4
$\begingroup$

I begin by defining the Weil pairing in general (as in Oda's 1969 paper). My question is about an explicit formula for this pairing in the case of an elliptic curve over complex numbers.

Let $\lambda\colon A \rightarrow B$ be an isogeny between abelian schemes over a base scheme $S$ and let $\lambda^*\colon \mathrm{Pic}^0_{B/S} \rightarrow \mathrm{Pic}^0_{A/S}$ be the dual isogeny. Define the Weil pairing $$< , >_{\lambda}\colon\mathrm{Ker}(\lambda) \times \mathrm{Ker}(\lambda^*) \rightarrow \mathbb{G}_m$$ as follows. For a line bundle $\mathscr{L} \in \mathrm{Ker}(\lambda^*)(S)$, choose an isomorphism $$ \alpha\colon \lambda^* \mathscr{L} \xrightarrow{\sim} \mathscr{O}_{A},$$ note that for an $S$-point $x \in \mathrm{Ker}(\lambda)$ one has $\lambda^* = T_x^* \lambda^*$ canonically, where $T_x$ is translation by $x$, and define $<x, \mathscr{L}>_{\lambda}$ to be the element of $\mathbb{G}_m(S)$ that gives rise to the composite isomorphism $$\mathscr{O}_{A} \xrightarrow{\alpha^{-1}} \lambda^* \mathscr{L} = T_x^* \lambda^* \mathscr{L} \xrightarrow{T_x^*\alpha} T_x^* \mathscr{O}_A = \mathscr{O}_A.$$

Now suppose that $S = \mathrm{Spec} (\mathbb{C})$ and $\lambda$ is the multiplication by $n$ isogeny on an elliptic curve. The above pairing then is (by definition) the Weil pairing $$ E[n] \times E[n] \rightarrow (\mathbb{G}_m)_{\mathbb{C}}.$$ Suppose that $E$ is presented via a complex uniformization: $E = \mathbb{C}/(\mathbb{Z} + \mathbb{Z} \cdot (x + iy))$ with $y > 0$. My question is: why does the Weil pairing agree with the pairing $$ \left(\frac{a}{n}, \frac{b}{n} \cdot (x + iy)\right) \mapsto e^{-\frac{2\pi i}{n} ab}$$ (say, as opposed to its complex conjugate or some other pairing).

P.S. I am not interested in suggestions to use a different "easier" definition of the Weil pairing that works only for elliptic curves over fields (unless, of course, the suggestion is accompanied by a proof of the agreement between the pairings). The point is to understand the correct definition, not to avoid using it.

$\endgroup$
  • 1
    $\begingroup$ This is clearly explained in §24 of Mumford's Abelian varieties. $\endgroup$ – abx Feb 13 '15 at 16:36
1
$\begingroup$

In the case of Jacobian varieties, the Weil pairing has an alternative (and beautiful) description in terms of Weil reciprocity for curves, and also in terms of Hilbert symbols. For elliptic curves, the Weil reciprocity part is an exercise in Silverman's book (#3.16). For general Jacobians (and both Weil reciprocity & Hilbert symbols) see Howe's paper "The Weil pairing and the Hilbert symbol" (Math Ann 305, 1996).

$\endgroup$
  • 1
    $\begingroup$ Beware that the autoduality $\phi$ in Silverman's book that underlies his definition of the Weil self-pairing is not a polarization (in the intrinsic sense that the $(1,\phi)$-pullback of the Poincare bundle is anti-ample rather than ample), whereas the definition in Mumford's book (as recommended by abx) does rest on polarizations. Likewise with Jacobians one has to be careful about the definitions to ensure one is working with a principal polarization rather than the negative of one. $\endgroup$ – user74230 Feb 14 '15 at 6:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.