The Fourier expansion of Hecke operators for vector-valued modular forms
type: Item
instance of: scholarly article
title: The Fourier expansion of Hecke operators for vector-valued modular forms
zbMATH Open document ID: 1398.11073
author: Oliver Stein
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2018-04-20T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1434650879
review text: To discuss and obtain the main results of the paper, the author first reviews theorems about the Weil transformation and vector-valued modular forms transforming with the mentioned transformation. Then, he recalls how Hecke operators on vector-valued modular forms can be defined. He considers cases of even and odd signatures of the corresponding lattice \(L\). Particularly if it is odd, it suffices to consider vector valued modular forms of half-integral weight. The author studies general Gauss sums too. The sums and explicit formulas for them are not new, but the paper obtains more simple and more convenient form of the sums for the paper's necessary evaluations. He defines and studies Hecke operators of special type. He provides an explicit formula for the Fourier expansion of the Hecke operator \(T(p^2l)\) for the prime \(p\) and a positive integer \(l\) for the Weil representation associated to a lattice \(L\), i.e., in more generalized conditions than it was obtained earlier.
reviewed by: Guram Gogishvili
Mathematics Subject Classification ID: 11F25
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 11E08
zbMATH DE Number: 6862260
zbMATH Keywords: Fourier expansion
zbMATH Keywords: Hecke operators
zbMATH Keywords: vector-valued modular forms
zbMATH Keywords: Weil representation
zbMATH Keywords: Gauss sums
MaRDI profile type: Publication
OpenAlex ID: W1555956412
cites work: Q4382830
cites work: Automorphic forms with singularities on Grassmannians
cites work: Reflection groups of Lorentzian lattices
cites work: Borcherds products on \(O(2,l)\) and Chern classes of Heegner divisors
cites work: Heegner divisors, \(L\)-functions and harmonic weak Maass forms
cites work: Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms
cites work: The Weil representation and Hecke operators for vector valued modular forms
cites work: Q4780771
cites work: Q4219187
cites work: Q3151402
cites work: The theory of Jacobi forms
cites work: Die irreduziblen Darstellungen der Gruppen \(SL_2(Z_p)\), insbesondere \(SL_2(Z_2)\). II
cites work: On the classification of automorphic products and generalized Kac-Moody algebras
cites work: The Weil Representation of and Some Applications
cites work: Q5625265
cites work: On modular forms of half integral weight
cites work: The special values of the zeta functions associated with Hilbert modular forms
cites work: On construction of holomorphic cusp forms of half integral weight
cites work: Hecke operators and vector-valued modular forms associated with the Weil representation
cites work: Weil representations associated with finite quadratic modules
DOI: 10.7169/FACM/2015.52.2.4
description: scientific article; zbMATH DE number 6862260