The infinitude of $\mathbb {Q}(\sqrt {-p})$ with class number divisible by 16

Abstract

Abstract. The density of primes p such that the class number h of Q( √ −p) is divisible by 2 k is conjectured to be 2 −k for all positive integers k. The conjecture is true for 1 ≤ k ≤ 3 but still open for k ≥ 4. For primes p of the form p = a 2 + c 4 with c even, we describe the 8-Hilbert class field of Q( √ −p) in terms of a and c. We then adapt a theorem of Friedlander and Iwaniec to show that there are infinitely many primes p for which h is divisible by 16, and also infinitely many primes p for which h is divisible by 8 but not by 16.

Spins of prime ideals and the negative Pell equation

“…Nevertheless, we are able to show, in Theorem 2, that the density of primes such that exists and is equal to . It is proved unconditionally in [Mil17a] that there are infinitely many primes such that , but that result implies nothing about the density as in Theorem 1.…”

On the 8-rank of narrow class groups of ℚ(−4pq), ℚ(−8pq), and ℚ(8pq)

“…Then, putting together the formulas above, we deduce that , and let f (p, q; r, s, e) be defined as in (23). If e = (0, 0), then…”

“…Finally, given a vector e = (e 1 , e 2 ) ∈ F 2 2 and p, q, r, and s as above, define (23) f (p, q) = f (p, q; r, s, e) := c(p, q; r, s)χ p (q) e1 ε(p, q) e2 .…”

“…See Section A for more details. We also note that the 16-and higher 2-power-ranks appear to be much harder to study from an analytic perspective, and there are only a few results in this direction [23,24,19,20,29] The proof of the Theorem 1 exploits a new type of lower bound for the 8-rank. In [7], Fouvry and Klüners define a quantity λ D conducive to analytic techniques which gives a good upper bound for the 8-rank of the narrow class group Cl(D) for a special class of positive discriminants D. This upper bound λ D actually coincides with rk 8 Cl(D) when rk 4 Cl(D) = 1.…”