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.     

Rank determines semi-stable conductor

Suppose that E₁ and E₂ are elliptic curves over the rational field, , such that for all quadratic fields . We prove that their conductors N(E₁), and N(E₂) are equal up to squares. If for all quadratic fields , then the same conclusion holds, provided the 2-parts of their Tate-Shafarevich groups are finite.     

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 .     

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 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.     

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 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 .     

p-Catalan numbers and squarefree binomial coefficients

In this paper, we consider the generalized Catalan numbers , which we call s-Catalan numbers. For p prime, we find all positive integers n such that p^q divides F(p^q,n), and also determine all distinct residues of , q?1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers by 4. In the second part of the paper we prove that if pq?99999, then is not squarefree for n?τ₁(p^q) sufficiently large (τ₁(p^q) computable). Moreover, using the results of the first part, we find n<τ₁(p^q) (in base p), for which may be squarefree. As consequences, we obtain that is squarefree only for n=1,3,45, and is squarefree only for n=1,4,10.     

Abelian subgroups of any order in class groups of global function fields

Let be a finite field with q elements, and T a transcendental element over . In this paper, we construct infinitely many real function fields of any fixed degree over 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 , there are infinitely many imaginary fields of fixed degree m such that the class group contains a subgroup isomorphic to .     

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.     

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      

A construction of covers of arithmetic schemes

Let X be a regular arithmetic scheme, i.e. a regular integral separated scheme flat and of finite type over SpecZ. Assume that for all closed irreducible subschemes C⊆X of dimension 1 with normalisation there are given open normal subgroups NC of , which fulfil the following compatibility condition: For all the pre-images of NC₁ and NC₂ in coincide. If the indices of the NC are bounded, then these data uniquely determine an open normal subgroup of π₁(X), whose pre-image in is NC for all C.     

Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial

Using the theory of elliptic curves, we show that the class number h(−p) of the field appears in the count of certain factors of the Legendre polynomials , where p is a prime >3 and m has the form (p−e)/k, with k=2,3 or 4 and . As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y²+αxy+y=x³ and find an elementary expression for the supersingular polynomial ss_p(x) whose roots are the supersingular j-invariants of elliptic curves in characteristic p. As a corollary we show that the class number h(−p) also shows up in the factorization of certain Jacobi polynomials.     

Uniform bounds on the number of rational points of a family of curves of genus 2

We exhibit a genus-2 curve defined over which admits two independent morphisms to a rank-1 elliptic curve defined over . We describe completely the set of -rational points of the curve and obtain a uniform bound on the number of -rational points of a rational specialization of the curve for a certain (possibly infinite) set of values . Furthermore, for this set of values we describe completely the set of -rational points of the curve . Finally, we show how these results can be strengthened assuming a height conjecture of Lang.