TITLE: Conceptualizing Weil Pairing for elliptic curves ( and number fields) 
QUESTION [22 upvotes]: There are two explanations in Silverman ( Arithmetic of Elliptic Curves), one in exercises developing the Weil reciprocity law ( for algebraic curves) and then generalizing, and then there is a different, somewhat computational (in my opinion) proof in one of the chapters. 
[I should also point out that in case of elliptic curves over complex numbers there is a rather simple description of this pairing in terms of determinant of a matrix!, see Ribet Stein : Hecke Operators ... for example] 
While I do understand the proofs, I have heard that there is a conceptual explanation of this pairing?
Is there a uniform construction for the Hilbert Symbols in number field case, which is again a "form" of Weil pairing?

REPLY [13 votes]: Here is how I like to understand the Weil pairing.
The dual of an abelian variety $A$ is the scheme $\hat{A} = \mathrm{Hom}(A, B\mathbf{G}_m)$.  Here $B\mathbf{G}_m$ is the stack of line bundles and $\mathrm{Hom}$ refers to homomorphisms of group stacks.  One therefore has a perfect pairing
$A \times \hat{A} \rightarrow B\mathbf{G}_m$.
This actually means something relatively concrete:  For each pair of points $(a, a') \in A \times \hat{A}$ one has a one dimensional vector space $L(a,a')$, together with isomorphisms
$L(a_1 + a_2, a') \simeq L(a_1, a') \otimes L(a_2, a')$,
$L(a, a'_1 + a'_2) \simeq L(a, a'_1) \otimes L(a, a'_2)$
satisfying some compatibility conditions (you can find the details of this definition under the heading of biextensions; there is a nice explanation in SGA7).  One of these compatibilities is that the two isomorphisms
$L(a_1 + a_2, a'_1 + a'_2) \simeq L(a_1, a'_1) \otimes L(a_1, a'_2) \otimes L(a_2, a'_1) \otimes L(a_2, a'_2)$
should coincide.
One consequence of the definition is that $L(0, a') = L(0 + 0, a') \simeq L(0, a') \otimes L(0, a')$ which means that $L(0, a')$ is canonically trivialized, as is $L(a, 0)$ by symmetry.  Moreover, the two trivializations of $L(0,0)$ must be the same.
If we choose an $n$-torsion point $a$ of $A$ then $L(a, a')$ will be a line bundle with a trivialization of its $n$-th tensor power.  Similarly, if $a'$ is also an $n$-torsion point then $L(a,a')^{\otimes n}$ will come with two trivializations coming from its identification with the canonically trivialized line bundles $L(na, a') = L(0,a')$ and $L(a, na') = L(a,0)$.  Comparing these two trivializations, we get an element of $\mathbf{G}_m$ and therefore a pairing
$A[n] \times \hat{A}[n] \rightarrow \mathbf{G}_m$.
However, we notice that the induced trivializations of $L^{\otimes n^2} \simeq L(na, na')$ must coincide.  Therefore the image of this map actually lands in $\mathbf{G}_m[n] = \mu_n$.  This gives the Weil pairing
$A[n] \times \hat{A}[n] \rightarrow \mu_n$.
I don't know of a way to interpret the Hilbert symbol in quite this way, but I'd be very interested if someone could suggest one!