Finite linear groups, lattices, and products of elliptic curves
| dc.creator | Popov, Vladimir L. | |
| dc.creator | Zarhin, Yuri G. | |
| dc.date | 2005-05-26 | |
| dc.date | 2005-09-20 | |
| dc.date.accessioned | 2026-07-07T07:49:32Z | |
| dc.date.available | 2026-07-07T07:49:32Z | |
| dc.description | Let $V$ be a finite dimensional complex linear space and let $G$ be an irreducible finite subgroup of $\GL(V)$. For a $G$-invariant lattice $Λ$ in $V$ of maximal rank, we give a description of structure of the complex torus $V/Λ$. In particular, we prove that for a wide class of groups, $V/Λ$ is isogenous to a self-product of an elliptic curve, and that in many cases $V/Λ$ is isomorphic to a product of mutually isogenous elliptic curves with complex multiplication. We show that there are $G$ and $Λ$ such that the complex torus $V/Λ$ is not an abelian variety but one can always replace $Λ$ by another $G$-invariant lattice $Δ$ such that $V/Δ$ is a product if elliptic curves with complex multiplication. We amplify these results with a criterion, in terms of the character and the Schur $\mathbf Q$-index of $G$-module $V$, of the existence of a nonzero $G$-invariant lattice in $V$. | |
| dc.description | 25 pages. Several examples are added | |
| dc.identifier | https://arxiv.org/abs/math/0505571 | |
| dc.identifier | http://arxiv.org/abs/math/0505571 | |
| dc.identifier | J. Algebra 305 (2006), no. 1, 562--576. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/124888 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Group Theory | |
| dc.subject | 14K20; 14K22; 22E40; 32J18; 22E40 | |
| dc.title | Finite linear groups, lattices, and products of elliptic curves | |
| dc.type | text |