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On the existence of fields governing the $ 2$-invariants of the classgroup of $ {\bf Q}(\sqrt{dp})$ as $ p$ varies

Authors: H. Cohn and J. C. Lagarias
Journal: Math. Comp. 41 (1983), 711-730
MSC: Primary 11R11; Secondary 11R29, 11R45
MathSciNet review: 717716
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Abstract: This paper formulates general conjectures relating the structure of the 2-classgroup $ {C_2}(dp)$ associated to $ {\mathbf{Q}}(\sqrt {dp} )$ to the splitting of the ideal (p) in certain algebraic number fields. Here $ d\,\nequiv\,2$ $ \pmod 4$ is a fixed integer and p varies over primes. The conjectures assert that there exists an algebraic number field $ {\Omega _j}(d)$ such that the Artin symbol $ [({\Omega _j}(d)/{\mathbf{Q}})/(p)]$ determines the first j 2-invariants of the group $ {C_2}(dp)$, i.e. it determines $ {C_2}(dp)/{C_2}{(dp)^{{2^j}}}$. These conjectures imply that the set of primes p for which $ {C_2}(dp)$ has a given set of 2-invariants has a natural density which is a rational number. Existing results prove the conjectures whenever $ j = 1$ or 2 and also for an infinite set of d with $ j = 3$. The smallest open case is $ j = 3$, $ d = - 21$. This paper presents evidence concerning these conjectures for $ d = - 4$, 8 and $ - 21$. Numerical evidence is given that $ {\Omega _3}( - 21)$ exists, and that natural densities which are rational numbers exist for the sets of primes with $ {2^j}\vert h(dp)$ for $ d = - 4$ and 8, for $ 1 \leqslant j \leqslant 7$. A search for the hypothetical field $ {\Omega _4}( - 4)$ ruled out the simplest candidate fields: $ {\Omega _4}( - 4)$ is not a normal extension of Q of degree 16 ramifying only at (2).

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Article copyright: © Copyright 1983 American Mathematical Society