Plane algebraic curves with Singer automorphisms

Non-singular plane algebraic curves over with a Singer group of PGL(3,q) in their automorphism group are classified. Apart from three distinguished points, the set of -rational points of such curves can be partitioned into 2−(q²+q+1,q+1,1) designs each isomorphic to the finite projective plane .     

On a theorem of Chowla

Let be a finite field with q=p^felements, where p is a prime number and f is a positive integer. For a nonprincipal multiplicative character χ and a nontrivial additive character ψ on , it is well known that Gauss sum has absolute value . In this paper, we investigate when is a root of unity.     

Quadratic addition rules for quantum integers

For every positive integer n, the quantum integer [n]_q is the polynomial [n]_q=1+q+q²+?+q^n-1. A quadratic addition rule for quantum integers consists of sequences of polynomials , , and such that for all m and n. This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials that satisfy the associated functional equation .     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

The distribution of prime ideals in a real quadratic field with units having a given index in the residue class field

Let k be a real quadratic number field and the ring of integers and the group of units in k. Denote by the subgroup represented by elements of E of for a prime ideal in k. We show that for a given positive rational integer a, the set of prime numbers p for which the residual index of for lying above p is equal to a has a natural density c under the Generalized Riemann Hypothesis. Moreover, we give the explicit formula of c and conditions to c=0.     

The ring of quasimodular forms for a cocompact group

We describe the additive structure of the graded ring of quasimodular forms over any discrete and cocompact group Γ⊂PSL(2,R). We show that this ring is never finitely generated. We calculate the exact number of new generators in each weight k. This number is constant for k sufficiently large and equals where I and are the ideals of modular forms and quasimodular forms, respectively, of positive weight. We show that is contained in some finitely generated ring of meromorphic quasimodular forms with i.e., the same order of growth as      

The distribution of patterns of inverses modulo a prime

We investigate the distribution of the numbers x∈[1,p] for which all lie in a subset of the set of multiplicative inverses modulo a prime p. Here the a_i are integers coprime to p and the numbers are distinct .     

The density of 0's in recurrence double sequences

Let be a double sequence over a finite field satisfying a linear recurrence with constant coefficients, with at most finitely many nonzero elements on each row. Given a nonzero element g of , we show how to obtain an explicit formula for the number of g's in the first qⁿ rows of A. We also characterize the cases when the density of 0's is 1.     

An extension to overpartitions of the Rogers-Ramanujan identities for even moduli

We study a class of well-poised basic hypergeometric series , interpreting these series as generating functions for overpartitions defined by multiplicity conditions on the number of parts. We also show how to interpret the as generating functions for overpartitions whose successive ranks are bounded, for overpartitions that are invariant under a certain class of conjugations, and for special restricted lattice paths. We highlight the cases (a,q)→(1/q,q), (1/q,q²), and (0,q), where some of the functions become infinite products. The latter case corresponds to Bressoud's family of Rogers-Ramanujan identities for even moduli.     

Quartic residues and binary quadratic forms

Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let ε_d be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that ε_d is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {u_n(a,b)} is the Lucas sequence defined by and .     

Générateurs de l'anneau des entiers d'une extension cyclotomique

Let p be an odd prime and q=pm, where m is a positive integer. Let ζq be a qth primitive root of 1 and Oq be the ring of integers in Q(ζq). 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 α∈Oq is a generator of Oq (in other words Z[α]=Oq) either α is equals to a conjugate of an integer translate of ζq or is an odd integer. In this paper we show that we can remove the hypothesis over . In other words we show that if α∈Oq is a generator of Oq then either α is a conjugate of an integer translate of ζq or is an odd integer.     

Arithmetical properties of wendt's determinant

Wendt's determinant of order n is the circulant determinant W_n whose (i,j)-th entry is the binomial coefficient , for 1?i,j?n, where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then and . If q is another prime, distinct from p, and h any positive integer, then . Furthermore, if p is odd, then . In particular, if p?5, then . Also, if m and n are relatively prime positive integers, then W_mW_n divides W_mn.     

On real quadratic function fields

Let q be a power of an odd prime number p and K:=Fq(T) be the rational function field with a fixed indeterminate T. For P a prime of K, let be the maximal real subfield of the Pth-cyclotomic function field and its ring of integers. We prove that there exists infinitely many primes P of even degree such that the cardinal of the ideal class group is divisible by q. We prove also an analogous result for imaginary extensions.     

Volumes and diophantine inequalities associated with decomposable forms

For homogeneous decomposable forms in n variables with real coefficients, we consider the associated volume of all real solutions to the inequality . We relate this to the number of integral solutions to the Diophantine inequality in the case where F has rational coefficients. We find quantities which bound the volume and which yield good upper bounds on the number of solutions to the Diophantine inequality in the rational case.