The origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation)

Question

The motivation for this question is the same as in my previous question in MO: https://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over

I am just curious to know the origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation $f(s)=w(C/ℚ)f(2-s)=εf(2-s)$) of the curve $C$. I read several papers on this topic, but I cannot find where this root number come from. I wonder if this number has some relation with $a$ and $b$ in the equation of the curve $C$.

Answer 1

I am not sure I understand your question. The functional equation $f(s)=\epsilon f(1-s)$ applied twice yields $\epsilon^2=1$, hence $\epsilon=\pm 1$.

Now, finding the root number $\epsilon$ of a self-dual modular $L$-function can be a tricky business, and the meaning of "finding" is of course subjective. It is known that $$ \epsilon =i^k\eta, $$ where $k$ is the weight of the underlying modular form (i.e. $k=2$ for a rational elliptic curve), and $\eta=\pm 1$ is the eigenvalue of the Atkin-Lehner involution on the form. If the level $N$ (i.e. the conductor for a rational elliptic curve) is square-free, then $$ \eta=\mu(N)\lambda(N)N^{1/2}, $$ where $\lambda(N)$ is the $N$-the Hecke eigenvalue of the form. In the case of a rational elliptic curve $E$, one can readily express $\lambda(N)$ from the number of points on $E$ over $\mathbb{F}_p$ for the various primes $p$ dividing $N$. If $N$ is not square-free, then I am not aware of any way to express the root number from Hecke eigenvalues.

There exist algorithms to find the root number of an elliptic curve. They usually proceed by calculating the local root numbers and taking their product (which is the global root number). See for example Helfgott's paper On the behaviour of root numbers in families of elliptic curves how to do this in general.

Answer 2

In the specific context of elliptic curves $C$ over (any finite extension of) $\mathbf{Q}$, the root number has been analysed by

Rohrlich (David), Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), no. 3, 311–349. Numdam.

Kobayashi (Shin-ichi), The local root number of elliptic curves with wild ramification, Math. Ann. 323 (2002), no. 3, 609–623. Springer.

and

Dokchitser (Tim) & Dokchitser (Vladimir), Root numbers of elliptic curves in residue characteristic 2, Bull. Lond. Math. Soc. 40 (2008), no. 3, 516–524. Oxford Journals.