On the Spiegelungssatz for the 4-rank

Abstract

Let d be a nonsquare positive integer. We give the value of the natural probability that the narrow ideal class groups of the quadratic fields ‫(ޑ‬

Spins of prime ideals and the negative Pell equation

“…The first significant achievement for families with arbitrary discriminants was made by Fouvry and Klüners [FK07], who translated Rédei’s theory on -ranks of class groups to sums of characters conducive to analytic techniques and then successfully dealt with these sums, basing some of their work on the techniques developed by Heath-Brown in [Hea93, Hea94]. Fouvry and Klüners subsequently developed their methods in various settings [FK10a, FK10b, FK10c, FK11], most notably obtaining impressive upper and lower bounds for the solvability of the negative Pell equation for general squarefree integers . When specialized to the one-prime-parameter family with prime, their results are as strong as the bounds in (2.1), so Theorem 4 can be viewed as the next natural step in the line of work initiated by Fouvry and Klüners.…”

“…Just as in [11], the implied constant above does not depend on β because Conjecture 1, and so also Corollary 7, encompasses all incomplete character sums of length ≪ x 1 8 , regardless of the endpoints of the interval being summed over.…”

“…Hence, we see that the inner sum in (4.1) can be rewritten as at most 8 sums, each of which runs over an arithmetic progression of modulus k in a single segment of length ≪ x 1/8 . As q = N(q) is squarefree, • q is the real primitive Dirichlet character of modulus q, and hence we have at most 8 incomplete character sums of length ≪ x 1 8 and modulus q ≪ x. When the modulus q of the Dirichlet character divides the modulus k of the arithmetic progression, one can not expect to get cancellation.…”

“…8) where α is any element of O M such that α ≡ ρ mod F ; this is well-defined by Lemma 3.2.Suppose α ∈ O M is coprime to F , and suppose β ∈ O × M . Again, it is clear that ψ M ′ (αβ 2 mod F ) = ψ M ′ (α mod F ).…”