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Sums and differences of quartic norms

Part of: Multiplicative number theory

Published online by Cambridge University Press:  26 February 2010

Henryk Iwaniec  and
Jacek Pomykala
Henryk Iwaniec
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA.
Jacek Pomykala
Affiliation:
Institute of Mathematics, Warsaw University, 00-950 Warsaw, Poland.
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Extract

Let K be a number field of degree k > 1. We would like to know if a positive integer N can be represented as the sum, or the difference, of two norms of integral ideals of K. Suppose K/ℚ is abelian of conductor Δ. Then from the class field theory (Artin's reciprocity law) the norms are fully characterized by the residue classes modulo Δ. Precisely, a prime number p ∤ Δ (unramified in K) is a norm (splits completely in K), if, and only if,

where k is a subgroup of (ℤ/Δℤ)* of index k. Accordingly we may ask N to be represented as the sum

or the difference

of positive integers a, b each of which splits completely in K. For N to be represented in these ways the following congruences

must be solvable in α β є k, respectively. Moreover the condition

must hold. Presumably the above local conditions are sufficient for (−) to have infinitely many solutions and for (+) to have arbitrarily many solutions, provided N is sufficiently large in the latter case.

MSC classification

Secondary: 11N36: Applications of sieve methods
Type
Research Article
Information
Mathematika , Volume 40 , Issue 2 , December 1993 , pp. 233 - 245
Copyright
Copyright © University College London 1993

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