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I think the

The unifying picture you're looking for is probably most transparent the other way around -- by re-writing the Weil pairing on elliptic curves (in fact, this works more generally for Jacobians) to make it look like Hilbert symbols. Indeed, once you view the Weil pairing as a class-field-theoretic construction and pass it through the standard function-field-to-number-field analogy, you get exactly the Hilbert symbols. This is made very explicit in, for example, Everett Howe's "The Weil Pairing and the Hilbert Symbol." With notation in the paper, compare the Weil pairing formula

\begin{equation*} e_m([X],[Y])=\prod_{p}(-1)^{m(\text{ord}_P(D))(\text{ord}_P(E))}\frac{g^{\text{ord}_P(D)}}{f^{\text{ord}_P(E)}}(P) \end{equation*}

(here, $X$ and $Y$ are $m$-torsion divisors on the Jacobian of a curve with $mX=div(f)$ and $mY=div(g)$, with $P$ running over geometric points of the curve) with Schmidt's formula for the Hilbert symbol, reveals a striking similarity.

I'm not sure if I have anything coherent to say about an improved conceptual explanation other than that the class-field-theoretic approach makes the Weil pairing appear as a natural and canonical construction, whereas the standard divisor construction feels rather ad hoc at first.

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I think the unifying picture you're looking for is most transparent the other way around -- by re-writing the Weil pairing on elliptic curves (in fact, this works more generally for Jacobians) to make it look like Hilbert symbols. Indeed, once you view the Weil pairing as a class-field-theoretic construction and pass it through the standard function-field-to-number-field analogy, you get exactly the Hilbert symbols. This is made very explicit in, for example, Everett Howe's "The Weil Pairing and the Hilbert Symbol."

I'm not sure if I have anything coherent to say about an improved conceptual explanation other than that the class-field-theoretic approach makes the Weil pairing appear as a natural and canonical construction, whereas the standard divisor construction feels rather ad hoc at first.