Local root number

Question

I am reading about the L-functions of elliptic curves and I was thinking about the root number as the product of local root numbers. So my question is how to think about the local root numbers geometrically or arithmetically. I have also read that even though the functional equation is conjectural (in different cases) but root number is a well-defined concept. Is it somehow related to the l-adic representation attached to my objects?

Answer 1

Yes, if $X$ is a variety over an extension $K$ of $\mathbb Q_p$, then the $\ell$-adic cohomology spaces $H^i(X,\mathbb Q_{\ell})$ are $\ell$-adic representations of $G_{K}$, which give rise to Weil--Deligne representations. (See Tate's Corvallis article, for example.) The resulting Weil--Deligne representation is conjectured to be independent of the choice of $\ell$ (as long as $\ell \neq p$). This is known when $X$ has a smooth proper model over the ring of integers of $K$ (although from the point of view of root numbers this case is not so interesting; in this good reduction situation the associated Weil--Deligne representation is unramified, so the local root number is 1, for the right choice of additive character). It is also known when $X$ is an elliptic curve (in which case $i = 1$ is the interesting choice; this gives the contragredient of the Tate module).

Actually computing root numbers is a non-trivial business, especially for instances of very bad reduction at small primes. For examples, see recent work by Mazur and Rubin, and by the Dokchitsers.

[Added in response to Arijit's comment below:]

One of the most interesting applications of root numbers is when you can prove that the product of the local root numbers (which is the global root number) of an elliptic curve is -1. Then BSD predicts that there will be a rational point of infinite order.

In the Heegner point situation studied by Gross--Zagier, one is in this context (the global root number of the elliptic curve over a quad. imag. field is -1), and if the order of vanishing is precisely one, they produce a point of infinite order. In general we don't have the technology yet to control Mordell--Weil groups, but can control Selmer groups (which morally should be the same thing, since, following Shafarevic and Tate, one conjectures that Sha has finite order, and hence that the Selmer rank and Mordell--Weil rank coincide).

In the work of Mazur and Rubin, and also of Nekovar, and of the Dokchitsers, the goal is, under assumptions which makes the global root number -1, to produce the predicted rank in the Selmer group. (They succeed in doing in this in many cases!) Perhaps developing some understanding of this question, and some sense of what people are doing about it, will help give you motivation. After all, it's pretty surprising that just by computing a sign, you can predict whether a Diophantine equation will have an interesting solution, and even more surprising that you can prove non-trivial results in this direction!