# Regional Zeta Functions

### Associated to Norm 1 Height Zeta Functions

#### Fractional Ideals of Norm 1

Let $K$ be a number field of degree $d$, Galois over $\mathbb Q$. Let $\mathfrak o$ be the ring of integers. Then a fractional ideal $\mathfrak x$ is a $\mathfrak o$-module in $K$ such that there exists $\alpha \in \mathfrak o$ with $\alpha \mathfrak x \subseteq \mathfrak o$.

Alternately, a fractional ideal is the quotient $\mathfrak x = \mathfrak a/\mathfrak b$ of integral ideals $\mathfrak a, \mathfrak b \subseteq \mathfrak o$. The set of non-zero fractional ideals $\mathfrak F$ is a group under multiplication. The set of principal ideals $\mathfrak F’$ is a finite index subgroup of $\mathfrak F$, and the (finite) quotient is called the ideal class group of $K$.

Each fractional ideal has a factorization into prime ideals, $$\mathfrak x = \mathfrak p_1^{m_1} \cdots \mathfrak p_L^{m_L}$$ where the prime ideals $\mathfrak p_1, \ldots, \mathfrak p_L$ and the integers $m_1, \ldots, m_L$ are uniquely determined. We denote the set of prime ideals by $\mathfrak P$.

Each integral ideal in $\mathfrak a$ has finite rank in $\mathfrak o$, and we define the (ideal) norm of $\mathbb N \mathfrak a = [\mathfrak a : \mathfrak o]$. The ideal norm is multiplicative, and we can extend its definition to the group of fractional ideals. That is $\mathbb N : \mathfrak F \rightarrow \mathbb Z_{>0}$ is a multiplicative homomorphism. If $\mathfrak x$ has the same factorization as before, then $$\mathbb N \mathfrak x = (\mathbb N \mathfrak p_1)^{m_1} \cdots (\mathbb N \mathfrak p_L)^{m_L}.$$

The group of fractional ideals of norm 1, $\mathfrak N$ is a subgroup of $\mathfrak F$. The set of principal ideals of norm 1 $\mathfrak N’$ is a finite index subgroup of $\mathfrak N$ and the quotient is the norm 1 ideal class group of $K$.

#### Norms of Prime Ideals

If $\mathfrak p \subset \mathfrak o$ is a prime ideal, then $\mathfrak p \cap \mathbb Z$ is a prime ideal in $\mathbb Z$. That is, there exists rational prime $p$ such that $\mathfrak p \cap \mathbb Z = p \mathbb Z$, and $\mathfrak o / \mathfrak p$ is a module over $\mathbb Z/p \mathbb Z$. We say $\mathfrak p$ is a prime (ideal) over $p$ and write $\mathfrak p | p$. There exists a positive integer $f_{\mathfrak p}$, called the inertial degree of $\mathfrak p$, such that $\mathbb N \mathfrak p = p^{f_{\mathfrak p}}$.

The principal ideal $p \mathfrak o$ has norm $p^d$. And, since $p$ may not be prime, we have prime ideals $\mathfrak p_1, \ldots, \mathfrak p_J$ such that $p \mathfrak o = \mathfrak p_1^{e_1} \cdots \mathfrak p_J^{e_J}$. The prime ideals are exactly those which restrict to $p \mathbb Z$. The exponents $e_1, \ldots, e_J$ are called the ramification degrees of $\mathfrak p_1, \ldots, \mathfrak p_J$. We sometimes denote the ramification degree of an arbitrary prime $\mathfrak p$ by $e_{\mathfrak p}$. If $e_{\mathfrak p} > 1$ then $\mathfrak p$ is called ramified. Let $f_1, \ldots, f_J$ be the inertial degrees of $\mathfrak p_1, \ldots, \mathfrak p_J$. It follows that $$\mathbb N p \mathfrak o = p^d = \prod_{j=1}^J p^{f_j e_j},$$ and hence $$e_1 f_1 + \cdots + e_J f_J = d.$$ The latter sometimes abbreviated $\sum_{\mathfrak p | p} e_{\mathfrak p} f_{\mathfrak p} = d$. There are only finitely many ramified primes, and thus for all but finitely many rational primes $p$, $\sum_{\mathfrak p | p} f_{\mathfrak p} = d$

The action of Galois induces a transitive action on the $\mathfrak p | p$. This implies that, because $K$ is Galois, the inertial degrees of all primes above $p$ are equal, and the ramification degrees of all primes above $p$ are equal. That is, there exists $f_p$ and $e_p$ such that $f_{\mathfrak p} = f_p$ and $e_{\mathfrak p} = e_p$ for all $\mathfrak p | p$. In which case the number of primes above $p$ is $J = J_p = d/(e_p f_p)$.

#### Archimedean Absolute Values

Because $K$ is Galois, there are either $d$ embedings into $\mathbb R$ or $d/2$ complex conjugate pairs of embeddings into $\mathbb C$. In the former case we say $K$ has local archimedean degree $d_{\infty} = 1$, and in the complex case $d_{\infty} = 2$. Both $\mathbb R$ and $\mathbb C$ come equipped with their usual absolute values, and these induce $d/d_v$ archimedean absolute values on $K$ (complex conjugate pairs of embeddings produce the same embedding). We denote by $| \cdot |_1, \ldots, |\cdot |_{d/d_v}$ the absolute values on $K$ induced by the usual real absolute value when $d_v = 1$ and induced by the usual absolute value squared on $\mathbb C$.

The absolute (field) norm of an element $\alpha \in K$ is given by $$|N_K(\alpha)| = \prod_{m=1}^{d/d_v} |\alpha|_M.$$ It is a theorem that if $\mathcal a = \alpha \mathfrak o$ is the fractional ideal generated by $\alpha$, then $\mathbb N \mathfrak a = | N_K(\alpha) |$. In particular, for our purposes, if $x$ has field norm 1, and $\mathfrak x = x \mathfrak o$ is the associated principal ideal, then $\mathbb N \mathfrak x = 1$. Let $X \subseteq K$ be the subgroup with absolute field norm 1.

Integers that have absolute field norm 1 are called units, and we denote the group of units by $U \subseteq \mathfrak o$. The ideal generated by a unit is all of $\mathfrak o$.

#### The Log Map

Let $\log : K \rightarrow \mathbb R^{d/d_v}$ be the map given by $\log(\alpha) = ( \log|\alpha|_1, \ldots, \log |\alpha|_{d/d_v}$. And let $\Sigma : \mathbb R^{d_v/d} \rightarrow \mathbb R$ be the linear map $\Sigma(z_1, \ldots, z_{d/d_v}) = z_1 + \cdots + z_{d/d_v}$. Then, $\log X \subset \ker \Sigma$.

Theorem (Dirichlet): $\log U$ is a full rank lattice in $\ker \Sigma$.

It follows that $T = \ker \Sigma / \log U$ is a torus of dimension $d/d_v – 1$, and the image of $X$ in this torus is in bijection with the principal ideals of norm 1, $\mathfrak N’$.

The torus being a compact abelian group, there is a natural Haar probability measure $\lambda$ on $T$. This measure is kind of like Lebesgue measure in that it is translation invariant and Borel. Indeed it is induced by Lebesgue measure on $\ker \Sigma \cong \mathrm R^{d_v/d – 1}$ restricted to a fundamental domain of $\log U$. To make this a probability measure we must normalize by the Lebesgue measure of a fundamental domain of $\log U$, and this normalizing constant is called the regulator.

#### The Ideal Height

If $\mathfrak x = \mathfrak a/ \mathfrak b$ is given in lowest terms ($(\mathfrak a, \mathfrak b) = \mathfrak o$), then the ideal height of $\mathfrak x$ is given by $$H(\mathfrak x) = \max\{ \mathbb N \mathfrak a, \mathbb N \mathfrak b \}.$$ If, in fact $\mathfrak x \in \mathfrak N$ then $H(\mathfrak x) = \mathbb N \mathfrak a = \mathbb N \mathfrak b.$

We denote by $\mathfrak N'(t) = \{ \mathfrak x \in \mathfrak N’ : H(\mathfrak x) \leq t\}.$

Big Question: How does $\mathfrak N'(t)$ distribute in $T$ as $t \rightarrow \infty$?

#### Equidistribution

Definition: $\mathfrak N’$ is equidistributed in $T$ with respect to $H$ if for any open set $B \subset T$, $$\lim_{t \rightarrow \infty} \frac{\# \mathfrak N'(t) \cap B}{\# \mathfrak N'(t)} = \lambda(B).$$ That is, if the asymptotic relative frequency of $\mathfrak N’$ on $B$, as ordered by height, approaches the Haar measure of $B$ as the height gets large.

Conjecture: $\mathcal N’$ is equidistributed in $T$ with respect to $H$.

A continuous homomorphism $\chi: T \rightarrow \mathbb T$ into the unit circle $\mathbb T \subset \mathbb C$ is called a character of $T$. The set of characters $\widehat T$ is a group under pointwise multiplication.

The topology on $T$ is generated by $\widehat T$, and this implies Weyl’s Criterion.

Theorem (Weyl): $\mathcal N’$ is equidistributed in $T$ with respect to the height if and only if for any non-trivial character $\chi \in \widehat T$,$$\lim_{t \rightarrow \infty} \frac{1}{\# \mathfrak N'(t)} \sum_{\mathfrak x \in \mathfrak N'(t)} \chi(\mathfrak x) = 0.$$

#### Associated Global $\zeta$ Functions

Definition: The function $t \mapsto \sum_{\mathfrak x \in \mathfrak N'(t)} \chi(\mathfrak x)$ is the summatory function for the global, norm 1, twisted partial height $\zeta$-function $$Z'(\chi; s) = \sum_{\mathfrak x \in \mathfrak N’} \frac{\chi(\mathfrak x)}{H(\mathfrak x)^s}.$$ Note the prime here indicates the sum if over principal ideals of norm one (not the derivative of $Z$). The complete twisted height $\zeta$-function is given by $$Z(\chi; s) = \sum_{\mathfrak x \in \mathfrak N} \frac{\chi(\mathfrak x)}{H(\mathfrak x)^s}.$$

Some care must be taken to define things correctly because the character $\chi$ is defined initially for principal ideals by the image of their generator on the torus $T$. The ideals of norm 1 are not naturally embedded in that torus, and we must make sense what $\chi(\mathfrak x)$ means.

While the ideal height is not multiplicative on $\mathfrak N’$, it nonetheless partially factors into terms indexed by the rational primes. To see this, we can factor $\mathfrak x \in \mathfrak N$ as $$\mathfrak x = \frac{\mathfrak{a}}{\mathfrak b} = \prod_{\mathfrak p} \mathfrak p^{m_{\mathfrak p}(\mathfrak x)},$$ where $m_{\mathfrak p}(\mathfrak x)$ is an integer which is equal to zero for all but finitely many primes ideals $\mathfrak p \in \mathfrak P$. The primes with positive $m_{\mathfrak p}(\mathfrak x)$ correspond to primes in the factorization of $\mathfrak a$, while those with negative $m_{\mathfrak p}(\mathfrak x)$ correspond to primes in the factorization of $\mathfrak b$. This implies that $$H(\mathfrak x) = \mathbb N \mathfrak a = \mathbb N \mathfrak b = (\mathbb N \mathfrak a \mathbb N \mathfrak b)^{1/2} = \prod_{\mathfrak p} \mathbb N \mathfrak p^{|m_{\mathfrak p}(\mathfrak x)|/2}.$$

Because $H(\mathfrak x)$ is an integer it also has a factorization into primes, and because $p | \mathbb N \mathfrak p$ if and only if $\mathfrak p | p$, we have $$H(\mathfrak x) = \prod_{p} H_p(\mathfrak x) \quad \mbox{where} \quad H_p(\mathfrak x) = \prod_{\mathfrak p | p} p^{f_{\mathfrak p} |m_{\mathfrak p}(\mathfrak x)|/2}$$ is the local height. Because $\mathfrak x$ has norm 1, $\sum_{\mathfrak p | p} m_p(\mathfrak x) = 0$, and the exponent of $p$ in the height is $$\frac{1}{2} \sum_{\mathfrak p | p} f_{\mathfrak p} |m_{\mathfrak p}(\mathfrak x)|.$$

Because $H$ factors over rational primes, there is a partial Euler product for $Z(\chi; s)$ over rational primes. A bit of notation is in order. By $\mathfrak N_p$ we mean set of ideals of norm 1 which factor into primes only lying above $p$. Given any $\mathfrak x \in \mathfrak N$ there are unique ideals $\mathfrak x_p \in \mathfrak N_p$, almost all of which are equal to $\mathfrak o$ such that $\mathfrak x = \prod_{p} \mathfrak x_p$ and $H(\mathfrak x) = \prod_{\mathfrak p | p} H_p(\mathfrak x_p)$ and $\chi(\mathfrak x) = \prod_{\mathfrak p | p} \chi_p(\mathfrak x_p)$. Then, $$Z(\chi; s) = \prod_p \sum_{\mathfrak x_p \in \mathfrak N_p} \frac{\chi(\mathfrak x_p)}{H(\mathfrak x_p)^{s}}.$$ Because the function $H$ is not completely multiplicative, there is no natural Euler product over $\mathfrak p \in \mathfrak P.$

#### Regional Zeta Functions

We call $$Z_p(\chi; s) = \sum_{\mathfrak x_p \in \mathfrak N_p} \frac{\chi(\mathfrak x_p)}{H_p(\mathfrak x_p)^{s}},$$ the twisted regional zeta function for the rational prime $p$.

We are going to look at properties of regional zeta functions, but we do not yet have the machinery to prove the equidistribution result we are after. We remain in the Galois situation.

There are $J_p = d/(e_p f_p)$ primes $\mathfrak p | p$, and for $\mathbf x \in \mathbb R^{J_p}$ we define $$\|\mathbf m \|_p = f_p e_p \sum_{j=1}^J |m_j |.$$ Except for the factor of $f_p e_p$, this is just the 1-norm of $\mathbf m$. It follows that $H_p(\mathfrak x_p) = p^{\| \mathbf m(\mathfrak x_p) \|_p},$ where $\mathbf m(\mathfrak x_p) = \left(m_{\mathfrak p}(\mathfrak x_p) \right)_{\mathfrak p |p}.$

We have a correspondence between ideals of norm $1$ lying above $p$ and $M_p = \{ \mathbf m : m_1 + \cdots + m_J = 0$ given by $\mathbf m \mapsto \prod_{\mathfrak p | p} \mathfrak p^{e_p m_p}.$ We write $\chi(\mathbf m) := \chi\left(\prod_{\mathfrak p | p} \mathfrak p^{e_p m_p}\right)$. We can therefore reduce our regional twisted zeta function to $$Z_p(\chi; s) = \sum_{\mathbf m \in M_p} \chi(\mathbf m) p^{\| \mathbf m \|_p }.$$

#### Ehrhart-Macdonald Reciprocity

To simplify the situation, suppose $p$ is unramified (there are only finitely many ramified primes) and let $\mathbf f_p = ( f_{\mathfrak p})_{\mathfrak p | p} \in \mathbb Z^{J_p}$, and let $\mathbf f_p^{\perp} = \{\mathbf x \in \mathbb R^{J_p} : \mathbf x \cdot \mathbf f_p = 0 \}$. When $K$ is Galois, $\mathbf f_p^{\perp} = \boldsymbol 1^{\perp}$ where $\boldsymbol 1 = (1, 1, \ldots, 1) \in \mathbb Z^{J_p}.$

In any case $M_p$ is a lattice (free $\mathbb Z$-module) in $\mathbf f_p^{\perp}$. In this case, $\| \cdot \|_p$ is simply the 1-norm. Note that $M_p \otimes_{\mathbb R} = \mathbb R^{J-1}$ and $M_p$ can be taken to be a full rank sub-lattice of the integer lattice in $\mathbb R^{J-1}$.

We define the closed unit ball $B_p$ in $\mathbb R^{J-1}$ by intersecting the unit ball of $\| \cdot \|_p$ in $\mathbb R^J$ with $\mathbf f_p^\perp = \mathbb R^{J-1}$. The open unit ball is denote $B_p^\circ$. The ball of radius $t > 0$ is denoted $\mathbb B_p(t)$. Denote $$L(t) = \# B_p(t) \cap \mathcal M_p \quad \mbox{and} \quad L_\circ(t) = \# B_p(t) \cap \mathcal M_p,$$ denote the number of lattice points in $B_p(t)$ and $B_p^\circ(t)$ respectively.

The Ehrhart series for $B_p$ and $B_p^\circ$ are the generating functions for $B_p(t)$ and $B_p^\circ(t)$ as $t$ takes on positive integer values. That is, $$E(z) = 1 + \sum_{t=1}^{\infty} L(t) z^t \quad \mbox{and} E_\circ(t) \sum_{t=1}^{\infty} L_\circ(t) z^t.$$. Note that $$E(z) – E_\circ(z) = \sum_{\mathbf m \in M_p} z^{\| \mathbf m \|_p}.$$

Theorem (Ehrhart-Macdonald Reciprocity}: $E(z)$ and $E_{\circ}(z)$ are rational functions satisfying $E(1/z) = (-1)^{J-1} E_{\circ}(z)$.

This is useful because the (non-twisted) regional zeta function is $$Z_p(1, s) = E(p^{-s}) – E_{\circ}(p^{-s})$$. There is a more general form of Ehrhart-Macdonald reciprocity, called Stanley reciprocity, and from this we can likewise get a functional equation for the twisted regional $\zeta$-functions.

Theorem: $Z_p(\chi; s)$ is a rational function in $p^{-s}$ satisfying $Z_p(\chi; s) = (-1)^{J-1} Z_p(\overline \chi,-s)$.

#### $Z_p(\chi; s)$ as a Non-archimedean Integral

Finally we write $Z_p(\chi; s)$ as an integral over a non-archimedean space. Give $\mathfrak p | p$ let $K_{\mathfrak p}$ be the completion of $K$ with respect to $| \cdot |_{\mathfrak p}$. Let $\mu^{\times}_{\mathfrak p}$ be Haar measure on $K_{\mathfrak p}^\times$ normalized so that the local group of units $U_{\mathfrak p}$ has $\mu^\times_{\mathfrak p}(U_{\mathfrak p}) = 1.$ We define $\mu^{\times}_p = \prod_{\mathfrak p | p} \mu^{\times}_p$ to be product measure on $\prod_{\mathfrak p | p} K_{\mathfrak p}^\times.$ The maximal ideal in $K_{\mathfrak p}$ is the closure of $\mathfrak p$, which we denote again by $\mathfrak p$. The local integers in $K_{\mathfrak p}$ can be denote $\mathfrak p^0$, and the group of units is $U_{\mathfrak p} = \mathfrak p^0 \setminus \mathfrak p^1.$ More generally we set $U_{\mathfrak p}^n$ to be the $n$th annulus given by $U^n_{\mathfrak p} = \mathfrak p^{n-1} \setminus \mathfrak p^n.$

For each $\mathfrak p | p$ we select an annulus $U_{\mathfrak p}^{n_{\mathfrak p}}$ and we define $D_p$ to be the subset of $prod_{\mathfrak p | p} K_{\mathfrak p}^\times.$ given by $$D_p = \bigsqcup_{\mathbf n \cdot \mathbf f_p = 0} \prod_{\mathfrak p | p} U_{\mathfrak p}^{\mathbf n_{\mathfrak p}}.$$ where the union is over all vectors of non-negative integers $\mathbf n = (n_{\mathfrak p})_{\mathfrak p | p}$ orthogonal to $\mathbf f_p$.

Theorem: $$Z_p(\chi; s) = \int_{D_p} \bigg\{ \prod_{\mathfrak p | p} \chi_{\mathfrak p}(x_{\mathfrak p}) \max\{| x_{\mathfrak p} |_{\mathfrak p}, | x_{\mathfrak p} |^{-1}_{\mathfrak p} \}^{-2s} \bigg\} \, d\mu^{\times}_p(\mathbf x).$$