Revistes Catalanes amb Accés Obert (RACO)

Regulators and total positivity

E. Friedman


The classical regulator $R$ of a number field $K$ is given [F1] by a rapidly convergent series of the form
\frac{R}{w}= \sum_{m=1}^\infty a_m g( m^2/ |D| ),
where $w$ is the number of roots of unity in $K$, $D$ is the discriminant of $K$, $a_m$ counts certain integral ideals in $K$ of absolute norm $m$, and $g\colon(0,\infty)\mapsto \mathbb{R}$ is defined as
g(x):=\frac{1}{2^{r_1}4\pi i}\int_{2-i\infty}^{2+i\infty} (4^{r_2}
\pi^{[K:\mathbb{Q}]}x)^{-s/2}(2s-1)\Gamma\bigl({\textstyle{\frac{s}{2}}}\bigr)^{r_1} \Gamma(s)^{r_2}\,ds,
$r_1$ and $r_2$ being, respectively, the number of real and complex places of $K$. If the unit group of $K$ is infinite, it is known that $g(x)$ tends to $-\infty$ as $x\to0^+$, and that $g(x)$ is positive and vanishes exponentially fast for large $x$. Using classical results from the theory of total positivity we
prove that $g$ has the simplest possible behavior compatible with these asymptotic data. Namely, $g(x)$ has a unique zero in $(0,\infty)$, and the same holds for each derivative of $g$. This leads to a new lower bound for the regulator
R > w \,g(1/|D|),
which is useful for certain ranges of $D$.

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