# Local root numbers of the Hecke character associated with some specific CM elliptic curves, should they be some roots of unity?

## TL;DR.

Some local root numbers of the Hecke character associated with our specific CM elliptic curve by $$\mathbf{Q}(i)$$ seem to have value in $$\mu_4$$. But apparently our computation via Rohrlich's local root number formula says otherwise. What am I missing here?

## Settings.

Let $$K = \mathbf{Q}(\sqrt{-1})$$ the quadratic ground field and $$E$$ an elliptic curve over $$K$$ having complex multiplication by the ring of integers $$O_K$$ of $$K$$ defined by an affine Weierstrass equation $$\begin{equation*} y^2 = x^3 - Dx \end{equation*}$$ for some $$D \in O_K$$. To $$E/K$$ we attach a Hecke character $$\psi=\psi_{E/K} = \prod' \psi_v: \mathbf{I}_K/K^\times \to \mathbf{C}^\times$$ as usual (c.f. e.g. [Sil], sections II.9--10), of conductor $$\mathfrak{f}$$ which have the same prime factors as the conductor of the elliptic curve $$E/K$$.

The local root number at $$v$$ of the Hecke character ($$\psi$$ in our case) is defined as the signature of its epsilon-factor of the Weil--Deligne representation induced from $$\psi_v$$, by the local Artin map. The formulae for the local root number are also well-known thanks to Rohrlich [Rohr], pp. 32--33. In particular, when $$v$$ is a finite and ramified place (i.e.~$$v$$ divides the conductor $$\mathfrak{f}$$) not dividing $$2$$ (i.e.~$$v$$ is relatively prime to the different ideal of $$K/\mathbf{Q}$$), the formula for $$\chi = \psi_v$$ can read: $$\begin{equation*} W(\chi) = \chi_{u}(\beta) \cdot q^{-a(\chi)/2} \cdot \sum_{x \in (O_v/\mathfrak{f}_v)^\times} \chi^{-1} (x) \exp (2 \pi i \operatorname{tr}^{K_v}_{\mathbf{Q}_p}(x/ \beta) ), \end{equation*}$$ where

• $$\beta \in K_v^\times$$ is any element with $$v(\beta) = a(\chi)$$,
• $$a(\chi)$$ is the conductor exponent of $$\psi$$ at the place $$v$$,
• $$q$$ is the size of the residue field of $$K$$ modulo $$v$$,
• $$\chi_v$$ denotes the unitary part of the character $$\chi$$.

## Property 1.

Let $$\ell$$ be a rational prime that splits completely in $$K$$ and $$v \not\mid \ell$$ be any finite prime of $$K$$. If $$V_\ell E$$ is the $$\ell$$-adic Tate module of $$E$$ over $$\mathbf{Q}_\ell$$, then we have $$V_\ell E \otimes_{\mathbf{Q}_\ell} \mathbf{C} \cong \psi_v \oplus \psi_v$$. See e.g. [Rubin], Proposition 5.4 and Theorem 5.15(ii).

## Property 2.

It is well-known that the infinity type $$\psi_\infty: \mathbf{C}^\times \to \mathbf{C}^\times$$ of $$\psi$$ is given by $$z \mapsto z^{-1}$$, so that for any $$\alpha \in K_{\mathfrak{f},1}$$ (the trivial ray), we have $$\psi(\alpha O_K) = \psi_\infty^{-1}(\alpha)$$ (of course with a fixed embedding $$K \hookrightarrow \mathbf{C}$$).

From now on, we assume $$p$$ is a rational prime $$\equiv 1 \pmod{4}$$ so that $$p O_K = \mathfrak{p} \overline{\mathfrak{p}}$$. Since $$K$$ has class number $$1$$, we can find generators $$\pi$$ and $$\overline\pi$$ in $$O_K$$ of $$\mathfrak{p}$$ and of $$\overline{\mathfrak{p}}$$, respectively, so that $$p = \pi \overline\pi$$. We may and do assume that $$\pi$$ and $$\overline\pi$$ are primary, i.e. $$\equiv 1 \pmod{2(1+i)}$$. Put $$D = \pi$$ and we may assume $$E$$ has good reduction modulo $$(1+i)$$. The last condition is surely satisfied when $$D = 1 + 2i$$, etc. This implies that our Hecke character $$\psi$$ is of conductor $$\mathfrak{f} = \mathfrak{p}$$. We also take $$\beta = p$$.

## Property 3.

Recall that $$\chi = \psi_v: K_v^\times \to \mathbf{C}^\times$$ is the local part of the Hecke character $$\psi$$. From [Rohr], Proposition 2.1, we can see that \begin{align*} \chi | O_K^\times &= \epsilon^{-1} \\ \chi(\pi) &= \psi_\infty^{-1}(\pi) \end{align*} since there are no other primes dividing $$\mathfrak{f}$$ except $$\mathfrak{p}$$. As values of the $$\epsilon$$-type lie in the group of 4th roots of unity $$\mu_4$$ (recall $$O_K/\mathfrak{f} \cong O_K/\mathfrak{p} \cong \mathbf{F}_5$$), we see that $$\begin{equation*} \chi_u(\beta) = \chi_u(\pi) \chi_u(\bar{\pi}) \equiv \psi_\infty^{-1}(\pi)/|\psi_\infty^{-1}(\pi)| = \pi/|\pi| \pmod{\mu_4}. \end{equation*}$$

## Property 4.

Write the Gauss sum part'' in the formula of $$W(\chi)$$ as $$G$$, i.e. $$\begin{equation*} G = \sum_{x \in (O_v/\mathfrak{f}_v)^\times} \chi^{-1} (x) \exp (2 \pi i \operatorname{tr}^{K_v}_{\mathbf{Q}_p}(x/ \beta) ) \end{equation*}$$ In our case, this is just $$\begin{equation*} G = \sum_{x \in \mathbf{F}_5^\times} \epsilon(x) \exp ( 2 \pi i x / p). \end{equation*}$$ Note that there are only four characters $$\epsilon: \mathbf{F}_5^\times \to \mathbf{C}^\times$$. When $$\epsilon$$ is the quartic residue symbol, then it is well-known that $$G^2 = \pm\pi \sqrt{p}$$.

## Question.

From Property 1, since the root number of elliptic curves is in $$\{ \pm 1 \}$$, it follows that the root number of $$\chi = \psi_v$$ lies in the group $$\mu_4$$. However, from the properties 3--4, we see that $$\begin{equation*} W(\chi) \equiv \frac{\pi}{\sqrt{p}} \cdot \frac{1}{\sqrt{p}} \cdot \sqrt{\pi} \sqrt[4]{p} \pmod{\mu_4} \neq 1 \pmod{\mu_4} \end{equation*}$$ and hence $$W(\chi) \not\in \mu_4$$. Why do such kinds of inconsistency occur? We will be very gratitude if someone find out any flaw/mistake in our consideration.

## References.

[Rohr] Rohrlich, D. E., Root numbers.

[Sil] Silverman, Advanced topics in the arithmetic of elliptic curves.

• Why do you expect the root number of $\psi$ to be $\pm1$? The $L$-function of $E$ is a product of two $L$-functions, one of $\psi$ and its conjugate. Dec 7, 2018 at 13:09
• Oh sorry, my bad. Ignore my first comment. I think $V_{\ell} E$ is $\psi \oplus \bar\psi$ as a $G_{\mathbb{Q}}$ module. Your reference is about $G_F$-modules where $E$ has cm defined over $F$. Now the root numbers will be complex conjugates that multipy to the root number of $E$, which is $\pm 1$, but I don't think there is a reason that they are in $\mu_4$. Dec 7, 2018 at 14:24
• @ChrisWuthrich Thank you very much for your comment. In our situation, the elliptic curve and the corresponding character $\psi$ is defined over $K$. The property $V_\ell E \cong \psi_v \oplus \psi_v$ remains valid in this case, because the $\ell^n$-torsion points of $E$ decompose into $E[L^n] \oplus E[L'^n]$ where $L, L'$ are the two primes in $K$ lying over $\ell$ ([Rubin] Proposition 5.4 and the Chinese Remainder Theorem) and the Galois action on these components are given by $\psi_v$ ([Rubin] Theorem 5.15(ii)). Dec 9, 2018 at 7:38
• @ChrisWuthrich For example once Cesnavicius used this argument before (Proof of Proposition 6.3). Since the root number of $E/K$ is $\pm 1$, we expect that the root number of $\psi_v$ must be in $\mu_4$. I suspect that we surely make mistakes somewhere though. Dec 9, 2018 at 7:44

As a specific example, take $$D=5$$ and the Magma command RootNumbers(Grossencharacter(EllipticCurve([-5,0]))); gives the primes above 5 as having root numbers as $$\pm0.52573+0.85065i$$, whose product is $$-1$$ (and are not roots of unity).