Let $K$ be a number field of degree $d$ over $\mathbb Q$. Let $\mathfrak F$ be the group of fractional ideals of $K$, and let $\mathcal N$ and $\mathfrak N$ be respectively the elements of absolute norm 1 in $K$ and the fractional ideals of norm 1 of $K$. We will use primes to denote principal ideals, so that for instance $\mathfrak N’$ is the subgroup of principal ideals of norm 1. The ring of integers of $K$ will be denote $\mathcal O = \mathfrak o$ depending on whether we want to view it as a set of elements, or as a fractional ideal. The group of units will be denoted $\mathcal U$ and the group of roots of unity (which we will prompty ignore) will be denoted $\mathcal W$.

**Claim:** If $\xi \in \mathcal N$ there exists a unit $u \in \mathcal U$, an algebraic integer $\alpha$ and rational $r > 0$ such that $r \xi = u \alpha$.

*Proof.* Set $\mathfrak x = \xi \mathcal O$ and write $\mathfrak x = \prod_p \mathfrak x_p$, where $\mathfrak x_p$ consists of the prime factors of $\mathfrak x$ that lie over the same rational prime $p$. In this case, $$\mathfrak x_p = \prod_{\mathfrak p | p} \mathfrak p^{r_{\mathfrak p}}$$ where $\mathbf r_p = (r_{\mathfrak p})_{\mathfrak p | p}$ is a vector of integers. It should also be remarked that $\mathbf r_p \neq \boldsymbol 0$ for only finitely many rational primes. The norm-1 condition implies that the $\mathbf r_p$ dotted with the vector of inertial degrees $\mathbf f_p = (f_{\mathfrak p})_{\mathfrak p | p}$ is equal to 0. Write $\mathbf s_p = ( f_{\mathfrak p} e_{\mathfrak p} )_{\mathfrak p | p}$ where $e_{\mathfrak p}$ is the ramification degree of $\mathfrak p | p$. Then, consider $$\{\mathbf r_p + \ell \mathbf s_p : \ell \in \mathbb Z, r_{\mathfrak p} + s_{\mathfrak p} \geq 0 \}.$$ Let $\ell_p$ be the minimum of this set. In which case, $p^{\ell_p} \mathfrak x_p $ is an integral ideal in $\mathcal O$. Set $r = \prod_p p^{\ell p} \in \mathbb Q$. Then $r \mathfrak x$ is an integral ideal in the same ideal class as $\mathfrak x$. Finally, $\mathfrak x = \xi \mathcal O$ is principal, in which case there exists $\alpha \in \mathfrak O$ such that $r \xi \mathcal O = \alpha \mathcal O$ and the claim follows by choosing the appropriate unit. $\square$.

**Definition:** With $\xi \in \mathcal N$ if $\alpha \in \mathcal O$ is as in the proof of the previous claim, then we call $\alpha$ a *visible integer* for $\xi$. We denote the set of visible integers by $\mathcal V$. A principal ideal $\mathfrak a = \alpha \mathcal O$ generated by a visible integer is called a *visible ideal*. Every generator of $\mathfrak x = \xi \mathcal O$ has the same visible ideal, and so we say $\mathfrak a$ is the visible ideal for $\mathfrak x$. We define the *morm* of $\mathfrak x$, $\mathbb M\mathfrak x$ to be the (ideal) norm of $\mathfrak a$, and we define the *morm* of $\xi$, $M(\xi)$ to be the (field) norm of $\xi$ over $\mathbb Q$. That is $$\mathbb M \mathfrak x = \mathbb N \mathfrak a; \qquad M(\xi) = N_{K/\mathbb Q}(\alpha).$$ Given $r \in [0, \infty)$ we define $\mathfrak N'(r) = \{ \mathfrak x \in \mathfrak N’ : \mathbb M \mathfrak x \leq r \}$ to be the *morm-exhaustion* of $\mathfrak N’$.

**Claim:** If $\alpha$ is a visible integer, then the norm of $\alpha$ is the $d$th power of a rational integer.

*Proof.* With $\alpha, \xi$ and $r$ as in the previous claim, $N_{K/\mathbb Q}(\alpha) = N_{K/\mathbb Q}(r \xi) = r^d.$ The claim follows because $N_{K/\mathbb Q}$ is a rational integer.

**Definition:** If $n > 1$ is a rational integer not equal to $0, 1, -1$, and $\alpha$ is a visible integer, then $n \alpha$ is called a *hidden* integer. The corresponding integral ideal $n \mathfrak a = n \alpha \mathcal O$ is called a *hidden* ideal. An integer or integral ideal which is not visible and not hidden is said to be *invisible*. Non-invisible ideals can be extended to the group of non-invisible fractional ideals $\mathfrak R$. The quotient group $\mathrm{Inv} = \mathfrak F/\mathfrak R$ is called the *invisibility group* of $K$.

The *local* invisibility group over $p$ is the subgroup of $\mathbb Z/d \mathbb Z$ given by $$\mathrm{Inv}_p = \{ \mathbf a \cdot \mathbf f_p \bmod d : \mathbf a \in \mathbb Z^{J_p} \}.$$ Here $J_p$ is the number or primes $\mathfrak p | p$. $p$ is inert exactly when $J_p = 1$, in which case $\mathrm{Inv}_p = \{ 0 \}.$ In simple situations, for instance when $K$ is Galois, the $\mathrm{Inv}_p$ can be explicitly determined in terms of the splitting classes of $p$.

**Claim:** $\mathrm{Inv}$ is the restricted direct sum of the $\mathrm{Inv}_p$ over the rational primes, with respect to the identity subgroups $\{0\} \in \mathrm{Inv}_p$. That is, given any fractional ideal $\mathfrak c$, there exists a rational $r$ and a vector of non-negative integers indexed by the rational primes $(m_p)$, with $0 \leq m_p < d$ and $m_p = 0$ for all but finitely many $p$, such that $\mathbb N (r \mathfrak c) = \prod_p p^{m_p}$.

For each $p$ and $r \in \mathrm{Inv}_p$ we select a representative $\mathbf n_p(r)$ such that $\mathbf f \cdot \mathbf n_p(r) \equiv r \bmod d$. We write $\mathfrak n_p(r) = \prod_{\mathfrak p | p} \mathfrak p^{n_{\mathfrak p}(r)}$. If $\mathbf r = (r_p) \in \mathrm{Inv}$, then we define $$\mathfrak n(\mathbf r) = \prod_p \mathfrak n_p(r_p),$$ and call $\mathfrak n(\mathbf r)$ a *representative ideal* for the invisibility class $\mathbf r$. We take the representative ideal for the trivial invisibility class $\mathbf 0$ to be the ring of integers $\mathfrak 0$.

Claim: If $\gamma \in K^{\times}$ is such that $\gamma \mathcal O \not \in \mathfrak R$ (that is, if $\gamma \not in \mathfrak R’$) then there exists $\mathfrak x \in \mathfrak N$ and a rational number $r \in \mathbb Q$ such that $\gamma \mathcal O =

### Characters on the Torus Containing $\mathfrak N’$.

The principal ideals of norm 1, $\mathfrak N’$ embeds as a countable set in a torus $S$ of dimension $r_1 + r_2 – 1$, where $r_1$ is the number of real embeddings of $K$ and $r_2$ is the number of complex conjugate embeddings. This torus is a quotient of $(\mathbb R^{\times})^{r^1} \times (\mathbb C^{\times})^{r_2}$ which we call the *multiplicative Minkowski space* in which $K^\times$ embeds densely. Given a character $\chi$ on $S$, we wish to lift our character to a character $\widetilde \chi$ on multiplicative Minkowski space. We do this by first lifting $\chi$ to character on $\mathcal R’$ by specifying that $\chi(r \mathfrak x) = \chi(\mathfrak x)$ for all $r \in \mathbb Q^{\times}$. That is, if $\widehat \chi$ is trivial on $\mathbb Q{\times}$ then it will have the appropriate projective property that allows us to extend the $\chi$ from ideals of norm 1 to the associated visible ideal and all integral ideals that are hidden by it.

The morm $L$-series for $K$ is defined to be $$L_K(s; \chi) = \sum_{\mathfrak x \in \mathfrak N’} \frac{\chi(\mathfrak x)}{\mathbb M \mathfrak x^{-s}} = \frac{1}{\zeta(ds)} \sum_{\mathfrak a \in \mathfrak R’} \frac{\chi(\mathfrak a)}{\mathbb N\mathfrak a^{-s}},$$ where the last sum is over principal integral ideals in $\mathfrak R$ (i.e. non-invisible, principal integral ideals).