On the existence of fields governing the 2-invariants of the classgroup of 𝑄(√𝑑𝑝) as 𝑝 varies

Cohn, H.; Lagarias, J. C.

doi:10.1090/S0025-5718-1983-0717716-8

On the existence of fields governing the $2$-invariants of the classgroup of $\textbf {Q}(\sqrt {dp})$ as $p$ varies
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Math. Comp. 41 (1983), 711-730 Request permission

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 ${2^j}|h(dp)$ 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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