Hecke operators and non congruence subgroups

Group Theory ◽  
1989 ◽  
pp. 215-224
Keyword(s):  
Hecke Operators ◽  
Congruence Subgroups

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.

scholarly journals Vectorial Drinfeld modular forms over Tate algebras

2018 ◽  
Vol 14 (06) ◽  
pp. 1729-1783 ◽  
Author(s):  
Federico Pellarin ◽  
Rudolph B. Perkins
Keyword(s):  
Modular Forms ◽  
Hecke Operators ◽  
Roots Of Unity ◽  
Congruence Subgroups ◽  
Complete Structure ◽  
Drinfeld Modular Forms ◽  
Special Case

In this paper, we develop the theory of vectorial modular forms with values in Tate algebras introduced by the first author, in a very special case (dimension two, for a very particular representation of [Formula: see text]). Among several results that we prove here, we determine the complete structure of the modules of these forms, we describe their specializations at roots of unity and their connection with Drinfeld modular forms for congruence subgroups of [Formula: see text] and we prove that the modules generated by these forms are stable under the actions of Hecke operators.

scholarly journals An elementary proof of the Eichler–Selberg trace formula

2020 ◽  
Vol 2020 (762) ◽  
pp. 105-122 ◽  
Author(s):  
Alexandru A. Popa ◽  
Don Zagier
Keyword(s):  
Trace Formula ◽  
Modular Forms ◽  
Modular Group ◽  
Algebraic Property ◽  
Double Coset ◽  
Kernel Functions ◽  
Hecke Operators ◽  
Congruence Subgroups ◽  
The Right ◽  
Number Relation

AbstractWe give a purely algebraic proof of the trace formula for Hecke operators on modular forms for the full modular group{\mathrm{SL}_{2}(\mathbb{Z})}, using the action of Hecke operators on the space of period polynomials. This approach, which can also be applied for congruence subgroups, is more elementary than the classical ones using kernel functions, and avoids the analytic difficulties inherent in the latter (especially in weight two). Our main result is an algebraic property of a special Hecke element that involves neither period polynomials nor modular forms, yet immediately implies both the trace formula and the classical Kronecker–Hurwitz class number relation. This key property can be seen as providing a bridge between the conjugacy classes and the right cosets contained in a given double coset of the modular group.

scholarly journals Twisted 6d (2, 0) SCFTs on a circle

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Zhihao Duan ◽  
Kimyeong Lee ◽  
June Nahmgoong ◽  
Xin Wang
Keyword(s):  
Lie Groups ◽  
Point Of View ◽  
Partition Functions ◽  
Congruence Subgroups ◽  
Gauge Groups ◽  
Simple Lie Groups ◽  
Instanton Contribution ◽  
Elliptic Genera ◽  
Supersymmetric Gauge ◽  
The One

Abstract We study twisted circle compactification of 6d (2, 0) SCFTs to 5d $$ \mathcal{N} $$ N = 2 supersymmetric gauge theories with non-simply-laced gauge groups. We provide two complementary approaches towards the BPS partition functions, reflecting the 5d and 6d point of view respectively. The first is based on the blowup equations for the instanton partition function, from which in particular we determine explicitly the one-instanton contribution for all simple Lie groups. The second is based on the modular bootstrap program, and we propose a novel modular ansatz for the twisted elliptic genera that transform under the congruence subgroups Γ0(N) of SL(2, ℤ). We conjecture a vanishing bound for the refined Gopakumar-Vafa invariants of the genus one fibered Calabi-Yau threefolds, upon which one can determine the twisted elliptic genera recursively. We use our results to obtain the 6d Cardy formulas and find universal behaviour for all simple Lie groups. In addition, the Cardy formulas remain invariant under the twist once the normalization of the compact circle is taken into account.

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