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(%Edited after abx comment%)

I seek explicit linear integral representations $\rho: Aut(X,\omega) \to Sp_{2g}(\mathbb{Z})$, when $(X,\omega)$ is complex $g$-dimensional PPAV. I prefer explicit matrix representations because I want to compute some orbits on vector spaces.

A list of interesting automorphism groups (with integral symplectic representations) of principally polarized abelian varieties in real dimensions 4, 6, 8 would be useful. These automorphism groups are, of course, related to maximal finite subgroups of the integral symplectic groups $Sp(\mathbb{Z}^4, \omega)$, $Sp(\mathbb{Z}^6, \omega)$, etc.

Some lists and representations are available for dimension 2 (abelian surfaces) in Birkenhake-Lange's book "Complex Abelian Varieties", c.f. $\S 13.4$ and 13.4.4,5,9. Reportedly there is some sort of table available for abelian surfaces in Kenji Ueno's paper ``On fibre spaces of normally polarized abelian varieties of dimension 2", J. Fac. Science, Univ. Tokyo, Sect. IA 18 37-95 1971 . However I cannot locate Ueno's paper online.

The comment of @abx below is useful: in low dimensions, PPAVs are Jacobians of Riemann surfaces:

For complex surfaces, the Bolza surface is the maximally symmetric genus 2 Riemann surface with automorphism group $\approx PGL(2,3)$. Here my question amounts to:

Question: can anybody specify an explicit faithful linear representation of $PGL(2,3) \to Sp(\mathbb{Z}^4, \omega)$ ?

For complex dimension 3, the Appendix 1 of Conway/Sloane in Buser/Sarnak's "On the period matrix of a Riemann surface of large genus" is useful: the automorphism group of the Barnes-Wall lattice $A_6 ^{(2)}$ is the ``most interesting" automorphism group, and my question amounts to:

Question: Can anybody specify an explicit faithful linear representation of $Aut(A_6 ^{(2)}) \to Sp(\mathbb{Z}^6, \omega)$ ?

For complex dimension 4, the $E_8$ lattice has the largest automorphism group, and here our question amounts to:

Question: Can anybody specify an explicit faithful linear representation of $Aut(E_8) \to Sp(\mathbb{Z}^8, \omega)$ ?

I am searching through the various Atlas/GAP/MAGMA/SAGE databases on finite group representations, but have not yet located the desired representations.

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    $\begingroup$ 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. $\endgroup$
    – abx
    Commented Feb 2, 2019 at 15:05
  • $\begingroup$ @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. $\endgroup$
    – JHM
    Commented Feb 2, 2019 at 17:26
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    $\begingroup$ 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. $\endgroup$
    – abx
    Commented Feb 3, 2019 at 14:16

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