Heilbronn characters

A variant of Heilbronn characters

In this paper, we introduce arithmetic Heilbronn characters that generalize the notion of the classical Heilbronn characters, and discuss several properties of these characters. This formalism has several arithmetic applications. For instance, we obtain the holomorphy of suitable quotients of L-functions attached to elliptic curves, which is predicted by the Birch–Swinnerton–Dyer conjecture, an...

The Inverse Erdös-Heilbronn Problem

The famous Erdős-Heilbronn conjecture (first proved by Dias da Silva and Hamidoune in 1994) asserts that if A is a subset of Z/pZ, the cyclic group of the integers modulo a prime p, then |A +̂ A| > min{2 |A| − 3, p}. The bound is sharp, as is shown by choosing A to be an arithmetic progression. A natural inverse result was proven by Karolyi in 2005: if A ⊂ Z/pZ contains at least 5 elements and |...

The Newton Polygon of Gauss-Heilbronn Sums

Let p be a prime, Zp the ring of p-adic integers, Qp the field of p-adic numbers, and Qp the algebraic closure of Qp. Let Fp be the algebraic closure of the finite field Fp, and ω : Fp → Qp the Techmüller lifting. Let q = pa, χ a multiplicative character of Fq into Q, and ψ a character of Zp[μq−1] into Q of exact order pn > 1, where μr is the set of r-th roots of unity in Qp. For simplicity, we...

Hans Arnold Heilbronn 1908-1975 Hans

Error Estimates for the Davenport–heilbronn Theorems

We obtain the first known power-saving remainder terms for the theorems of Davenport and Heilbronn on the density of discriminants of cubic fields and the mean number of 3-torsion elements in the class groups of quadratic fields. In addition, we prove analogous error terms for the density of discriminants of quartic fields and the mean number of 2-torsion elements in the class groups of cubic f...