Root number of the $L$-function of $y^2 = x^3 - n^2x$ and $n \pmod 8$.

Question

Root number definition. Let $E_n$ be the elliptic curve $y^2 = x^3 - n^2 x$ where $n$ is a positive squarefree integer. It is known that the $L$-function of $E_n$, denoted $L(E_n,s)$, can be extended to an entire function $\Lambda(s)$ satisfying the functional equation $$ \Lambda (s) = \epsilon(E_n) \cdot \Lambda(2-s) $$ The number $\epsilon(E_n) = \pm 1$ is called the root number.

Question. It is known that the root number of $E_n$ can be simply expressed as $$ \epsilon(E_n) = \begin{cases} 1 & \text{ if } n \equiv 1, 2, 3 \pmod 8 \\ -1 & \text{ if } n \equiv 5, 6, 7 \pmod 8 \end{cases} $$

Is there not-too-difficult proof of this question? If anyone has a proof or a reference, that would be greatly appreciated!

I have found a Theorem mentioning this on p84 of Koblitz's book Introduction to Elliptic Curves and Modular Forms, Second edition. However, that proof is mostly about proving the existence of the analytic extension of $L(E_n, s)$ to $\Lambda(s)$, and I also cannot find where he proves the statement about the root number.

There is also this post on MO: https://mathoverflow.net/q/157631/167513. It related the root number of $E$ with the quadratic twist $E^D$. If we let $E: y^2 = x^3 - x$, then $E^D: y^2 = x^3 - D^2 x$. I have found that conductor $N_E = 32$. So if one could find an expression for $$ \psi_D(-N_E) = \psi_D(-32) $$ where $\psi_D$ is the quadratic character of $\mathbb{Q}(\sqrt D)$, then it would solve the question for $2 \nmid D$.

Answer 1

Determining root numbers is not easy in general. Here are some general approaches.

For a specific elliptic curve:

• approximate the $L$-function and use this to numerically determine the root number for a specific elliptic curves

• express it as a product of local root numbers, which you can compute with local theory

• determine the associated modular form, and compute its root number, say using Atkin-Lehner theory

The specific curves you are interested in are quite special:

1. They are CM, so their L-functions factor as a product of Dirichlet L-functions. Thus the epsilon factors can be determined by Dirichlet epsilon factors.

2. They are a family of quadratic twists. It is relatively easy to determine how root numbers vary in quadratic twists (though it is more complicated if the conductor of the twist is not coprime to the conductor of your starting curve). This makes use of writing the root number in terms of local root numbers.

However proving any of these things requires more background in elliptic curves and modular forms. But if you just want to check what the root number is for specific curves, you can use a computational algebra package like Sage or Magma, or look things up in tables: e.g., see the LMFDB entry for $y^2 = x^3-x$.