# relationship between pairings on principally polarized abelian varieties

Let $X$ be a $g$-dimensional principally polarized abelian variety over $\mathbb{C}$, for example the jacobian of a curve of genus $g$. Let $X = \mathbb{C}^{g}/\Lambda$ where $\Lambda$ is a full $\mathbb{Z}$-lattice. Then we may examine three alternating 2-forms on $\Lambda$ or $n\Lambda/\Lambda$ which are apparently closely related:

The intersection pairing on $\Lambda$ (viewing $X$ as a complex manifold and viewing elements of $\Lambda$ as homology classes)

The Riemann form $e_{\lambda}$ associated to the polarization $\lambda$ which takes integer values on $\Lambda$

The Weil pairing, also denoted $e_{\lambda}$, defined on $X_{n} \times \hat{X}_{n} \cong n\Lambda/\Lambda \times n\Lambda/\Lambda$ (applying the polarization to the second argument).

From what I am told, these are related in the following ways: by choosing a simplectic basis for the Riemann form, we get an intersection pairing, and (as seems much less obvious to me), the Weil pairing can be considered as the exponential of the Riemann form on $n\Lambda$ (which is 1 everywhere in $\Lambda$ itself). The problem is that I haven't been able to find a discussion of this in any of the main literature, not even Mumford's Abelian Varieties. Where can I find confirmation and proofs of this?

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Have you tried checking J. S. Milne's notes? Maybe Chapter I, section 13? jmilne.org/math/CourseNotes/av.html –  Parsa Oct 14 '11 at 20:31
The Weil pairing should surely be defined on $\frac{1}{n}\Lambda/\Lambda$, or perhaps $\Lambda/n\Lambda$, rather than $n\Lambda/\Lambda$. –  Martin Orr Nov 25 '11 at 14:17