relationship between pairings on principally polarized abelian varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T21:36:34Z http://mathoverflow.net/feeds/question/77876 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77876/relationship-between-pairings-on-principally-polarized-abelian-varieties relationship between pairings on principally polarized abelian varieties Jeff 2011-10-11T23:21:08Z 2011-10-14T15:31:42Z <p>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:</p> <p>The <em>intersection pairing</em> on $\Lambda$ (viewing $X$ as a complex manifold and viewing elements of $\Lambda$ as homology classes)</p> <p>The <em>Riemann form</em> $e_{\lambda}$ associated to the polarization $\lambda$ which takes integer values on $\Lambda$</p> <p>The <em>Weil pairing</em>, 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).</p> <p>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 <em>Abelian Varieties</em>. Where can I find confirmation and proofs of this?</p>