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The 16-rank of \(\mathbb{Q}(\sqrt{-p})\). (English) Zbl 1451.11120
The author obtains a density result for the \(16\)-rank of the imaginary quadratic fields \({\mathbb Q}(\sqrt{-p})\) when \(p\) runs through all the prime numbers. The study of the \(2\)-part of the class group \(Cl(K)\) of a quadratic field \(K\) is often done via so-called governing fields, a concept first defined by H. Cohn and J. C. Lagarias [in: Topics in classical number theory, Colloq. Budapest 1981, Vol. I, Colloq. Math. Soc. János Bolyai 34, 257–280 (1984; Zbl 0541.12002)] and which is defined in the following way. Let \(k\) be an integer \(\geq 1\) and let \(d\) be an integer \(\not \equiv 2\pmod 4\). For a finite abelian group \(A\), define the \(2^k\)-rank of \(A\) to be the dimension of \(2^{k-1}A/2^k\, A\) over \({\mathbb Z}/2{\mathbb Z}\). Then a governing field \(M_{d,k}\) is a normal field extension of \({\mathbb Q}\) such that the \(2^k\)-rank of \(\mathrm{Cl}({\mathbb Q}(\sqrt{dp})\) is determined by the splitting of \(p\) in \(M_{d,k}\). Not knowing that \(M_{d,4}\) exists has been a handicap for proving density results for the \(16\)-rank. In previous papers, the author and Milovic managed to obtain results under certain hypotheses. In the paper under review, the author proves that on the average \(1/{16}\) of the imaginary quadratic fields \(\mathbb Q(\sqrt{-p})\) verify the divisibility \(16\vert h(-p)\). Koymans obtains his result unconditionally by using a criterion of P. A. Leonard and K. S. Williams [Can. Math. Bull. 25, 200–206 (1982; Zbl 0431.12007)] which is a product of quartic residue symbol and a quadratic one. This prevents Koymans from using the method of J. Friedlander and H. Iwaniec [Ann. Math. (2) 148, No. 3, 945–1040 (1998; Zbl 0926.11068)] described in the paper “The polynomial \(X^2+Y^4\) captures its primes” and forced him to use ad hoc bright arguments to achieve his goal.

11R29 Class numbers, class groups, discriminants
11N45 Asymptotic results on counting functions for algebraic and topological structures
11R45 Density theorems
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