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Let say you have two number fields, that are tamely ramified, and suppose that the $p$-part of their Dedekind zeta functions coincide for all prime $p$ which is ramified in either field. Suppose additionally that both fields have the same number of real and complex embeddings, that is to say that the infinity parts of their complete Dedekind zeta functions also agree.


i) Is is true that both fields have the same local root numbers for all $p$?

ii) What about normalized local root numbers?

In the case quadratic fields both questions have positive answers, in fact under the conditions imposed the fields turn out to be isomorphic. In the cubic case I can show that ii) has a positive answer so, as explained just below, the same is true about i).

Addendum: It turns out i) and ii) are equivalent; the point is that the tame ramification hypothesis plus the conditions on the ramified local factors imply that both fields have the same discriminant, and this in turn implies that the normalized local root numbers are equal iff the local root numbers are equal.

In hope to be very precise about what I'm asking, and in case my terminology is not standard, below there is a more thorough version of the questions I just posed.

Questions. Let $K,L$ be tamely ramified number fields. Suppose that $L_{\ell}(s,\rho_{K})=L_{\ell}(s,\rho_{L})$ for all prime $\ell$ that ramifies in either $K$ or $L$, and for $\ell=\infty$.

i) Does it follow that $W_{p}(\rho_{K})=W_{p}(\rho_{L})$ for all $p$?

ii)Same question about normalized root numbers; do we have that $\displaystyle \frac{W_{p}(\rho_{K})}{W_{p}(\det \rho_{K})}=\frac{W_{p}(\rho_{L})}{W_{p}(\det \rho_{L})}$ for all $p$?

Brief reminder of notation:

Let $L$ be a number field with Galois closure $\tilde{L}$. Let $G(L):=\text{Gal}(\tilde{L}/\mathbb{Q})$ and $H(L):=\text{Gal}(\tilde{L}/L)$. By composing the natural map $G_{\mathbb{Q}} \to G(L)$ with the natural action $G(L) \to \text{Sym}(G(L)/H(L))$ one gets a permutation representation $\pi_{L} \in H^{1}(G_{\mathbb{Q}}, S_{\text{deg}(L)})$, in other words $\pi_{L} =\text{Inf}_{G(L)}^{G_{\mathbb{Q}}}(\text{Ind}_{H(L)}^{G(L)}1)$. The usual inclusions $S_{\text{deg}(L)} \hookrightarrow \text{O}_{\text{deg}(L)}(\mathbb{C}) \hookrightarrow \text{GL}_{\text{deg}(L)}(\mathbb{C})$, together with $\pi_{L}$, yield an Artin representation $\rho_{L} \in H^{1}_{cont}(G_{\mathbb{Q}}, \text{GL}_{\text{deg}(L)}(\mathbb{C}))$. The Artin L-function $L(s,\rho_{L})$ associated to $\rho_{L}$, which in general is defined as a product of local functions where the local parts are defined by restricting $\rho_{L}$ to a decomposition subgroup $G_{\mathbb{Q}_{p}}$, is nothing else but the Dedekind zeta function $\zeta_{L}(s)$. The local factors $L_{p}(s,\rho_{L})$ are given by $\displaystyle \prod_{i=1}^{g}\left(\frac{1}{1-(p^{-s})^{f_{i}}}\right)$, where the $f_{i}'s$ are the residue degrees of a rational prime $p$ in its decomposition in $L$. There is also a local infinite factor $L_{\infty}(s,\rho_{L})$ which in this case is equal to $\Gamma_{\mathbb{R}}^{r_1}(s)\Gamma_{\mathbb{C}}^{r_2}(s)$ where $r_{1}$ (resp $r_{2}$) is the number of real (resp complex) embeddings of $L$ and the functions $\Gamma_{\mathbb{R}}, \Gamma_{\mathbb{C}}$ are fixed complex functions, defined in terms of the usual Gamma function, independent of the field $L$. The completed Artin L-function associated to $\rho_{L}$ is defined by $\Lambda(s,\rho_{L}):= A(\rho_{L})^{s/2}L_{\infty}(s,\rho_{L})L(s,\rho_{L})$, where $A(\rho_{L})$ is a positive integer divisible only by the primes ramifying in $L$. The completed L-function satisfies a functional equation $\Lambda(s,\rho_{L})= W(\rho_{L})\Lambda(1-s,\rho^{\vee}_{L})$, where $\rho^{\vee}_{L}$ is the contragradient representation and $W(\rho_{L})$ is the root number of $\rho_{L}$. Since $\rho_{L}$ is an orthogonal representation it is a result of Fr\"ohlich and Queyrut that $W(\rho_{L})=1$ hence the usual functional equation of the Dedekind zeta function. It is known that the root number can be written as product of local root numbers $\displaystyle W(\rho_{L})=\prod_{p}W_{p}(\rho_{L})$. The local root numbers $W_{p}(\rho_{L})$ are complex numbers of norm $1$, and if $\rho_{L}$ is unramified at $p$ then $W_{p}(\rho_{L})=1.$ The normalized local root numbers of $\rho_{L}$ are $\frac{W_{p}(\rho_{L})}{W_{p}(\det \rho_{L})}.$

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