3
$\begingroup$

Let $(T,E)$ be a polarized abelian variety ($T=V/L$, $\dim_\mathbb{C} V=g$, $E:V\times V\to\mathbb{R}$ a nondegenerate real alternating bilinear form, with $E(L\times L)\subseteq\mathbb{Z}$ and $E(iv,iw)=E(v,w)$). I'm trying to prove the existence of a basis for $L$ such that the matrix of $E$ has the form $E=\left(\begin{array}{cc}0&D\\-D&0\end{array}\right),$ where $D$ is a diagonal matrix with diagonal $d_1,\ldots,d_g$, and $d_i\mid d_{i+1}$.

Now what I've tried so far is considering $\min\{E(v,w):v,w\in L,E(v,w)>0\}$, and taking $e_1,e_{g+1}$ that achieve this minimum. Then I would like to decompose $V$ into $V=\langle e_1,e_{g+1}\rangle\oplus\langle e_1,e_{g+1}\rangle^\perp$ (this can be done since $E$ is nondegenerate). I would like to repeat this same process in $\langle e_1,e_{g+1}\rangle^\perp$, but I have no way of knowing "how many" lattice points are in $\langle e_1,e_{g+1}\rangle^\perp$. Am I attacking this problem the wrong way, or is there something I'm missing?

  • 0
    Thanks, you made it a lot clearer. I'm able to complete the proof now.2011-09-06

0 Answers 0