Search

Abstract: The paper gives proofs of some results just claimed in [R. Massy, Galois averages, J. Number Theory 113 (2005) 244–275]. For instance, it is proved that for a finite non-trivial separable extension , , of Hilbertian fields finitely generated over their prime field, the quotient group , for the corresponding multiplicative groups of non-zero elements, cannot be a torsion group of finite exponent. [Copyright &y& Elsevier]

Abstract: We shall make a slight improvement to a result of p-adic logarithms, which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient . [Copyright &y& Elsevier]

Abstract: Remark on Polickýʼs paper on circular units of a compositum of quadratic number fields is given. [Copyright &y& Elsevier]

Abstract Let n be a non-negative integer and put p n (x) = ∏ i = 0 n (x + i). In the first part of the paper, for given n , we study the existence of integer solutions of the Diophantine equation y m = p n (x) + ∑ i = 1 k p a i (x) , where m ∈ N ≥ 2 and a 1 < a 2 < ... < a k < n. This equation can be considered as a generalization of the Erdős–Selfridge Diophantine equation y m = p n (x). We present some general finiteness results concerning the integer solutions of the above equation. In particular, if n ≥ 2 with a 1 ≥ 2 , then our equation has only finitely many solutions in integers. In the second part of the paper we study the equation y m = ∑ i = 1 k p a i (x i) , for m = 2 , 3 , which can be seen as an additive version of the equation considered by Erdős and Graham. In particular, we prove that if m = 2 , a 1 = 1 or m = 3 , a 2 = 2 , then for each k − 1 tuple of positive integers (a 2 , ... , a k) there are infinitely many solutions in integers. [ABSTRACT FROM AUTHOR]

This paper investigates the relations between modular graph forms, which are generalizations of the modular graph functions that were introduced in earlier papers motivated by the structure of the low energy expansion of genus-one Type II superstring amplitudes. These modular graph forms are multiple sums associated with decorated Feynman graphs on the world-sheet torus. The action of standard differential operators on these modular graph forms admits an algebraic representation on the decorations. First order differential operators are used to map general non-holomorphic modular graph functions to holomorphic modular forms. This map is used to provide proofs of the identities between modular graph functions for weight less than six conjectured in earlier work, by mapping these identities to relations between holomorphic modular forms which are proven by holomorphic methods. The map is further used to exhibit the structure of identities at arbitrary weight. [ABSTRACT FROM AUTHOR]

Text A set A of nonnegative integers is said to be complete if every sufficiently large natural number is the sum of distinct terms taken from A . In 1959, Birch confirmed a conjecture of Erdős by proving that the set { p n q m : n , m = 0 , 1 , … } is complete, where p and q are two coprime integers greater than 1. In this paper, we study extensively the set P ( S p 1 , … , p r ) of all vectors ( m 1 , … , m r ) with integer coordinates which can be represented as ∑ ( p 1 α 1 , … , p r α r ) , where p 1 , … , p r are integers greater than 1 and α 1 , … , α r are nonnegative integers. We find many regular parts in P ( S p 1 , … , p r ) , for example, discrete rectangles. To study the problem, we find a new matching principle. Video For a video summary of this paper, please visit https://youtu.be/qXUe1E4jlLQ . [ABSTRACT FROM AUTHOR]

Text Let k , l , a , b , c be positive integers such that gcd ⁡ ( k a , l b ) = 1 , min ⁡ { a , b , c } > 1 , a ≠ 3 , b ≠ 3 and 2 ∤ c . In this paper, we prove that there are at most four solutions in positive integers ( x , y , z ) to the equation k a x + l b y = c z and at most two solutions when 2 ∤ ( u ( l / k ) ) , where u ( m ) is the least positive integer t with m t ≡ 1 ( mod c ) . Video For a video summary of this paper, please visit https://youtu.be/Dt3Y7TDMxlg . [ABSTRACT FROM AUTHOR]

The N th linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large N th linear complexity. However, in this paper we show that for q -automatic sequences over F q the converse is not true. We prove that any (not ultimately periodic) q -automatic sequence over F q has N th linear complexity of order of magnitude N . For some famous sequences including the Thue–Morse and Rudin–Shapiro sequence we determine the exact values of their N th linear complexities. These are non-trivial examples of predictable sequences with N th linear complexity of largest possible order of magnitude. [ABSTRACT FROM AUTHOR]

Text In a series of papers C. Dartyge and A. Sárközy (partly with other coauthors) studied pseudorandom measures of subsets. In this paper we extend the theory of C. Dartyge and A. Sárközy to several dimensions. We introduce measure for multi-dimensional pseudorandom subsets, and study the connection between measures of different orders. Large families of multi-dimensional pseudorandom subsets are given by using the squares in F q . Video For a video summary of this paper, please visit https://youtu.be/7iH9x7nyyfQ . [ABSTRACT FROM AUTHOR]

In this note, we study properties of certain parabolas which were recently defined by Hutchinson and Zaharescu in their paper on parabolas infiltrating the Ford circles. We examine this connection between Ford circles and these parabolas. In particular, we characterize the intersection points of these parabolas in terms of tangent Ford circles. Further, we provide a finer analysis of area under the arcs of these parabolas with end points joining the centers of two neighboring Ford circles, than discussed in their paper. [ABSTRACT FROM AUTHOR]

Text In this paper we study the structure of a class of cubic tame towers having finite ramification locus. These towers were defined by Garcia, Stichtenoth and Thomas in 1997. We also prove some extensions of a well known result of H. Lenstra Jr. on the infiniteness of the ramification locus of these towers. Video For a video summary of this paper, please visit https://youtu.be/NYcajjrrVLI . [ABSTRACT FROM AUTHOR]

Text In this note, among other results, we find the optimal constants of the generalized Bohnenblust–Hille inequality for m -linear forms over R and with multiple exponents ( 1 , 2 , … , 2 ) , sometimes called mixed ( ℓ 1 , ℓ 2 ) -Littlewood inequality. We show that these optimal constants are precisely ( 2 ) m − 1 and this is somewhat surprising since a series of recent papers have shown that similar constants have a sublinear growth. This result answers a question raised by Albuquerque et al. in a paper published in 2014 in the Journal of Functional Analysis . Video For a video summary of this paper, please visit https://youtu.be/KnKtjbvsbW0 . [ABSTRACT FROM AUTHOR]

In 1975, J. Leitzel, M. Madan and C. Queen listed 7 function fields over finite fields (up to isomorphism) with positive genus and class number one. They claimed to prove that these were the only ones such. Recently, Claudio Stirpe found an 8th one! In this paper, we fix the argument in the former paper to show that this 8th example could have been found by their method and is the only one, so that the list is now complete. [ABSTRACT FROM AUTHOR]

Text In this paper we present and analyze a construction of irreducible polynomials over odd prime fields via the transforms which take any polynomial f ∈ F p [ x ] of positive degree n to ( x k ) n ⋅ f ( k ( x + x − 1 ) ) , for some specific values of the odd prime p and k ∈ F p . Video For a video summary of this paper, please visit http://youtu.be/Lmw5m_c-i8s . [ABSTRACT FROM AUTHOR]

Text In this paper, we have studied the third order variations on the Fibonacci universal code and we have displayed tables GH a ( 3 ) ( n ) we have defined for − 20 ≤ a ≤ − 2 and 1 ≤ n ≤ 100 . Also, we have compared with the third order variations on the Fibonacci universal code and the second order variations on the Fibonacci universal code [2] in terms of cryptography and we have found that the third order variations on the Fibonacci universal code are more advantageous than the second order variations on the Fibonacci universal code. Video For a video summary of this paper, please visit http://youtu.be/i2A4LZl9nNE . [ABSTRACT FROM AUTHOR]

Let q ∈ ( 1 , 2 ) and x ∈ [ 0 , 1 q − 1 ] . We say that a sequence ( ϵ i ) i = 1 ∞ ∈ { 0 , 1 } N is an expansion of x in base q (or a q -expansion) if x = ∑ i = 1 ∞ ϵ i q − i . Let B ℵ 0 denote the set of q for which there exists x with exactly ℵ 0 expansions in base q . In [5] it was shown that min ⁡ B ℵ 0 = 1 + 5 2 . In this paper we show that the smallest element of B ℵ 0 strictly greater than 1 + 5 2 is q ℵ 0 ≈ 1.64541 , the appropriate root of x 6 = x 4 + x 3 + 2 x 2 + x + 1 . This leads to a full dichotomy for the number of possible q -expansions for q ∈ ( 1 + 5 2 , q ℵ 0 ) . We also prove some general results regarding B ℵ 0 ∩ [ 1 + 5 2 , q f ] , where q f ≈ 1.75488 is the appropriate root of x 3 = 2 x 2 − x + 1 . Moreover, the techniques developed in this paper imply that if x ∈ [ 0 , 1 q − 1 ] has uncountably many q -expansions then the set of q -expansions for x has cardinality equal to that of the continuum, this proves that the continuum hypothesis holds when restricted to this specific case. [ABSTRACT FROM AUTHOR]

Text The aim of this paper is to present a method to calculate the rate of growth of divergent series. In the case of the harmonic series, we study an alternate of other classical methods presented by L. Comtet, S.M. Zeyman, W. DeTemple and S.-H. Wang. Moreover, the method presented in this paper is suitable for obtaining similar results on other divergent series, as we will show in the final part of our investigation. Video For a video summary of this paper, please visit http://youtu.be/SFR7zY4pvG4 . [ABSTRACT FROM AUTHOR]

This paper is a continuation of a recent paper (see [10] ), in which for a fixed Diophantine pair { a , b } with a < b , we gave an upper bound for minimal c such that { a , b , c , d } is an irregular Diophantine quadruple with b < c < d . Here, we provide three very good examples to support the theory developed in [10] . [ABSTRACT FROM AUTHOR]

Text: For any positive integer n, let n=q1 ... qs be the prime factorization of n with q1 ≥ ... ≥ qs > 1. A positive integer n is said to be ordinary if the smallest positive integer with exactly n divisors is p1q1 - 1 ... psqs - 1, where pk denotes the kth prime. Let x be the largest integer not exceeding x. In 2006, Brown proved that all square-free integers are ordinary and the set of all ordinary integers has asymptotic density one. In this paper, we prove that, if q s ≥ 9(logs)2, then n is ordinary. Furthermore, the set of such integers n has asymptotic density one. We also determine all ordinary integers which are not divisible by any fifth power of a prime. Video: For a video summary of this paper, please visit http://youtu.be/UeIMWjRFUnA. [ABSTRACT FROM AUTHOR]

Abstract: This paper proves a result concerning linear forms on the Sinnott module. This is perhaps of intrinsic interest, and it is needed in another paper of the same authors. We obtain a congruence which can be interpreted as a strengthening of a congruence of Anthony Hayward. [Copyright &y& Elsevier]

Abstract: Text: This paper concerns the function , where is the argument of the Riemann zeta-function along the critical line. The main result is that which holds for all . Video: For a video summary of this paper, please click here or visit http://youtu.be/FldP0idE0aI. [Copyright &y& Elsevier]

Abstract: Text: Let the prime factorization of n be with . A positive integer n is said to be ordinary if the smallest positive integer with exactly n divisors is , where denotes the kth prime. In this paper I prove that all integers of the form ql are ordinary, where l is a square-free positive integer and q is a prime. This confirms a conjecture of Yong-Gao Chen. Video: For a video summary of this paper, please click here or visit http://youtu.be/WTY4wr8L_U0. [Copyright &y& Elsevier]

Abstract: Text: The aim of this paper is to establish some sharp bounds for the Glaisher–Kinkelin and Bendersky–Adamchik constants. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=LmxI_0AW1vk. [Copyright &y& Elsevier]

Abstract: Text: For any given two positive integers and , and any set A of nonnegative integers, let denote the number of solutions of the equation with . In this paper, we determine all pairs of positive integers for which there exists a set such that for all . We also pose several problems for further research. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EnezEsJl0OY. [Copyright &y& Elsevier]

Abstract: In the present paper we investigate the sum-of-digits function for canonical number systems. We are interested in its re-partition in arithmetic progressions and similar results for sum sets. The original problem goes back to Gelfond, who proved the independence of the distribution of the digits and their sum of digits. The present paper extends results by Thuswaldner and Mauduit and Sárközy to this kind of numeration systems. [Copyright &y& Elsevier]

Abstract: In a recent paper (Buium et al., 2011 ), Buium et al. proved that f is a locally analytic function from the p-adic integers, to itself if and only if it is written as a restricted power series over with finitely many iterates of the Fermat quotient operator. In this paper we establish the function field analogs of their result. In addition, we deduce Wagnerʼs result, which is the function field analog of Mahlerʼs result for continuous functions on . [Copyright &y& Elsevier]

Abstract: Text: Let be the set of all positive integers pqr, where p, q, r are primes such that at least two of them are not equal. Denote by the largest prime factor of n. For , define . In 2006, Wushi Goldring proved that for any , there exists an i such that , where and (). If , then m is called a parent of n. Let be the set of all positive integers of . In this paper, we study the function w extensively. For example, one of our results is that there exist infinitely many which have at least parents in . Several open problems are posed. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=8FWvR8_KoHA. [Copyright &y& Elsevier]

Abstract: Text: In this paper we investigate higher order dimensional Dedekind–Rademacher sums given by the expression where is a positive integer, are positive integers prime to and are non-negative integers. We study arithmetical properties of these sums. For instance, we prove that these sums are rational numbers and we explicit good bounds for their denominators. A reciprocity law is given generalizing a theorem of Rademacher for the classical Dedekind sums and a theorem of Zagier for higher dimensional Dedekind–Rademacher sums. On the other hand, our reciprocity results can be viewed as complements to the Beck reciprocity theorem. Taking , we derive the reciprocity and rationality theorems of Zagier. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=J1_5H28fgAg. [Copyright &y& Elsevier]

Abstract Let F (z) be a Hecke–Maass form for S L (3 , Z) (or S L (2 , Z)), A F (n , 1) (or A F (n)) be its Fourier coefficients, and L (s , F) the Godement–Jacquet L -function attached to F. And let μ F (n) be the coefficients of the reciprocal of L (s , F) in the absolutely convergent region ℜ s > 1. In this paper we prove that μ F (n) is strongly orthogonal to 1-step nilsequence. As applications, we prove that the Möbius function μ (n) is strongly orthogonal to A F (n , 1) e (n α) and A F (n) 2 e (n α). More precisely, we establish ∑ n ≤ N μ (n) A F (n , 1) e (n α) ≪ c N exp ⁡ (− c log ⁡ N) , and ∑ n ≤ N μ (n) A F (n) 2 e (n α) ≪ A N (log ⁡ N) − A. Moreover, we also show that μ F (n) is asymptotically orthogonal to any A C 0 (d) function, and all MMA and MMLS flows. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we seek to evaluate the most general form of moments of L (1 , χ) twisted by character value χ (c) as χ ranges over non-principal Dirichlet characters mod f. And we use this result to model the joint distribution for χ (c) and L (1 , χ). [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss the variation of the numbers of the isomorphic classes of stable lattices when the weight and the level vary in a Hida deformation by using the Kubota–Leopoldt p -adic L -function. Then in Corollary 1.7 , we give a sufficient condition for the numbers of the isomorphic classes of stable lattices in Hida deformation to be infinite. [ABSTRACT FROM AUTHOR]

Abstract: Text: This paper proposes new explicit formulas for the doubling and addition steps in Miller''s algorithm to compute the Tate pairing on elliptic curves in Weierstrass and in Edwards form. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in addition and doubling. The Tate pairing on Edwards curves can be computed by using these functions in Miller''s algorithm. Computing the sum of two points or the double of a point and the coefficients of the corresponding functions is faster with our formulas than with all previously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also improve the formulas for Tate pairing computation on Weierstrass curves in Jacobian coordinates. Finally, we present several examples of pairing-friendly Edwards curves. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=nideQo-K9ME/. [Copyright &y& Elsevier]

Abstract: Text: In this paper, we study the mean value of the index of composition of an integral ideal, and get an asymptotic formula for it. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=oAdWsFz3gKM. [Copyright &y& Elsevier]

Abstract: Marvin Knopp showed that entire automorphic integrals with rational period functions satisfy a Hecke correspondence theorem, provided the rational period functions have poles only at 0 or ∞. For other automorphic integrals the corresponding Dirichlet series has a functional equation with a remainder term that arises from the nonzero poles of the rational period function. In this paper we prove a Hecke correspondence theorem for a class of automorphic integrals with rational period functions on the Hecke groups. We restrict our attention to automorphic integrals of weight that is twice an odd integer and to rational period functions that satisfy a symmetry property we call “Hecke-symmetry.” Each remainder term satisfies two relations (the second of which is new in this paper) corresponding to the two relations for the rational period function. [Copyright &y& Elsevier]

Abstract: Text: The aim of this paper is to provide some estimates about the Somos'' quadratic recurrence constant, using its relation with the generalized Euler constant. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=3QjmHit3mC4. [Copyright &y& Elsevier]

Abstract: Text: Let G be a finite cyclic group. Every sequence S over G can be written in the form where and , and the index of S is defined to be the minimum of over all possible such that . The problem regarding the index of sequences has been studied in a series of papers, and a main focus is to determine sequences of index 1. In the present paper, we show that if G is a cyclic of prime power order such that , then every minimal zero-sum sequence of length 4 has index 1. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=BC7josX_xVs. [Copyright &y& Elsevier]

Abstract: In this paper we continue our study, begun in G. Harman and A.V. Kumchev (2006) , of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious oversight in our first paper, but make further progress on the exponential sums estimates needed, together with an embellishment of the previous sieve technique employed. This leads to an improvement in our bounds for the maximal size of the exceptional sets. [Copyright &y& Elsevier]

Abstract: The classical Dedekind sums were found in transformation formulae of η-functions. It is known that these sums have some properties, especially a reciprocity law Sczech (1984) expanded his theory for analogies of Dedekind sums and in imaginary quadratic fields. And in my paper (Okada, 1989) we studied analogies of Dedekind sums in function fields. In this paper we correct some mistakes, change some notations and rewrite some lines in the previous paper. And also in this paper we consider another analogies of Dedekind sums and a reciprocity law for these Dedekind sums. [Copyright &y& Elsevier]

Abstract: This paper is the first of several papers in which we prove, for the case where the fields of coefficients are of characteristic zero, four open problems posed in the work of Melvyn Nathanson (2003) concerning the solutions of a functional equation arising from multiplication of quantum integers . In this paper, we prove one of the problems. The next papers, namely by Lan Nguyen, contain the solutions to the other 3 problems. [Copyright &y& Elsevier]

Abstract: This paper deals with two problems arising in the study of Drinfeld quasi-modular forms. The first problem is to find the maximal order of vanishing at infinity of a non-zero Drinfeld quasi-modular form and leads to the notion of “extremal” quasi-modular form (highest possible order of vanishing for fixed weight and depth). The second problem is determining differential properties of extremal forms, leading to the notion of “differentially extremal” form. From our investigations, we will obtain an upper bound for the order of vanishing at infinity of non-zero Drinfeld quasi-modular forms of small depths. The paper ends with a collection of tools used in the previous parts. The notion of “extremal” form is similar to one introduced by Kaneko and Koike in [M. Kaneko, M. Koike, On extremal quasimodular forms, Kyushu J. Math. 60 (2006) 457–470]. [Copyright &y& Elsevier]

Abstract: Let be an integer and be the sum of digits in base g of the positive integer m. In this paper, we study the positive integers n such that and satisfy certain relations for a fixed, or arbitrary positive integer k. In the first part of the paper, we prove that if n is not a power of g, then there exists a nontrivial multiple of n say kn such that . In the second part of the paper, we show that for any the set of the integers n satisfying for all is of asymptotic density 0. This gives an affirmative answer to a question of W.M. Schmidt. [Copyright &y& Elsevier]

Abstract: For a set A of nonnegative integers the representation functions , are defined as the number of solutions of the equation with , , respectively. Let and let denote the number of ones in the binary representation of a. Let be the set of all nonnegative integers a with even and be the set of all nonnegative integers a with odd . In this paper we show that (a) if for all , then for all except for or ; (b) if for all , then for all . Several problems are posed in this paper. [Copyright &y& Elsevier]

Abstract: Text: The problem of determining the maximum size of coefficients of cyclotomic polynomials has been studied extensively. Let be the maximum absolute value of a coefficient of , the nth cyclotomic polynomial. This paper further investigates a variation of introduced by Pomerance and Ryan. We consider the function , the maximum absolute value of a coefficient of any divisor of . In the first part of this paper we give an analogue of a well-known result for proved independently by Felsch and Schmidt, and by Justin, and give an upper bound for for a general n. In the second part of the paper we resolve a question of Pomerance and Ryan giving an explicit formula for . We then give upper and lower bounds for where are primes. Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=KLAdOHS_Ekw. [Copyright &y& Elsevier]

Abstract: Text: It is a theorem of Kaplansky that a prime is representable by both or none of and , whereas a prime is representable by exactly one of these binary quadratic forms. In this paper five similar theorems are proved. As an example, one theorem states that a prime is representable by both or none of and , whereas a prime is representable by exactly one of these forms. A heuristic argument is given why there are no other results of the same kind. This argument relies on the (plausible) conjecture that there are exactly 485 negative discriminants Δ such that the class group has exponent 4. Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=l_yRq0oqKx4. [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: Text: Finding a function which generates a sequence via iteration whose values at one or many points in its domain satisfy certain prescribed properties, i.e., finding a function such that the Picard orbit(s) of one or many points in its domain which possess some given properties, is an interesting problem. Given any positive integer n greater than one, we construct in this paper families of functions on the natural numbers such that the sequence of the iterations of each of these functions at any positive integer s contains infinitely many perfect n-powers. In terms of Picard sequences, this amounts to constructing a function whose Picard orbit at every point in its domain contains infinitely many perfect n-powers. Video: For a video summary of this paper, please visit http://www.youtube.com/watch?v=wJqaXyB2pdo. [Copyright &y& Elsevier]

Abstract: In this paper we deal with monogenic and k-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors–Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation between k-hypermonogenic automorphic forms and Maaß wave forms. In particular, we show how one can construct from any arbitrary non-vanishing monogenic automorphic form a Clifford algebra valued Maaß wave form. In the second part of the paper we compute the Fourier expansion of the k-hypermonogenic Eisenstein series which provide us with the simplest non-vanishing examples of k-hypermonogenic automorphic forms. [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 give an explicit from formula for the local density number of representing a two by two 2-integral matrix T by a quadratic 2-integral lattice L over Z2. The non-dyadic case was dealt in a previous paper. The special case when L is a (maximal) lattice in the space of trace zero elements in a quaternion algebra over Q2 yields a clean and interesting formula, which matches up perfectly with the non-dyadic case in terms of the Gross–Keating invariants. This work is used to compare the central derivative of a genus two Eisenstein series with certain generating function of arithmetic 0-cycles on certain Shimura curve, in a joint work with Kudla and Rapoport. [Copyright &y& Elsevier]