Jacobians among Abelian threefolds: a formula of Klein and a question of Serre
| dc.creator | Lachaud, Gilles | |
| dc.creator | Ritzenthaler, Christophe | |
| dc.creator | Zykin, Alexey | |
| dc.date | 2008-02-27 | |
| dc.date.accessioned | 2026-07-07T09:23:35Z | |
| dc.date.available | 2026-07-07T09:23:35Z | |
| dc.description | Let k be a field and f be a Siegel modular form of weight h \geq 0 and genus g>1 over k. Using f, we define an invariant of the k-isomorphism class of a principally polarized abelian variety (A,a)/k of dimension g. Moreover when (A,a) is the Jacobian of a smooth plane curve, we show how to associate to f a classical plane invariant. As straightforward consequences of these constructions, when g=3 and k is a subfield of the complex field, we obtain (i) a new proof of a formula of Klein linking the modular form χ_{18} to the square of the discriminant of plane quartics ; (ii) a proof that one can decide when (A,a) is a Jacobian over k by looking whether the value of χ_{18} at (A,a) is a square in k. This answers a question of J.-P. Serre. Finally, we study the possible generalizations of this approach for g>3. | |
| dc.description | 15 pages | |
| dc.identifier | https://arxiv.org/abs/0802.4017 | |
| dc.identifier | http://arxiv.org/abs/0802.4017 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/155783 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11G10 ; 14K15 | |
| dc.title | Jacobians among Abelian threefolds: a formula of Klein and a question of Serre | |
| dc.type | text |