The ring of modular forms of degree two in characteristic three
- PDF / 265,436 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 71 Downloads / 148 Views
Mathematische Zeitschrift
The ring of modular forms of degree two in characteristic three Gerard van der Geer1,2 Received: 28 March 2020 / Accepted: 4 September 2020 © The Author(s) 2020
Abstract We determine the structure of the ring of Siegel modular forms of degree 2 in characteristic 3. Mathematics Subject Classification 11F03 · 14J15 · 14G35 · 11G18
1 Introduction Let Ag be the moduli space of principally polarized abelian varieties of dimension g. It is a Deligne-Mumford stack over Z. It carries a natural vector bundle of rank g, the Hodge bundle Eg . We write L for its determinant line bundle. The vector bundle Eg extends in a natural way over any compactification A˜ g of Faltings-Chai type and we will denote the extension of Eg and L again by the same symbols. Sections of L ⊗k over A˜ g are called modular forms of weight k. It is known that for g ≥ 2 any section of L k over Ag extends to a section of L k over A˜ g , a fact usually referred to as the Koecher principle, see [7, Prop. 1.5, p. 140]. If F = Z or Z p or a field one has the graded ring Rg (F) = ⊕k H 0 (A˜ g ⊗ F, L k ) .
It is known by [7] that it is a finitely generated F-algebra. In the case of F = C the ring Rg (C) is the ring of scalar-valued Siegel modular forms of degree g. It is well-known known that R1 (C) = C[E 4 , E 6 ] is freely generated over C by the Eisenstein series E 4 and E 6 of weights 4 and 6. In the 1960s Igusa [11] determined the structure of R2 (C): 2 R2 (C) = C[ψ4 , ψ6 , χ10 , χ12 , χ35 ]/ χ35 −P , where the indices of the generators indicate the weights and P is a polynomial in ψ4 , ψ6 , χ10 and χ12 . Moreover, the ideal of cusp forms is generated by χ10 , χ12 and χ35 . For g = 3,
B
Gerard van der Geer [email protected]
1
Korteweg-de Vries Instituut, Universiteit van Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
2
Université du Luxembourg, Unité de Recherche en Mathématiques, 4364 Esch-sur-Alzette, Luxembourg
123
G. van der Geer
Tsuyumine showed in [20] that R3 (C) is generated by 34 elements; recently the number of generators was reduced to 19 by Lercier and Ritzenthaler [14]. For F = F p , a finite field with p elements, the ring R1 (F p ) was described by Deligne [5]. Besides giving the structure of the ring over Z R1 (Z) = Z[c4 , c6 , ]/ c43 − c62 − 1728 , he showed that R1 (F2 ) = F2 [a1 , ]
and R1 (F3 ) = F3 [b2 , ] ,
where is of weight 12 and a1 (resp b2 ) is of weight 1 (resp. 2). For p ≥ 5 we have R1 (F p ) = F p [c4 , c6 ]. For g = 2, Igusa determined in [13] also the ring of modular forms over Z; it is generated by elements of weight 4, 6, 10, 12, 12, 16, 18, 24, 28, 30, 35, 36, 40, 42, 48 . For finite fields the structure of R2 (F p ) is known for p ≥ 5. For this we refer to Ichikawa’s paper [10]. For p ≥ 5 the ring is just as in characteristic zero generated by modular forms 2 = P(ψ , ψ , χ , χ ). Moreover ψ4 , ψ6 , χ10 , χ12 and χ35 with χ35 satisfying a relation χ35 4 6 10 12 for p ≥ 5 the reduction map R2 (Z p ) → R2 (F p ) is surjective. Nagaoka studied
Data Loading...
