On the problem of homology stability for congruence subgroups

For any prime number p, let Γn, p denote the congruence subgroup of SLn(ℤ) of level p, i.e. the kernel of the surjective homomorphism fp: SLn(ℤ) → SLn(p) induced by the reduction mod p (Fp is the field with p elements). We defineusing upper left inclusions Γn, p ↪ Γn+1, p. Recall that the groups Γn, p are homology stable with M-coefficients, for instance if M = ℚ, ℤ[1/p], or ℤ/q with q prime and q ╪ p: Hi(Γn, p; M) ≅ Hi(Γp; M) for n ≥ 2i + 5 from [7] (but the homology stability fails if M = ℤ or ℤ/p).

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

Journal of High Energy Physics ◽

10.1007/jhep07(2021)179 ◽

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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On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

Open Mathematics ◽

10.1515/math-2019-0115 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1631-1651

Author(s):

Ick Sun Eum ◽

Ho Yun Jung

Keyword(s):

Modular Forms ◽

Fourier Coefficients ◽

Singular Values ◽

Congruence Subgroups ◽

Maass Form ◽

Weakly Holomorphic Modular Forms ◽

Reciprocity Law ◽

Class Invariants ◽

Higher Genus ◽

Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

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