EICHLER COHOMOLOGY THEOREM FOR VECTOR-VALUED MODULAR FORMS

2013 ◽  
Vol 09 (07) ◽  
pp. 1765-1788 ◽  
Author(s):  
JOSE GIMENEZ
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Large Integer ◽  
Eichler Cohomology ◽  
Vector Valued

We prove the Eichler cohomology theorem for vector-valued modular forms of large integer weights on the full modular group.

scholarly journals Constructions of vector-valued modular forms of rank four and level one

2020 ◽  
Vol 16 (05) ◽  
pp. 1111-1152
Author(s):  
Cameron Franc ◽  
Geoffrey Mason
Keyword(s):  
Irreducible Representation ◽  
Differential Equations ◽  
Modular Forms ◽  
Modular Group ◽  
Tensor Products ◽  
Two Dimensional ◽  
Minimal Weight ◽  
Graded Module ◽  
Holomorphic Modular Forms ◽  
Vector Valued

This paper studies modular forms of rank four and level one. There are two possibilities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we describe when each case occurs for general choices of exponents for the [Formula: see text]-matrix. In the remaining sections we describe how to write down the corresponding differential equations satisfied by minimal weight forms, and how to use these minimal weight forms to describe the entire graded module of holomorphic modular forms. Unfortunately, the differential equations that arise can only be solved recursively in general. We conclude the paper by studying the cases of tensor products of two-dimensional representations, symmetric cubes of two-dimensional representations, and inductions of two-dimensional representations of the subgroup of the modular group of index two. In these cases, the differential equations satisfied by minimal weight forms can be solved exactly.

PARABOLIC GENERALIZED MODULAR FORMS AND THEIR CHARACTERS

2009 ◽  
Vol 05 (05) ◽  
pp. 845-857 ◽  
Author(s):  
MARVIN KNOPP ◽  
GEOFFREY MASON
Keyword(s):  
Riemann Surface ◽  
Modular Form ◽  
Finite Index ◽  
Modular Forms ◽  
Modular Group ◽  
Compact Riemann Surface ◽  
Eichler Cohomology

We make a detailed study of the generalized modular forms of weight zero and their associated multiplier systems (characters) on an arbitrary subgroup Γ of finite index in the modular group. Among other things, we show that every generalized divisor on the compact Riemann surface associated to Γ is the divisor of a modular form (with unitary character) which is unique up to scalars. This extends a result of Petersson, and has applications to the Eichler cohomology.

Vector-valued Siegel modular forms of weight detk ⊗ Sym(8)

2015 ◽  
Vol 26 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Tomoya Kiyuna
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Siegel Modular Forms ◽  
Vector Valued

We determine the structures of modules of vector-valued Siegel modular forms of weight det k ⊗ Sym (8) with respect to the full Siegel modular group of degree two.

EICHLER COHOMOLOGY THEOREM FOR GENERALIZED MODULAR FORMS

2011 ◽  
Vol 07 (04) ◽  
pp. 1103-1113 ◽  
Author(s):  
WISSAM RAJI
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Large Integer ◽  
Isomorphism Theorem ◽  
Classical Approach ◽  
Negative Weight ◽  
Eichler Cohomology ◽  
Automorphic Integrals

We show starting with relations between Fourier coefficients of weakly parabolic generalized modular forms of negative weight that we can construct automorphic integrals for large integer weights. We finally prove an Eichler isomorphism theorem for weakly parabolic generalized modular forms using the classical approach as in [3].

scholarly journals Fourier Coefficients of Vector-valued Modular Forms of Dimension 2

2014 ◽  
Vol 57 (3) ◽  
pp. 485-494 ◽  
Author(s):  
Cameron Franc ◽  
Geoffrey Mason
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Dimension 2 ◽  
Vector Valued

AbstractWe prove the following theorem. Suppose that F = ( f1, f2) is a 2-dimensional, vectorvalued modular form on SL2(ℤ) whose component functions f1, f2 have rational Fourier coefficients with bounded denominators. Then f1, f2 are classical modular forms on a congruence subgroup of the modular group.

scholarly journals A trace formula for vector-valued modular forms

2015 ◽  
Vol 17 (06) ◽  
pp. 1550069
Author(s):  
P. Bantay
Keyword(s):  
Trace Formula ◽  
Modular Forms ◽  
Weight Distribution ◽  
Hilbert Polynomial ◽  
Free Generators ◽  
Vector Valued

We present a formula for vector-valued modular forms, expressing the value of the Hilbert-polynomial of the module of holomorphic forms evaluated at specific arguments in terms of traces of representation matrices, restricting the weight distribution of the free generators.

scholarly journals Boundary behaviour of special cohomology classes arising from the Weil representation

2012 ◽  
Vol 12 (3) ◽  
pp. 571-634 ◽  
Author(s):  
Jens Funke ◽  
John Millson
Keyword(s):  
Modular Forms ◽  
Symmetric Spaces ◽  
Theta Functions ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Boundary Behaviour ◽  
Orthogonal Groups ◽  
Local Coefficients ◽  
Generating Series ◽  
Vector Valued

AbstractIn our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

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