elementary proofs of the irrationality of π and for Gaussian integers . [Copyright &y& Elsevier]

Abstract: In this paper, we determine the closure in the full topology over of the set , where is a nondegenerate binary recurrent sequence with integer coefficients whose characteristic roots are quadratic units. This generalizes the result for the case when was the nth Fibonacci number. [Copyright &y& Elsevier]

Abstract: In this paper we study generalised prime systems for which the integer counting function is asymptotically well behaved, in the sense that , where ρ is a positive constant and . For such systems, the associated zeta function is holomorphic for . We prove that for , for any , and also for for all such σ except possibly one value. The Dirichlet divisor problem for generalised integers concerns the size of the error term in , which is for some . Letting denote the infimum of such θ, we show that . [Copyright &y& Elsevier]

Abstract: We call a set A of positive integers an asymptotic basis of order h if every sufficiently large integer n can be written as a sum of h elements of A. If no proper subset of A is an asymptotic basis of order h, then A is a minimal asymptotic basis of that order. Erdős and Nathanson showed that for every there exists a minimal asymptotic basis A of order h with , where denotes the density of A. Erdős and Nathanson asked whether it is possible to strengthen their result by deciding on the existence of a minimal asymptotic bases of order such that . Moreover, they asked if there exists a minimal asymptotic basis with . In this paper we answer these questions in the affirmative by constructing a minimal asymptotic basis A of order 2 fulfilling a very restrictive condition [Copyright &y& Elsevier]

Abstract: The main theorem of this paper, proved using Mahler''s method, gives a necessary and sufficient condition for the values at any distinct algebraic points to be algebraically independent, where is an analogue of a certain q-hypergeometric series and generated by a linear recurrence whose typical example is the sequence of Fibonacci numbers. Corollary 1 gives taking algebraically independent values for any distinct triplets of nonzero algebraic numbers. Moreover, is expressed as an irregular continued fraction and is an analogue of q-exponential function as stated in Corollaries 3 and 4, respectively. [Copyright &y& Elsevier]

Abstract: Let D be the ring of integers of a number field K, P a prime of D for which is finite, the corresponding valuation of K and E a homogeneous subset of D with respect to P, i.e. a set with the property for some positive integer ℓ. Also let denote the ring of polynomials in which take values in D when evaluated at points of E. The characteristic sequence of E with respect to P is the sequence of integers where is the fractional ideal formed by 0 and the leading coefficients of elements of of degree ⩽n. In this paper we give a recursive method for computing the limit for any homogeneous set, apply it to the special case of the homogeneous sets for  , and show that in general the possible values of this limit as E ranges over all possible homogeneous subsets are dense in the interval . We also apply this method to certain infinite unions of homogeneous sets and obtain formulas for these limits as regular continued fractions. [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: 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: 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 p be an odd prime and , where m is a positive integer. Let be a qth primitive root of 1 and be the ring of integers in . In [I. Gaál, L. Robertson, Power integral bases in prime-power cyclotomic fields, J. Number Theory 120 (2006) 372–384] I. Gaál and L. Robertson show that if , where is the class number of , then if is a generator of (in other words ) either α is equals to a conjugate of an integer translate of or is an odd integer. In this paper we show that we can remove the hypothesis over . In other words we show that if is a generator of then either α is a conjugate of an integer translate of or is an odd integer. [Copyright &y& Elsevier]

Abstract: In this paper, we are able to sharpen Hua''s classical result by showing that each sufficiently large integer can be written as where and p, are primes. This result improves a previous result with replaced by . [Copyright &y& Elsevier]

Abstract: In this paper we will apply Biró''s method in [A. Biró, Yokoi''s conjecture, Acta Arith. 106 (2003) 85–104; A. Biró, Chowla''s conjecture, Acta Arith. 107 (2003) 179–194] to class number 2 problem of real quadratic fields of Richaud–Degert type and will show that there are exactly 4 real quadratic fields of the form with class number 2, where is a even square free integer. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the generalization of the Hecke''s integration formula for the Epstein zeta functions. We treat the Epstein zeta function as an Eisenstein series come from a degenerate principal series. For the Epstein zeta function of degree two, Siegel considered the Hecke''s formula as the constant term of a certain Fourier expansion of the Epstein zeta function and obtained the other Fourier coefficients as the Dedekind zeta functions with Grössencharacters of a real quadratic field. We generalize this Siegel''s Fourier expansion to more general Eisenstein series with harmonic polynomials. Then we obtain the Dedekind zeta functions with Grössencharacters for arbitrary number fields. [Copyright &y& Elsevier]

Abstract: In this paper, we apply Langlands–Shahidi method to exceptional groups, with the assumption that the cuspidal representations have one spherical tempered component. A basic idea is to use the fact that the local components of residual automorphic representations are unitary representations, and use the classification of the unitary dual. We prove non-unitarity of certain spherical representations of exceptional groups. We need to divide into five different cases, and in two cases we can prove that the completed L-functions are holomorphic except possibly at 0, 1/2, 1 under some local assumptions. [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 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: In this paper, we show the following theorems. Suppose are algebraically independent numbers and for . Then we have the joint t-universality for Lerch zeta functions for . Next we generalize Lerch zeta functions, and obtain the joint t-universality for them. In addition, we show examples of the non-existence of the joint t-universality for Lerch zeta functions and generalized Lerch zeta functions. [Copyright &y& Elsevier]

Abstract: The structure of the Galois group of the maximal unramified p-extension of an imaginary quadratic field is restricted in various ways. In this paper we construct a family of finite 3-groups satisfying these restrictions. We prove several results about this family and characterize them as finite extensions of certain quotients of a Sylow pro-3 subgroup of . We verify that the first group in the family does indeed arise as such a Galois group and provide a small amount of evidence that this may hold for the other members. If this was the case then it would imply that there is no upper bound on the possible lengths of a finite p-class tower. [Copyright &y& Elsevier]

Abstract: In this paper we study the Picard modular forms and show a new three terms arithmetic geometric mean (AGM) system. This AGM system is expressed via the Appell hypergeometric function of two variables. The Picard modular forms are expressed via the theta constants, and they give the modular function for the family of Picard curves. Our theta constants are “Neben type” modular forms of weight 1 defined on the complex 2-dimensional hyperball with respect to an index finite subgroup of the Picard modular group. We define a simultaneous 3-isogeny for the family of Jacobian varieties of Picard curves. Our main theorem shows the explicit relations between two systems of theta constants which are corresponding to isogenous Jacobian varieties. This relation induces a new three terms AGM which is a generalization of Borweins'' cubic AGM. [Copyright &y& Elsevier]