TITLE: Spherical Harmonics - a bunch of questions about them
QUESTION [5 upvotes]: Hi there,
Please tell me if I should divide these into individual questions next time.
Short intro:
Spherical Harmonics are a nice collection of functions. They are orthogonal and allow you to take a function $f$ and break it into a linear combination of spherical harmonic functions $Y_n^m$. This is very similar to how the Fourier Transform allows you to break a function into a linear combination, which is the Fourier Series.
Questions about Spherical Harmonics:

What kind of functions can be broken down using this linear combination?
I read in Wikipedia here about having the Laplacian of $f$ equal to zero. How is that connected to this linear combination (or is that an unrelated and has more to do with physics)?
For the following linear combination of $f$: $f=\displaystyle\sum_n^\infty\displaystyle\sum_{m=-n}^n\alpha_{nm}Y_{nm}(\theta,\phi)$ it seems as though $f$ takes only two parameters $\theta$ and $\phi$ but that doesn't make a lot of sense because (I think) there should be a radial parameter $r$ somewhere. Where am I wrong?

Thank you very much, help saves lives :)
Ofer

REPLY [18 votes]: If you are only interested in expanding polynomials then you can forgo Hilbert spaces and get an exact expansion into a finite linear combination.

 Spherically-invariant case  Let $f$ be a polynomial in $x_1,\ldots, x_n$ (the coefficients may be real, complex, or even a field of characteristic zero). If $f$ is invariant under isometries of the $n$-dimensional space, i.e. $f(Ax)=f(x)$ for any $n\times n$ orthogonal matrix $A,$ then $f=g(r^2),$ where $g$ is a polynomial in one variable of degree $\deg(f)/2$ and
$$r^2=x_1^2+x_2^2+\ldots+x_n^2.$$
This is intuitively clear: a rotationally invariant function depends only on the distance to the center and $g(t)=f(t,0,\ldots,0)$ is a polynomial. (For $n\geq 2$ it's sufficient that $f$ be invariant under rotations, i.e. the special orthogonal group $SO(n)$.)

The theory of spherical harmonics, which is closely related to the representation theory of the orthogonal group $O(n),$ generalizes this observations to polynomials (and even more general functions) that are not necessarily spherically invariant.

General case  The first key theorem states that a polynomial $f$ can be expanded as
$$ f(x)=\sum_{k=0}^d r^{2k}h_k(x) \qquad (*)$$
where $d\leq\deg(f)/2$ and $h_k$ is a harmonic polynomial of degree at most $\deg(f)-2k.$ Recall that a polynomial is $h$ is harmonic if $\Delta h=0,$ where $\Delta$ is the Laplace operator,
$$\Delta h=\frac{\partial^2 h}{\partial x_1^2}+\frac{\partial^2 h}{\partial x_2^2}+\ldots+\frac{\partial^2 h}{\partial x_n^2}.$$
Moreover, the harmonic polynomials $h_k$ in $(*)$ are uniquely determined. If $f$ is spherically invariant then $h_k$ is simply the coefficient of $t^k$ in the polynomial $g(t)$ from above, viewed as a constant (i.e. degree $0$) polynomial.
So now every polynomial can be expanded in terms of the powers of $r^2$ and harmonic polynomials. What can we say about the latter? Let $\mathcal{H_\ell}$ (respectively, $\mathcal{H}$) denote the space of homogeneous harmonic polynomials of degree $\ell$ (respectively, all harmonic polynomials). Every harmonic polynomial is uniquely decomposed into the sum of homogeneous polynomials of various degrees, and these components are themselves harmonic:
$$\mathcal{H}=\bigoplus_{\ell\geq 0}\mathcal{H}_\ell\qquad (**)$$
The second key theorem states that $\mathcal{H}_\ell$ is an irreducible representation of the orhtogonal group $O(n)$ of dimension $\binom{n+\ell-1}{n-1}-\binom{n+\ell-3}{n-1}$ (for $\ell=0$ or $1$, the second term is zero). For $n\geq 3,$ this representations remains irreducible upon restriction to the special orthogonal group $SO(n)$ (for $n=3$, this is the rotation group).
For small $n$, namely $n=2,3,4$, there are standard bases in the spaces $\mathcal{H}_\ell$ that have been tabulated and extensively studied. In particular, for $n=3,$ $\mathcal{H}_\ell$ is a $(2\ell+1)$-dimensional representation of $SO(3)$, or spin $\ell$ representation in physics language, which has a basis indexed by an integer $m,\ -\ell\leq m\leq \ell$ constructed using associate Legendre polynomials $P_\ell^m.$ Depending on your goals, you may want a suitable explicit description of these functions in rectangular or spherical coordinates (the Wikipedia article Spherical harmonics, including the references, is a good starting point).
Let me also mention a common point of confusion related to your question 3: in view of $(*),$ for any polynomial $f$ as above, there exists a unique harmonic polynomial $h$ such that the restrictions of $f$ and $h$ to the unit sphere
$$S^{n-1}: x_1^2+x_2^2+\ldots+x_n^2=1$$ coincide, $f|_{S^{n-1}}=h|_{S^{n-1}}.$ In other words, the space $\mathcal{H}$ of harmonic polynomials can be naturally identified with the space of polynomial functions on the unit sphere $S^{n-1}$ and the decomposition $(**)$ becomes the decomposition of the polynomial functions on the sphere into irreducible representations of $O(n),$ each of which occurs with multiplicity one. These functions are frequently called spherical harmonics.

Addendum Here is a high level view of polynomial spherical harmonics from a representation theory vantage point. The decomposition $(*)$ is related to the representation theory of the Lie algebra $\mathfrak{sl}_2.$ The operators
$$E=r^2/2,\ F=-\Delta/2,\ H=\deg+n/2$$
on the vector space of polynomials in $n$ variables commute with orthogonal transformations and form a representation of $\mathfrak{sl}_2.$ (I've described the skew-symmetric analogue in this answer.) Homogeneous harmonic polynomials are precisely lowest weight vectors for $\mathfrak{sl}_2,$ and the second key theorem amounts to saying that the lowest weight spaces are irreducible representations of $O(n).$ One consequence of this description is that the coefficients $h_k(x)$ in $(*)$ can be found inductively starting with $h_d(x)$ using repeated applications of the Laplace operator (the precise statement is omitted due to bulky notation). Furthermore, the first and second key theorems can be combined into the statement that the space $\mathcal{P}$ of polynomials in $n$ variables has the following decomposition into irreducible components under the joint $O(n)$ and $\mathfrak{sl}_2$ actions:
$$\mathcal{P}=\bigoplus_{\ell\geq 0} \mathcal{H}_\ell \otimes V_\ell, $$
where $V_\ell$ is the lowest weight $\mathfrak{sl}_2$-module with lowest weight $\ell+n/2.$
This is one of the starting points of Roger Howe's theory of reductive dual pairs. In the present case, the reductive dual pair is $(O(n),SL(2,\mathbb{R}))$ over the real numbers.
Remark The group $SL(2)$ is secretely a symplectic group, in fact, the isometry group of a 2-dimensional symplectic vector space, and the theory extends to reductive dual pairs consisting of an orthogonal group and a symplectic group of any rank. The same theory over $\mathbb{Q}$ and its adeles is behind some classical results in the theory of theta functions due to Siegel and Weil. In particular, it explains the importance of theta functions with harmonic coefficients.


TITLE: Behaviour of mass for currents with disjoint supports
QUESTION [3 upvotes]: I am sorry if this is a basic question, but I don't think in MSE I will receive any answers.
Let $(M^3,g)$ be a compact and oriented Riemannian $3$-manifold. Let $\alpha$ and $\beta$ be the integral currents supported on the compact, connected, oriented embedded surfaces $\Sigma_1$ and $\Sigma_2$ with multiplicities $\theta_1$ and $\theta_2$, that is:
$$ \alpha(\omega) = \int_{\Sigma_1} \langle \omega(x), \tau_1 (x) \rangle \theta_1(x) d \mathcal{H}^2(x), \quad \omega \in \Omega_2(M),  $$
and similarly for $\beta$. Here, $\tau_1$ is a $2$-plane field that orients $\Sigma_1$ and $\theta_1$ is an integrable positive integer-valued function on $\Sigma_1$.
Assume that $\Sigma_1$ and $\Sigma_2$ are disjoint. Is it true that the mass of $\alpha$ is smaller than the mass of the sum $\alpha + \beta$?

REPLY [3 votes]: Yes, this is true, from the fact that the supports of these currents are disjoint. In fact, I think the same argument should work for two general currents, as long as their supports have disjoint neighborhoods.
Write $M(\alpha)$ and $M(\alpha + \beta)$ for the masses of $\alpha$ and $\alpha + \beta$, respectively.
Then there is a sequence of 2-forms $\omega_k$ such that $|\omega_k|_{C^0(M)} = 1$ for every $k$ and
$$|\alpha(\omega_k)| \to M(\alpha).$$
Let $\chi: M \to \mathbb{R}$ be a smooth, non-negative cutoff function that is equal to $1$ in a neighborhood of $\Sigma_1$ and $0$ in a neighborhood of $\Sigma_2$. Then it follows that
$$\alpha(\omega_k) = \alpha(\chi\omega_k)$$
for every $k$, and moreover
$$\beta(\chi\omega_k) = 0$$
for every $k$.
It follows that
$$|(\alpha + \beta)(\chi\omega_k)| \to M(\alpha).$$
However, by definition,
$$|(\alpha + \beta)(\chi\omega_k)| \leq M(\alpha + \beta)$$
for every $k$.
It follows from the above two statements that we must have the inequality
$$M(\alpha) \leq M(\alpha + \beta).$$


text:
\begin{document}

\title{The Gauss higher relative class number problem}
\author{John Voight}
\address{Department of Mathematics and Statistics, 16 Colchester Avenue, University of Vermont, Burlington, Vermont 05401-1455}
\email{jvoight@gmail.com}

\begin{abstract}
Assuming the $2$-adic Iwasawa main conjecture, we find all CM fields with higher relative class number at most $16$: there are at least $31$ and at most $34$ such fields, and at most one is nonabelian.
\end{abstract}

\maketitle

The problem of determining all imaginary quadratic fields $K=\Q(\sqrt{d})$ of small class number $h(K)$ was first posed in Article 303 of Gauss' \emph{Disquisitiones Arithmeticae}.  It would take almost 150 years of work, culminating in the results of Stark \cite{Stark} and Baker \cite{Baker}, to determine those fields with class number at most two: there are exactly $27$, the last having discriminant $d=-427$.  (See Goldfeld \cite{Goldfeld} or Stark \cite{StarkHistory} for a history of this problem.)  Significant further progress has been made recently by Watkins \cite{Watkins}, who enumerated all such fields $K$ with class number $h(K) \leq 100$.

One interesting generalization of the Gauss class number problem is to replace $\Q$ by a totally real field $F$.  Let $K/F$ be a CM extension, i.e., $K$ is a totally imaginary quadratic extension of a totally real field $F$, and let $[F:\Q]=n$.  We have the divisibility relation $h(F) \mid h(K)$, and we denote by $h^-(K)=h(K)/h(F)$ the relative class number.  It is known that there are only finitely many CM fields with fixed relative class number \cite{StarkRelative}.  The complete list of CM fields of relative class number one is still unknown (see Lee-Kwon \cite{LeeKwon} for an overview).  However, many partial results are known: for example, there are exactly $302$ imaginary abelian number fields $K$ with relative class number one \cite{ChangKwon}, each having degree $[K:\Q] \leq 24$.

The integer $h^-(K)$ can be determined by the analytic relative class number formula, as follows.  Let $\chi:\Gal(K/F) \to \{\pm 1\}$ denote the nontrivial character associated to the extension $K/F$ and let $L(\chi,s)$ denote the Artin $L$-function associated to $\chi$.  Then
\[ L(\chi,0)=\frac{2^n}{Q(K)}\frac{ h^-(K)}{w(K)},  \]
where $w(K)=\#\mu(K)$ is the number of roots of unity in the field $K$ and $Q(K)=[\Z_K^*:\Z_F^*\,\mu(K)] \in \{1,2\}$ is the Hasse $Q$-unit index.

In this article, we consider the further generalization of the Gauss problem to higher relative class numbers of CM fields.   Let $E$ be a number field with ring of integers $\Z_E$ and let $m \in \Z_{\geq 3}$ be an odd integer.  We define the \emph{higher class group} of $E$ to be
\[ H^2(\Spec \Z_E,\Z(m)) = \prod_p H\sbet^2(\Spec \Z_E[1/p], \Z_p(m)). \]
The group $H^2(\Spec \Z_E,\Z(m))$ is finite, and we let $h_m(E)$ denote its order.  For $p \neq 2$, the Quillen-Lichtenbaum conjecture (which appears to have been proven by Voevodsky-Rust-Suslin-Weibel) implies that the \'etale $\ell$-adic Chern character
\[ K_{2m-2}(\Z_E) \otimes \Z_p \xrightarrow{\sim} H\sbet^2(\Spec \Z_E[1/p], \Z_p(m)) \]
is an isomorphism, thus $h_m(E)$ and $\#K_{2m-2}$ agree up to a power of two.  (See \S 1 for more detail.)

For the CM extension $K/F$, we define the \emph{higher relative class number} to be
\[ h_m^{-}(K) = \frac{h_m(K)}{h_m(F)} \in \Z. \]
In analogy with the usual class number, an analytic higher relative class number formula holds: up to a power of $2$, we have the equality
\begin{equation} \label{Lvalueconj}
L(\chi,1-m)=(-1)^{n\left(\frac{m-1}{2}\right)} \frac{2^{n+1}}{Q_m(K)} \frac{h_m^{-}(K)}{w_m(K)},
\end{equation}
where $w_m(K) \in \Z_{>0}$ and $Q_m(K) \in \{1,2\}$ are the number of higher roots of unity and the higher $Q$-index, respectively.  Assuming the Iwasawa main conjecture for $p=2$, the formula (\ref{Lvalueconj}) holds exactly (see Kolster \cite{Kolster} and the discussion below).

Our main result is as follows.

\begin{thm*}
Suppose that the Iwasawa main conjecture holds for $p=2$.  Then there are at least $31$ and at most $34$ pairs $(K/F,m)$ where $h_m^-(K) \leq 16$.
\end{thm*}

The extensions are listed in Tables 4.1--4.2 in Section 4, and explains the calculations of Henderson ($h_m^-(K)=1$ and $F=\Q$) reported in Kolster \cite[\S 3]{Kolster}.

We begin in Section 1 by giving the necessary background.  In Section 2, we estimate the size of the higher relative class number using the analytic formula in an elementary way, and prove a statement in the spirit of the Brauer-Siegel theorem.  In Section 3, we combine these estimates with the Odlyzko bounds to reduce the problem to a finite computation and then carry it out to prove our main result.  We conclude in Section 4 by tabulating the fields.

The author wishes to thank Manfred Kolster for suggesting this problem and for his helpful comments and the anonymous referee for a careful reading.

\section{Background}

In this section, we state in detail the higher relative class number formula, which determines analytically the order of the higher relative class number of a CM extension of number fields.

We begin by introducing the $L$-function and its functional equation---see Lang \cite[Chapter XIII]{Lang} or Neukirch \cite[Chapter VII]{Neukirch} for a reference.  Let $E$ be a number field with ring of integers $\Z_E$, absolute discriminant $d_E$, and degree $[E:\Q]=r_1+2r_2=n$, where $r_1,r_2$ denote the number of real and pairs of complex places of $E$.  Let $\zeta_E(s)$ be the Dedekind zeta function of $E$, defined by
\[ \zeta_E(s)=\sum_{\fraka \subset \Z_E} \frac{1}{(N\fraka)^s}=\prod_{\frakp \subset \Z_E} \left(1-\frac{1}{(N\frakp)^s}\right)^{-1} \]
for $s \in \C$ with $\repart s > 1$.  Then $\zeta_E(s)$ has an analytic continuation to $\C \setminus \{1\}$ with a simple pole at $s=1$.  Define the completed zeta function of $E$ by
\[ \xi_E(s)=\left(\frac{d_E}{4^{r_2}\pi^n}\right)^{s/2} \Gamma(s/2)^{r_1} \Gamma(s)^{r_2} \zeta_E(s). \]
Then $\xi_E$ satisfies the functional equation $\xi_E(1-s)=\xi_E(s)$, and it follows that
\begin{equation} \label{funceq}
\zeta_E(1-s)=\zeta_E(s)\left(\frac{d_E}{4^{r_2} \pi^n}\right)^{s-1/2}\frac{\Gamma(s/2)^{r_1}\Gamma(s)^{r_2}}{\Gamma((1-s)/2)^{r_1}\Gamma(1-s)^{r_2}}.
\end{equation}

Now let $K/F$ be a CM extension of number fields with $[F:\Q]=n$ and let $\chi$ denote the nontrivial character of $\Gal(K/F)$.  Then the Artin L-function
\begin{equation} \label{defofL}
L(\chi,s) = \frac{\zeta_K(s)}{\zeta_F(s)}
\end{equation}
has an analytic continuation to $\C$, and for $s \in \C$ with $\repart s > 1$ we have
\begin{equation} \label{Lprod}
L(\chi,s) =\sum_{\fraka \subset \Z_F} \frac{\chi(\fraka)}{(N\fraka)^s} = \prod_{\frakp \subset \Z_F} \left(1-\frac{\chi(\frakp)}{(N\frakp)^s}\right)^{-1}.
\end{equation}

Applying equation (\ref{funceq}) to (\ref{defofL}) we obtain after simplification that
\begin{equation} \label{expandlval}
L(\chi,1-s)=L(\chi,s)\left(\frac{1}{(4\pi)^n}\frac{d_K}{d_F}\right)^{s-1/2}
\left(\frac{\Gamma(s)\Gamma((1-s)/2)}{\Gamma(s/2)\Gamma(1-s)}\right)^n.
\end{equation}

We now define the higher relative class group, a group whose order is determined by values of $L(\chi,s)$ at negative even integers---see Kolster \cite{KolsterIntro,Kolster} for a more complete treatment.  Let $m \in \Z_{\geq 3}$ be odd.  For a prime $p$, we denote by $H\sbet^i(\Z_E[1/p], \Z_p(m))$ the $i$th \'etale cohomology group of $\Spec \Z_E[1/p]$ with $m$-fold twisted $\Z_p$-coefficients.  Define the ($m$th) \emph{higher class group} of $E$ by
\[ H^2(\Z_E,\Z(m)) = \prod_p H\sbet^2(\Z_E[1/p], \Z_p(m)). \]
There exists a homomorphism $K_{2m-2}(\Z_E) \to H^2(\Z_E,\Z(m))$ with finite cokernel \cite{DF, Soule} (in fact, supported at $2$), and it follows that $H^2(\Z_E,\Z(m))$ is finite, since $K_{2m-2}(\Z_E)$ is finite (a result of Borel \cite{Borel} and Quillen \cite{Quillen}).  We let $h_m(E)=\# H^2(\Z_E,\Z(m))$ denote the ($m$th) \emph{higher class number} of $E$.  We again have the divisibility $h_m(F) \mid h_m(K)$, and we define the \emph{higher relative class number} of the CM extension $K/F$ to be the quotient
\[ h_m^{-}(K) = \frac{h_m(K)}{h_m(F)} \in \Z. \]

Two other quantities appear in the higher class number formula.  First, let
\[ H^0(E,\Q/\Z(m))=\prod_p H^0(E, \Q_p/\Z_p(m)) \]
be the group of \emph{higher roots of unity}, given in terms of Galois cohomology (invariants), and let $w_m(E)=\#H^0(E,\Q/\Z(m))$ denote its order.  By definition, $w_m(E)$ is the largest integer $q$ such that $G=\Gal(E(\zeta_q)/E)$ has exponent dividing $m$, where $\zeta_q$ denotes a primitive $q$th root of unity.

We have the following lemma that characterizes $w_m(E)$.  If $q$ is the power of a prime, let $\Q(\zeta_q)^{(m)}$ denote the subfield of $\Q(\zeta_q)$ of index $\gcd(\phi(q),m)$.

\begin{lem} \label{qmidwm}
If $q$ is a power of a prime, then $q \mid w_m(E)$ if and only if $E$ contains $\Q(\zeta_q)^{(m)}$.
\end{lem}

\begin{proof}
Note that since $\Q(\zeta_q)$ is Galois, we have
\[ G=\Gal(E(\zeta_q)/E) \cong \Gal(\Q(\zeta_q)/(E \cap \Q(\zeta_q))) \subset \Gal(\Q(\zeta_q)/\Q) \cong (\Z/q\Z)^\times. \]
Since $m \geq 3$ is odd, if $q$ is even then $q \mid w_m(E)$ if and only if $\Q(\zeta_q) \subset E$, as claimed.  If $q$ is odd, then since $\Gal(\Q(\zeta_q)/\Q)$ is cyclic, we have $q \mid w_m(E)$ if and only if $G$ has order dividing $m$ if and only if $\Q(\zeta_q)^{(m)} \subset E$.
\end{proof}

Finally, we define the \emph{higher $Q$-unit index} of the CM extension $K/F$ by
\[ Q_m(K)=[H\sbet^1(K, \Z_2(m)) : H\sbet^1(F, \Z_2(m)) H^0(K, \Q_2/\Z_2(m))]. \]
Collecting results of Kolster \cite[\S 3]{Kolster}, we know the following facts about $Q_m$.

\begin{prop} \label{Qmtests}
\
\begin{enumalph}
\item $Q_m \in \{1,2\}$.
\item $Q_m=2$ if and only if $H^1(K/F, H\sbet^1(K,\Z_2(m)))=0$.
\item If an odd prime of $F$ ramifies in $K$, then $Q_m=1$.
\item If no odd prime of $F$ ramifies in $K$, the field $F$ has only one prime lying above $2$, and $h(F)$ is odd, then $Q_m=2$.
\end{enumalph}
\end{prop}

With these definitions, we now state the higher relative class number formula.

\begin{prop}[{Kolster \cite{Kolster}}] \label{hrcnf}
We have
\begin{equation} \label{hrcnfform}
L(\chi,1-m)=(-1)^{n\left(\frac{m-1}{2}\right)} \frac{2^{n+1}}{Q_m(K)} \frac{h_m^{-}(K)}{w_m(K)}
\end{equation}
up to a power of $2$.  If the $2$-adic Iwasawa main conjecture holds, then \textup{(\ref{hrcnfform})} holds.
\end{prop}

The proof of the Iwasawa main conjecture for $p$ odd by Wiles \cite{Wiles} implies that the formula (\ref{hrcnfform}) holds up to a power of $2$, and a proof of the $2$-adic main conjecture would imply it exactly.  Indeed, Kolster \cite{Kolster} has proven that (\ref{hrcnfform}) holds in the case that $K$ is abelian over $\Q$ by applying Wiles' proof of the $2$-adic main conjecture when $K$ is abelian.

\section{Estimates}

In this section, we estimate the size of the relative class number.  Throughout, we assume the truth of the higher relative class number formula (Proposition \ref{hrcnf}).  We retain the notation from the previous section, suppressing the dependence on $K$ whenever possible; in particular, we recall that $m \in \Z_{\geq 3}$ is odd.

We begin by substituting $s=m$ into (\ref{expandlval}).  From standard $\Gamma$-function identities, letting $m=2k-1$ we have
\[ \Gamma(m/2)=\frac{(2k)!}{4^k k!}\sqrt{\pi} \]
and
\[ \frac{\Gamma((1-m)/2)}{\Gamma(1-m)}=\frac{\Gamma(-k)}{\Gamma(-2k)}=(-1)^k\frac{(2k)!}{k!}. \]
Putting these together, we obtain
\[ \frac{\Gamma(m)\Gamma((1-m)/2)}{\Gamma(m/2)\Gamma(1-m)}=(-1)^{(m-1)/2}\frac{2^m(m-1)!}{\sqrt{\pi}} \]
hence
\begin{equation} \label{lchi1-m} L(\chi,1-m)=(-1)^{n\left(\frac{m-1}{2}\right)}L(\chi,m)\left(\frac{d_K}{d_F}\right)^{m-1/2}\left(\frac{2(m-1)!}{(2\pi)^m}\right)^n.
\end{equation}

Now, by the higher relative class number formula, we have
\begin{equation} \label{hmest}
h_m^{-}=|L(\chi,1-m)|\frac{w_m Q_m}{2^{n+1}} \geq \frac{|L(\chi,1-m)|}{2^{n+1}}
\end{equation}
since $Q_m,w_m \in \Z_{\geq 1}$.  From (\ref{lchi1-m}) and (\ref{hmest}) we obtain
\begin{equation} \label{est1}
h_m^{-} \geq \frac{1}{2}L(\chi,m)\left(\frac{d_K}{d_F}\right)^{m-1/2}\left(\frac{(m-1)!}{(2\pi)^m}\right)^n.
\end{equation}

We now estimate the value of $L(\chi,m)$.

\begin{lem} \label{Lchigamma}
We have $\gamma(m)^n \leq L(\chi,m) \leq \zeta(m)^n$, where
\[ \gamma(m)=\prod_p \left(1+\frac{1}{p^m}\right)^{-1}. \]
\end{lem}

\begin{proof}
From (\ref{Lprod}) we obtain
\[ L(\chi,m) = \prod_\frakp \left(1-\frac{\chi(\frakp)}{(N\frakp)^m}\right)^{-1}. \]
Organizing the product by primes $p$, we have
\[ \left(1+\frac{1}{p^m}\right)^{-n} \leq \prod_\frakp \left(1-\frac{\chi(\frakp)}{(N\frakp)^m}\right)^{-1} \leq \left(1-\frac{1}{p^m}\right)^{-n} \]
hence
\[ \gamma(m)^n \leq L(\chi,m) \leq \zeta(m)^n \]
as claimed.
\end{proof}

We compute easily that $\gamma(3) \geq 0.8463$, $\gamma(5) \geq 0.9653$, and $\gamma(m) \geq 0.9917$ if $m \geq 7$; clearly $\gamma(m) \to 1$ as $m \to \infty$.  Applying Lemma \ref{Lchigamma} to (\ref{est1}) we obtain
\begin{equation} \label{est!}
h_m^{-} \geq \frac{1}{2} \left(\frac{d_K}{d_F}\right)^{m-1/2} \left(\gamma(m)\frac{(m-1)!}{(2\pi)^m}\right)^n.
\end{equation}

We pause to prove the following proposition, which is in the spirit of the Brauer-Siegel theorem.  We have $d_K/d_F=d_F N\frakd_{K/F} \geq d_F$, where $\frakd_{K/F} \subset \Z_F$ denotes the relative discriminant of $K/F$.

\begin{prop}
Let $\{K_i/F_i\}_i$ be a sequence of CM extensions with
\[ [F_i:\Q] = o\left(\log(d_{K_i}/d_{F_i})\right). \]
Then
\[ \log h_m^{-}(K_i) \sim (m-1/2)\log(d_{K_i}/d_{F_i}) \]
as $i \to \infty$.
\end{prop}

\begin{proof}
Let $K/F$ be a CM extension with $[F:\Q]=n$.  From (\ref{est!}) we have
\begin{equation} \label{hmlower}
\log h_m^- \geq (m-1/2)\log(d_K/d_F) + n c(m)
\end{equation}
where $c(m)$ is a constant depending only on $m$.  On the other hand, by Lemma \ref{Lchigamma} we have $L(\chi,m) \leq \zeta(m)^n$.  Then applying equations (\ref{lchi1-m}) and (\ref{hmest}) with this estimate we obtain
\begin{equation} \label{hmupper}
\log h_m^{-} \leq \log w_m + (m-1/2)\log(d_K/d_F) + n \log \zeta(m).
\end{equation}
Putting together (\ref{hmlower}) and (\ref{hmupper}), since $[F_i:\Q]=o(\log(d_{K_i}/d_{F_i}))$ as $i \to \infty$, the result follows if we show that
\begin{equation} \label{wmK}
\log w_m(K_i) = o( \log(d_{K_i}/d_{F_i}) ).
\end{equation}

To prove (\ref{wmK}), we compare $w_m(K)$ and $d_K/d_F$ for a CM extension $K/F$.  Let $q=p^r$ be a power of a prime and suppose that $q \mid w_m$.  By Lemma \ref{qmidwm}, the field $K$ must contain $\Q(\zeta_q)^{(m)}$.  Suppose first that $q$ is odd.  It follows that $F$ must contain the field $\Q(\zeta_q)^{(2m)}$; since $d_{\Q(\zeta_q)^{(2m)}} \mid d_F$, by the conductor-discriminant formula \cite[Theorem 3.11]{Washington} we have
\[ p^{\phi(q)/(2m)-1} \mid d_{\Q(\zeta_q)^{(2m)}}. \]
If $q$ is even, then $F$ must contain the totally real subfield $\Q(\zeta_q)^+$ of $\Q(\zeta_q)$, and we similarly conclude that $p^{\phi(q)/2-1} \mid d_F$.  Define the multiplicative function $f$ with the value $f(p^r)=p^{\phi(p^r)/(2m)-1}$ for a prime power $p^r$.

Now let $K_i/F_i$ be a subsequence with $w_m(K_i) \to \infty$; if no such subsequence exists, then we are done.  We now show that for any sequence of positive integers $n_i \to \infty$, we have $\log n_i = o(\log f(n_i))$.  The result then follows as
\[ \log w_{m_i} = o(\log f(w_{m_i})) = o(\log d_{F_i}) = o(\log(d_{K_i}/d_{F_i})) \]
as desired.

So let $\eps > 0$.  We prove that
\[ r < \eps \left(\frac{\phi(p^r)}{2m}-1\right) \]
for all sufficiently large prime powers $p^r$, or also it is sufficient to show that
\begin{equation} \label{stupidpr}
\frac{\phi(p^r)}{4rm} > \frac{1}{\eps}.
\end{equation}
For $r=1$, clearly the inequality (\ref{stupidpr}) will be satisfied for $p$ sufficiently large; but then for the finitely many remaining primes $p$, the inequality holds for a sufficiently large power $r$.  Therefore it holds for any sufficiently large prime power $p^r$, and claimed.
\end{proof}

\begin{cor}
Let $F$ be a totally real field and $m \in \Z_{\geq 3}$ be odd.  Then for any $h \in \Z_{\geq 1}$, there are only finitely many CM extensions $K/F$ with $h_m^-(K) \leq h$.
\end{cor}

We return to estimating $h_m^{-}$ from below.  Rewriting (\ref{est!}) we obtain
\begin{equation} \label{est!!}
d_F \leq \frac{d_K}{d_F} \leq (2h_m^-)^{1/(m-1/2)} C(m)^n
\end{equation}
where
\[ C(m)=\left(\frac{(2\pi)^m}{\gamma(m)(m-1)!}\right)^{1/(m-1/2)} \]
depends only on $m$.  We compute that
\[ C(3) \leq 7.3517, \quad C(5) \leq 3.8332, \quad C(7) \leq 2.6336, \quad C(9) \leq 2.011. \]
It follows from Stirling's approximation and the fact that $\gamma(m)$ is increasing to $1$ that $C(m)$ is decreasing to $0$.

Let $\delta_F=d_F^{1/n}$ denote the root discriminant of $F$.   Taking $n$th roots we obtain from (\ref{est!!}) that
\begin{equation} \label{mainest}
\delta_F \leq (2h_m^-)^{2/\left((2m-1)n\right)} C(m).
\end{equation}

\section{Enumerating the list of extensions}

We now apply the results of \S 2 to enumerate CM extensions with small higher relative class number.

We list those extensions with $h_m^- \leq 16$; we have chosen this bound to capture the smallest higher relative class number of a nonabelian field, and we note that this bound can easily be increased, if desired.  Let $NF_m(n)$ denote the set of totally real fields $F$ of degree $n$ that satisfy the bound (\ref{mainest}) with $h_m^-=16$, and let $NF_m = \bigcup_n NF_m(n)$.

For $m=3$, the estimate (\ref{mainest}) then reads
\begin{equation} \label{deltaFest}
\delta_F \leq 32^{2/(5n)} C(3) \leq 7.3517 \cdot 4^{1/n}.
\end{equation}

By the (unconditional) Odlyzko bounds \cite{Martinet}, if $n \geq 7$ we have $\delta_F \geq 9.301$, but by (\ref{deltaFest}) we have $\delta_F \leq 8.962$, a contradiction.  In Table 2.3, for each degree $n \geq 2$ we list the upper bound on the root discriminant $\delta_F$, the corresponding Odlyzko bound $B_O$, the corresponding upper bound on the discriminant $d_F$, and the size of $NF_3(n)$.

\begin{center}
\vspace{1ex}
\textbf{Table 2.3.} Degree and Root Discriminant Bounds \\
\vspace{-2ex}
\[
\begin{array}{c|cccccc}
n & 2 & 3 & 4 & 5 & 6 & \geq 7 \\
\hline
\delta_F & \leq 14.703 & 11.670 & 10.397 & 9.701 & 9.263 & 8.962 \\
B_O & > 2.223 & 3.610 & 5.067 & 6.523 & 7.941 & 9.301 \\
\hline
d_F & \leq 216 & 1589 & 11684 & 85899 & 631505 & - \\
\#NF_3(n) & 65 & 48 & 64 & 8 & 6 & 0
\end{array}
\]
\vspace{0ex}
\end{center}
The fields $NF_3(n)$ of such small discriminant are well-known \cite{Tables}.

For $m=5$, arguing in a similar way we have
\[ \delta_F \leq 32^{2/(9n)} C(5) \leq 3.8332 \cdot 2.1602^{1/n}, \]
which already for $n \geq 4$ gives $\delta_F \leq 4.5273$, contradicting the Odlyzko bound $\delta_F > 5.067$.  In fact, we have $d_F \leq 28$ for $n=2$ and $d_F \leq 109$ for $n=3$, so that $\#NF_5(2)=8$ and $\#NF_5(3)=2$.  Proceeding in this way, we find:
\[
NF_m=
\begin{cases}
\{\Q,\Q(\sqrt{5}),\Q(\sqrt{8})\}, & \text{ if $m=7$}; \\
\{\Q,\Q(\sqrt{5})\}, & \text{ if $m=9$}; \\
\{\Q\}, & \text{ if $11 \leq m \leq 19$}; \\
\emptyset, & \text{ if $m \geq 21$}. \\
\end{cases} \]

Now for each such field $F$ and $m$, we have from (\ref{est!!}) that
\begin{equation} \label{boundk}
d_K \leq \lfloor 32^{2/(2m-1)} C(m)^n \rfloor d_F,
\end{equation}
leaving only finitely many possibilities for the CM extension $K/F$.  We can find these relative quadratic extensions explicitly by using a relative version of Hunter's theorem due to Martinet: see Cohen \cite[\S\S 9.2--9.3]{Cohen} for more details.  We obtain in this way $90,9,2,1$ extensions $K/F$ for $m=3,5,7,9$, and none for $m \geq 11$.

Next, using (\ref{lchi1-m}) and the higher relative class number formula (Proposition \ref{hrcnf}) we numerically compute the value $h_m^-/(w_m Q_m) \in \R$.  In order to recover this value exactly, it suffices to bound the size of the denominator.  We have $Q_m \in \{1,2\}$.

To determine $w_m$, we apply Lemma \ref{qmidwm}.  If $q$ is the power of a prime and $q \mid m$, then $q$ is odd and $K$ contains the unique subfield of index $m$ of $\Q(\zeta_q)$.  In particular, this implies that $\phi(q)/m \mid 2n=[K:\Q]$, which already gives a bound on $w_m$.  To reduce the size of this bound further, we note that we also have $d_{\Q(\zeta_q)^{(m)}} \mid d_K$, so in particular $q \mid d_K$ whenever $\phi(q) > m$; furthermore, for any prime $\frakp$ of $K$ that is prime to $qd_K$, it is easy to see that the order of $N\frakp \in (\Z/q\Z)^*$ must divide $m$.  A prime power $q$ that passes these tests, the latter for sufficiently many primes $\frakp$, is very likely to divide $w_m$.  To compute $w_m$ exactly and verify that indeed $q \mid w_m$, we simply check if the $q$th cyclotomic polynomial over $K$ factors into polynomials of degree at most $m$.

In this way, we compute $h_m^-/Q_m \in \frac{1}{2}\Z$.  If this value is not an integer, then we immediately know $Q_m=2$.  Otherwise, we may apply the tests of Proposition \ref{Qmtests} to determine the value $Q_m$ in almost all cases.  We are lucky that in many cases where we cannot determine the value of $Q_m$, we nevertheless have $h_m^- \geq h_m^-/Q_m > 16$.  We have $6$ remaining cases.  One of which we can resolve as follows: for $F=\Q(\sqrt{30})$ and $K=\Q(\sqrt{-15},\sqrt{-2})$ with $h_3^-/Q_3=12$, we apply the divisibility result $8=h_3^-(\Q(\sqrt{-15})) \mid h_3^-(K)$ \cite[Corollary 3.6]{Kolster}, which implies $Q_3=2$, and hence $h_3^-=24$.  We were unable to resolve, and we leave it an open problem to compute the higher $Q$-index $Q_m$ in the other $5$ cases.  We expect the problem to be nontrivial for the reason that already characterizing Hasse's $Q$-index is quite intricate (see Hasse \cite{Hasse}).

Of the $90,9,2,1$ CM extensions for $m=3,5,7,9$, respectively, $26-29,4,1,0$ have $h_m^- \leq 16$, and they are listed in Tables 4.1--4.2.

\section{Tables}

In this section, we present the tables of CM extensions with higher relative class number $h_m^- \leq 16$.  Below, we list the totally real field $F$ and its discriminant $d_F$, the CM field $K$, its absolute discriminant $d_K$ and the norm of the relative discriminant $N(\frakd_{K/F})$, an element $\delta \in F$ such that $K=F(\sqrt{\delta})$, and the higher class number $h_m^-$.  As usual, we let $\zeta_k$ denote a primitive $k$th root of unity, $\omega=\zeta_3$ and $i=\zeta_4$, and we define $\lambda_k = \zeta_k+1/\zeta_k$, so that $\Q(\lambda_k)=\Q(\zeta_k)^+$.

The computations were performed in \textsf{Magma} \cite{Magma} and \textsf{Sage} \cite{Sage}; total computing time was about $2$ minutes.

It is interesting to note that there is no CM extension with higher relative class number $h_m^-=2$.  Also, note that there is a misprint in the computation of $h_m^-(\zeta_{12})=1$ in Kolster \cite{Kolster}.

\begin{center}
\vspace{1ex}
\textbf{Table 4.1.} CM extensions with higher relative class number $h_m^- \leq 16$ for $m \geq 5$ \\
\vspace{-2ex}
\[
\begin{array}{cc|cccc|c}
d_F & F & d_K & K & N(\frakd_{K/F}) & \delta  & h_5^- \\ \hline \hline
\rule{0pt}{2.5ex}
1 & \Q             & 3     & \Q(\omega) & 3  & -3 & 1 \\ \hline
\rule{0pt}{2.5ex}
1 & \Q             & 4     & \Q(i) & 4  & -4 & 5 \\
12 & \Q(\sqrt{3})  & 144   & \Q(\zeta_{12}) & 1  & -1 & 5 \\ \hline
\rule{0pt}{2.5ex}
1 & \Q             & 7     & \Q(\sqrt{-7}) & 7  & -7 & 16 \\
\hline \ \\
\rule{0pt}{2.5ex} d_F & F & d_K & K & N(\frakd_{K/F}) & a  & h_7^- \\ \hline \hline
\rule{0pt}{2.5ex} 1 & \Q             & 3     & \Q(\omega) & 3  & -3 & 7
\end{array}
\]
\end{center}

\newpage

\begin{center}
\vspace{1ex}
\textbf{Table 4.2.} CM extensions with higher relative class number $h_3^- \leq 16$ \\
\vspace{-2ex}
\[
\begin{array}{cc|cccc|c}
d_F & F            & d_K   & K       & \hspace{-1ex}N(\frakd_{K/F}) & \delta  & h_3^- \\ \hline \hline
\rule{0pt}{2.5ex}
1 & \Q             & 3     & \Q(\omega)           & 3  & -3 & 1 \\
1 & \Q             & 4     & \Q(i)                & 4  & -1 & 1 \\
5 & \Q(\sqrt{5})   & 125   & \Q(\zeta_5)          & 5  & \hspace{-3ex}-\sqrt{5}(1+\sqrt{5})/2 & 1 \\
12 & \Q(\sqrt{3})  & 144   & \Q(\zeta_{12})       & 1  & -1 & 1 \\ \hline
\rule{0pt}{2.5ex}
1 & \Q             & 8     & \Q(\sqrt{-2})        & 8  & -2 & 3 \\
1 & \Q             & 11    & \Q(\sqrt{-11})       & 11 & -11 & 3 \\
8 & \Q(\sqrt{2})   & 256   & \Q(\zeta_8)          & 4  & -2 & 3 \\
24 & \Q(\sqrt{6})  & 576   & \Q(\sqrt{-2},\omega) & 1  & -2 & 3 \\
44 & \Q(\sqrt{11}) & 1936  & \Q(\sqrt{-11}, i)    & 1  & -1 & 3 \\ \hline
\rule{0pt}{2.5ex}
33 & \Q(\sqrt{33}) & 1089  & \Q(\sqrt{-11},\omega)& 1  & -3 & 3\text{ or }6? \\ \hline
\rule{0pt}{2.5ex}
60 & \Q(\sqrt{15}) & 3600  & \Q(\sqrt{-15}, i)    & 1  & -1 & 4\text{ or }8? \\ \hline
\rule{0pt}{2.5ex}
1 & \Q             & 7     & \Q(\sqrt{-7})        & 7  & -7 & 8 \\
1 & \Q             & 15    & \Q(\sqrt{-15})       & 15 & -15 & 8 \\
5 & \Q(\sqrt{5})   & 225   & \Q(\sqrt{-15},\omega)& 9  & -3 & 8 \\
28 & \Q(\sqrt{7})  & 784   & \Q(\sqrt{-7},i)      & 1  & -1 & 8 \\
49 & \Q(\lambda_7) & 16807 & \Q(\zeta_7)          & 7  & -7 & 8 \\ \hline
\rule{0pt}{2.5ex}
88 & \Q(\sqrt{22}) & 7744  & \Q(\sqrt{-11},\sqrt{-2})& 16 & -2 & 9 \\ \hline
\rule{0pt}{2.5ex}
1 & \Q             & 19    & \Q(\sqrt{-19})       & 19 & -19 & 11 \\
57 & \Q(\sqrt{57}) & 3249  & \Q(\sqrt{-19},\omega)& 1  & -3 & 11\text{ or }22? \\
76 & \Q(\sqrt{19}) & 5776  & \Q(\sqrt{-19},i)     & 1  & -1 & 11 \\ \hline
\rule{0pt}{2.5ex}

81 & \Q(\lambda_9) & 19683 & \Q(\zeta_9)          & 3  & -3 & 13 \\ \hline
\rule{0pt}{2.5ex}
1  & \Q            & 20    & \Q(\sqrt{-5})        & 20 & -5 & 15 \\
5  & \Q(\sqrt{5})  & 400   & \Q(\sqrt{-5},i)       & 16 & -1 & 15 \\
60 & \Q(\sqrt{15}) & 3600  & \Q(\sqrt{-5},\omega) & 1  & -1 & 15\text{ or }30? \\
2000 & \Q(\lambda_{20}) & 4000000 & \Q(\zeta_{20})& 1  & -1 & 15 \\ \hline
\rule{0pt}{2.5ex}
21 & \Q(\sqrt{21}) & 441   & \Q(\sqrt{-7},\omega) & 1  & -3 & 16 \\
8 & \Q(\sqrt{2})   & 1088  & \Q\left(\sqrt{2\sqrt{2}-5}\right) & 17 & 2\sqrt{2}-5 & 16 \\
17 & \Q(\sqrt{17}) & 2312  & \textstyle{\Q\left(\sqrt{-(5+\sqrt{17})/2}\right)} & 8 & \hspace{-3ex}-(5+\sqrt{17})/2 & 16\text{ or }32? \\
1125 & \Q(\lambda_{15}) & 1265625 & \Q(\zeta_{15}) & 1 & -15 & 16
\end{array}
\]
\vspace{0ex}
\end{center}

\newpage

config: arxiv