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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.

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