scholarly journals Generators for modules of vector-valued Picard modular forms

2013 ◽  
Vol 212 ◽  
pp. 19-57 ◽  
Author(s):  
Fabien Cléry ◽  
Gerard Van Der Geer
Keyword(s):  
Modular Forms ◽  
Unitary Group ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Picard Modular Forms ◽  
Vector Valued

AbstractWe construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2, 1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.

scholarly journals Generators for modules of vector-valued Picard modular forms

2013 ◽  
Vol 212 ◽  
pp. 19-57
Author(s):  
Fabien Cléry ◽  
Gerard Van Der Geer
Keyword(s):  
Modular Forms ◽  
Unitary Group ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Picard Modular Forms ◽  
Vector Valued

AbstractWe construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2, 1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.

scholarly journals Products of vector valued Eisenstein series

2017 ◽  
Vol 29 (1) ◽  
Author(s):  
Martin Westerholt-Raum
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Congruence Subgroups ◽  
Level 1 ◽  
Vector Valued ◽  
Level Aspect

AbstractWe prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin–Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context their previously mentioned result.

A TRACE FORMULA FOR HECKE OPERATORS ON VECTOR-VALUED MODULAR FORMS

2016 ◽  
Vol 59 (1) ◽  
pp. 143-165
Author(s):  
NICOLE RAULF ◽  
OLIVER STEIN
Keyword(s):  
Trace Formula ◽  
Modular Forms ◽  
Hecke Operators ◽  
Weil Representation ◽  
Cusp Forms ◽  
Dimension Formula ◽  
Integral Weight ◽  
Vector Valued

AbstractWe present a ready to compute trace formula for Hecke operators on vector-valued modular forms of integral weight for SL2(ℤ) transforming under the Weil representation. As a corollary, we obtain a ready to compute dimension formula for the corresponding space of vector-valued cusp forms, which is more general than the dimension formulae previously published in the vector-valued setting.

scholarly journals LOCAL BORCHERDS PRODUCTS FOR UNITARY GROUPS

2017 ◽  
Vol 234 ◽  
pp. 139-169
Author(s):  
ERIC HOFMANN
Keyword(s):  
Boundary Component ◽  
Unitary Group ◽  
Picard Group ◽  
Unitary Groups ◽  
Weil Representation ◽  
Cusp Forms ◽  
Modular Variety ◽  
Arithmetic Subgroup ◽  
Torsion Element ◽  
Vector Valued

For the modular variety attached to an arithmetic subgroup of an indefinite unitary group of signature $(1,n+1)$, with $n\geqslant 1$, we study Heegner divisors in the local Picard group over a boundary component of a compactification. For this purpose, we introduce local Borcherds products. We obtain a precise criterion for local Heegner divisors to be torsion elements in the Picard group, and further, as an application, we show that the obstructions to a local Heegner divisor being a torsion element can be described by certain spaces of vector-valued elliptic cusp forms, transforming under a Weil representation.

scholarly journals Hecke Operators on Vector-Valued Modular Forms

2019 ◽  
Author(s):  
Vincent Bouchard ◽  
◽  
Thomas Creutzig ◽  
Aniket Joshi ◽  
◽  
...  
Keyword(s):  
Modular Forms ◽  
Hecke Operators ◽  
Vector Valued

A magnetic modular form

2019 ◽  
Vol 15 (05) ◽  
pp. 907-924
Author(s):  
Yingkun Li ◽  
Michael Neururer
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Hecke Operators ◽  
Shimura Lift ◽  
Divisibility Property ◽  
Vector Valued

In this paper, we prove a conjecture of Broadhurst and Zudilin concerning a divisibility property of the Fourier coefficients of a meromorphic modular form using the generalization of the Shimura lift by Borcherds and Hecke operators on vector-valued modular forms developed by Bruinier and Stein. Furthermore, we construct a family of meromorphic modular forms with this property.

SYMMETRIC SQUARE L-FUNCTIONS AND SHAFAREVICH–TATE GROUPS, II

2009 ◽  
Vol 05 (07) ◽  
pp. 1321-1345 ◽  
Author(s):  
NEIL DUMMIGAN
Keyword(s):  
Modular Forms ◽  
Prime Order ◽  
Galois Representations ◽  
Critical Value ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Fourier Expansions ◽  
Symmetric Square ◽  
Vector Valued ◽  
Siegel Cusp Form

We re-examine some critical values of symmetric square L-functions for cusp forms of level one. We construct some more of the elements of large prime order in Shafarevich–Tate groups, demanded by the Bloch–Kato conjecture. For this, we use the Galois interpretation of Kurokawa-style congruences between vector-valued Siegel modular forms of genus two (cusp forms and Klingen–Eisenstein series), making further use of a construction due to Urban. We must assume that certain 4-dimensional Galois representations are symplectic. Our calculations with Fourier expansions use the Eholzer–Ibukiyama generalization of the Rankin–Cohen brackets. We also construct some elements of global torsion which should, according to the Bloch–Kato conjecture, contribute a factor to the denominator of the rightmost critical value of the standard L-function of the Siegel cusp form. Then we prove, under certain conditions, that the factor does occur.

scholarly journals Concomitants of ternary quartics and vector-valued Siegel and Teichmüller modular forms of genus three

2020 ◽  
Vol 26 (4) ◽  
Author(s):  
Fabien Cléry ◽  
Carel Faber ◽  
Gerard van der Geer
Keyword(s):  
Representation Theory ◽  
Modular Form ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Cusp Forms ◽  
Local Systems ◽  
Hyperelliptic Locus ◽  
Vector Valued

Abstract We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmüller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichmüller cusp forms on $$\overline{\mathcal {M}}_{g}$$ M ¯ g and the middle cohomology of symplectic local systems on $${\mathcal {M}}_{g}\,$$ M g . In genus 3, we make this explicit in a large number of cases.

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