On Deligne's determinant of motives

Question

This is a question about Deligne's conjecture on special values of L-functions. I have to confess that I've never understood the definition of the determinant which is supposed to give the right special value up to some power of $2\pi i$ and some rational number. Can anybody work out the following example?

Let $X$ be a (smooth projective) curve of genus $g$ over $\mathbb{Q}$ and $M=h^1(X)$ the Chow motive corresponding to $H^1(X)$. The factor at infinity is simply

$L_\infty(M, s)=[2 (2\pi)^{-s}\Gamma(s)]^g$

and the dual of $M$ being $h^1(X)(1)$, one has

$L_\infty(M^\vee, s)=L_\infty(M, s+1)=[2(2\pi)^{-s}\Gamma(s+1)]^g$

By definition, an integer $n$ is critical if neither $L_\infty(M, s)$ nor $L_\infty/M^\vee, 1-s)$ have poles at $s=n$. So the only critical integer is $s=1$. Deligne's conjecture says that $L(M, 1)$ is a rational multiple of $c^\pm(M)$.So...

What is $c^\pm(M)$ in this case?

Can one give a explicit description in terms of a basis of $H^0(X, \Omega^1_X)$ or $H^1(X, \mathcal{O}_X)$ and the homology $H_1(X(\mathbb{C}), \mathbb{Q})$?

What is known about the conjecture in this case?

Answer 1

You can compute $c^{\pm}(M)$ in this case by pairing $H_1(X(\mathbf{C}),\mathbf{Q})^{\pm}$ with $\Omega^1(X)$, which are all $\mathbf{Q}$-vector spaces of dimension $g$. Here $(H_1)^{\pm}$ means the $(\pm 1)$-eigenspace with respect to the action of complex conjugation on $X(\mathbf{C})$.

Moreover, the motive $M=h^1(X)$ coincides with the motive of the Jacobian $J$ of $X$. So Deligne's conjecture on $L(h^1(X),1)$ can be cast in terms of $J$, and in fact it amounts to the BSD conjecture for $J$ (more precisely, the BSD conjecture up to a non-zero rational factor). Note that the BSD conjecture is wide open for general abelian varieties over $\mathbf{Q}$.