Arithmetic theory of Siegel modular forms

1987 ◽  
pp. 101-108 ◽  
Author(s):  
Gerd Faltings
Keyword(s):  
Modular Forms ◽  
Siegel Modular Forms ◽  
Arithmetic Theory

scholarly journals Higher pullbacks of modular forms on orthogonal groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Brandon Williams
Keyword(s):  
Modular Forms ◽  
Differential Operators ◽  
Jacobi Form ◽  
Siegel Modular Forms ◽  
Orthogonal Groups

Abstract We apply differential operators to modular forms on orthogonal groups O ⁢ ( 2 , ℓ ) {\mathrm{O}(2,\ell)} to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature ( 2 , 2 ) {(2,2)} and ( 2 , 3 ) {(2,3)} , for which there are interpretations as Hilbert–Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

scholarly journals Lost chapters in CHL black holes: untwisted quarter-BPS dyons in the ℤ2 model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Fabian Fischbach ◽  
Albrecht Klemm ◽  
Christoph Nega
Keyword(s):  
String Duality ◽  
Modular Forms ◽  
Black Hole Entropy ◽  
Siegel Modular Forms ◽  
Partition Functions ◽  
Twisted Sector ◽  
Duality Symmetry ◽  
Iwahori Subgroup ◽  
M Theory ◽  
Bonnet Term

Abstract Motivated by recent advances in Donaldson-Thomas theory, four-dimensional $$ \mathcal{N} $$ N = 4 string-string duality is examined in a reduced rank theory on a less studied BPS sector. In particular we identify candidate partition functions of “untwisted” quarter-BPS dyons in the heterotic ℤ2 CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields meromorphic Siegel modular forms for the Iwahori subgroup B(2) ⊂ Sp4(ℤ) which generate BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition functions are shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as the black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our results are also compared with recently conjectured formulae of Bryan and Oberdieck for the partition functions of primitive DT invariants of the CHL orbifold X = (K3 × T2)/ℤ2, as suggested by string duality with type IIA theory on X.

Mixed Siegel Modular Forms and Special Valuesof Certain Dirichlet Series

2000 ◽  
Vol 131 (2) ◽  
pp. 109-122 ◽  
Author(s):  
YoungJu Choie ◽  
Min Ho Lee
Keyword(s):  
Dirichlet Series ◽  
Modular Forms ◽  
Siegel Modular Forms
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