Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures
| dc.creator | Bergström, Jonas | |
| dc.creator | Faber, Carel | |
| dc.creator | van der Geer, Gerard | |
| dc.date | 2008-03-06 | |
| dc.date | 2008-04-20 | |
| dc.date.accessioned | 2026-07-07T09:33:18Z | |
| dc.date.available | 2026-07-07T09:33:18Z | |
| dc.description | We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the étale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the contribution of the Eisenstein cohomology together with its S_6-action for the full level 2 structure and on the basis of our computations we make precise conjectures on the endoscopic contribution. We also make a prediction about the existence of a vector-valued analogue of the Saito-Kurokawa lift. Assuming these conjectures that are based on ample numerical evidence, we obtain the traces of the Hecke-operators T(p) for p < 41 on the remaining spaces of `genuine' Siegel modular forms. We present a number of examples of 1-dimensional spaces of eigenforms where these traces coincide with the Hecke eigenvalues. We hope that the experts on lifting and on endoscopy will be able to prove our conjectures. | |
| dc.description | Added a section on Harder's conjectural congruences. Some minor changes. 16 pages | |
| dc.identifier | https://arxiv.org/abs/0803.0917 | |
| dc.identifier | http://arxiv.org/abs/0803.0917 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/159088 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Number Theory | |
| dc.subject | 11F46; 11G18; 14G35; 14J15; 20B25 | |
| dc.title | Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures | |
| dc.type | text |