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.

[Rubin] Rubin, K., Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer.

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

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Taekyung Kim is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.
  • 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. – Chris Wuthrich Dec 7 at 13:09
  • 1
    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 $\mu_4$. – Chris Wuthrich Dec 7 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)). – Taekyung Kim Dec 9 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. – Taekyung Kim Dec 9 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).

  • Should be a comment, but the formatting didn't work well. – literature-searcher Dec 7 at 11:21

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