Efficient computation of root numbers and class numbers of parametrized families of real abelian number fields

Louboutin, Stéphane

doi:10.1090/S0025-5718-06-01888-6

Efficient computation of root numbers and class numbers of parametrized families of real abelian number fields
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by Stéphane R. Louboutin PDF

Math. Comp. 76 (2007), 455-473 Request permission

Abstract:

Let $\{K_m\}$ be a parametrized family of simplest real cyclic cubic, quartic, quintic or sextic number fields of known regulators, e.g., the so-called simplest cubic and quartic fields associated with the polynomials $P_m(x) =x^3 -mx^2-(m+3)x+1$ and $P_m(x) =x^4 -mx^3-6x^2+mx+1$. We give explicit formulas for powers of the Gaussian sums attached to the characters associated with these simplest number fields. We deduce a method for computing the exact values of these Gaussian sums. These values are then used to efficiently compute class numbers of simplest fields. Finally, such class number computations yield many examples of real cyclotomic fields $\textbf {Q}(\zeta _p)^+$ of prime conductors $p\ge 3$ and class numbers $h_p^+$ greater than or equal to $p$. However, in accordance with Vandiver’s conjecture, we found no example of $p$ for which $p$ divides $h_p^+$.

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