Timeline for List of Automorphism groups of Abelian Varieties for Dummies
Current License: CC BY-SA 4.0
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| Feb 3, 2019 at 14:16 | comment | added | abx | I don't see why there should be a representation $\operatorname{Aut}(E_{8})\rightarrow \operatorname{Sp}(8,\mathbb{Z}) $. The group which is the automorphism group of a 4-dimensional principally polarized abelian variety is the centralizer of $i$ in $\operatorname{Aut}(E_{8})$ (see the Appendix 2 in the paper you quote). It is much smaller than $\operatorname{Aut}(E_{8})$. Same in genus 3. | |
| Feb 3, 2019 at 13:45 | history | edited | JHM | CC BY-SA 4.0 |
elaborated question following abx comments.
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| Feb 2, 2019 at 17:26 | comment | added | JHM | @abx yes thank you that is useful observation. For complex dimension 2, i'm basically looking for representation of $GL(2,3)$ into $Sp(\mathbb{Z}^4)$. For $\dim=3$ a representation of the automorphisms of the $A_6^2$ lattice, and for $\dim=4$ want automorphisms of $E_8$ lattice. Will read Conway/Sloane closer. | |
| Feb 2, 2019 at 15:05 | comment | added | abx | In (complex) dimension $2$ and $3$, a principally polarized abelian variety is a Jacobian or a product of Jacobians. Using the Torelli theorem, you get directly the complete list in dimension 2 form Birkenhake-Lange, 11.7. It is possible, but probably rather tedious, to do the same in dimension 3, using the (known) classification of automorphisms of genus 3 curves. I believe that this is already hopeless in dimension 4. | |
| Feb 2, 2019 at 14:17 | history | asked | JHM | CC BY-SA 4.0 |