A Unit Norm Conjecture for Some Real Quadratic Number Fields: A Preliminary Heuristic Investigation

Abstract

In this paper we make a conjecture about the norm of the fundamental unit, N(e), of some real quadratic number fields that have the form k = Q(√(p1.p2) where p1 and p2 are distinct primes such that pi = 2 or  pi ≡ 1 mod 4, i = 1, 2. Our conjecture involves the case where the Kronecker symbol (p1/p2) = 1 and the biquadratic residue symbols (p1/p2)4 = (p2/p1)4 = 1, and is based upon Stevenhagen’s conjecture that if k = Q(√(p1.p2) is any real quadratic number field as above, then P(N(e) = -1)) = 2/3, i.e., the probability density that N(e) = -1 is 2/3. Given Stevenhagen’s conjecture and some theoretical assumptions about the probability density of the Kronecker symbols and biquadratic residue symbols, we establish that if k is as above with (p1/p2) = (p1/p2)4 = (p2/p1)4 = 1, then P(N(e) = -1)) = 1/3, and we support our conjecture with some preliminary heuristic data.