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.     

The twelfth moment of central values of Hecke series

Let H_j(s) be the Hecke L-function attached to the Maass wave form for the jth eigenvalue of the hyperbolic Laplacian acting in the Hilbert space of automorphic functions for the full modular group. The following mean value estimate for the central values is proved:

     

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 .     

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      

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 .     

Realizable classes of tetrahedral extensions

Let k be a number field and O_k its ring of integers. Let Γ be the alternating group A₄. Let be a maximal O_k-order in k[Γ] containing O_k[Γ] and its class group. We denote by the set of realizable classes, that is the set of classes such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Γ, for which the class of is equal to c, where O_N is the ring of integers of N. In this article we determine and we prove that it is a subgroup of provided that k and the 3rd cyclotomic field of are linearly disjoint, and the class number of k is odd.     

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 .     

2-Groupe des classes positives d'un corps de nombres et noyau sauvage de la K-théorie

We compute the 2-rank of the wild kernel WK₂(F) of a number field by constructing a 2-class group ad hoc. The main result generalizes in the more intricate case the canonical isomorphism established for odd primes under the assumption in a previous article (cf. (Acta. Arith. 67 (1994) 335; Math. Z. 238 (2001) 335)). It involves a criterium of triviality for the 2-part of the wild kernel of Galois number fields and, in the particular case of quadratic fields, it leads to a logarithmic interpretation of the diophantine conditions obtained by other authors.     

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.     

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.     

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

     

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.