scholarly journals Mathieu moonshine and Siegel Modular Forms

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Suresh Govindarajan ◽  
Sutapa Samanta
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Conjugacy Classes ◽  
Genus Two ◽  
Obvious Manner

Abstract A second-quantized version of Mathieu moonshine leads to product formulae for functions that are potentially genus-two Siegel Modular Forms analogous to the Igusa Cusp Form. The modularity of these functions do not follow in an obvious manner. For some conjugacy classes, but not all, they match known modular forms. In this paper, we express the product formulae for all conjugacy classes of M24 in terms of products of standard modular forms. This provides a new proof of their modularity.

scholarly journals On the computation of the determinant of vector-valued Siegel modular forms

2014 ◽  
Vol 17 (A) ◽  
pp. 247-256 ◽  
Author(s):  
Sho Takemori
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Vector Valued

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}A^{0}(\Gamma _{2})$ denote the ring of scalar-valued Siegel modular forms of degree two, level $1$ and even weights. In this paper, we prove the determinant of a basis of the module of vector-valued Siegel modular forms $\bigoplus _{k \equiv \epsilon \ {\rm mod}\ {2}}A_{\det ^{k}\otimes \mathrm{Sym}(j)}(\Gamma _{2})$ over $A^{0}(\Gamma _{2})$ is equal to a power of the cusp form of degree two and weight $35$ up to a constant. Here $j = 4, 6$ and $\epsilon = 0, 1$. The main result in this paper was conjectured by Ibukiyama (Comment. Math. Univ. St. Pauli 61 (2012) 51–75).

scholarly journals Covariants of binary sextics and vector-valued Siegel modular forms of genus two

2017 ◽  
Vol 369 (3-4) ◽  
pp. 1649-1669 ◽  
Author(s):  
Fabien Cléry ◽  
Carel Faber ◽  
Gerard van der Geer
Keyword(s):  
Modular Forms ◽  
Siegel Modular Forms ◽  
Genus Two ◽  
Vector Valued

On Siegel Modular Forms of Genus Two

1962 ◽  
Vol 84 (1) ◽  
pp. 175 ◽  
Author(s):  
Jun-Ichi Igusa
Keyword(s):  
Modular Forms ◽  
Siegel Modular Forms ◽  
Genus Two

scholarly journals Sturm’s operator for scalar weight in arbitrary genus

2017 ◽  
Vol 13 (10) ◽  
pp. 2677-2686 ◽  
Author(s):  
Kathrin Maurischat
Keyword(s):  
Modular Forms ◽  
Projection Operator ◽  
Siegel Modular Forms ◽  
Arbitrary Genus ◽  
Genus Formula ◽  
Genus Two ◽  
Holomorphic Projection

In contrast to the well-known cases of large weights, Sturm’s operator does not realize the holomorphic projection operator for lower weights. We prove its failure for arbitrary Siegel modular forms of genus [Formula: see text] and scalar weight [Formula: see text]. This generalizes a result for genus two in [K. Maurischat and R. Weissauer, Phantom holomorphic projections arising from Sturm’s formula, preprint (2016)].

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