Eisenstein cohomology for congruence subgroups of SO(n, 2)

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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Regular and residual Eisenstein series and the automorphic cohomology of Sp(2,2)

Compositio Mathematica ◽

10.1112/s0010437x09004266 ◽

2009 ◽

Vol 146 (1) ◽

pp. 21-57 ◽

Cited By ~ 1

Author(s):

Harald Grobner

Keyword(s):

Algebraic Group ◽

Eisenstein Series ◽

Symplectic Group ◽

Automorphic Forms ◽

General Theorem ◽

Simple Algebraic Group ◽

Ramanujan Conjecture ◽

Eisenstein Cohomology ◽

Derivatives Of

AbstractLetGbe the simple algebraic group Sp(2,2), to be defined over ℚ. It is a non-quasi-split, ℚ-rank-two inner form of the split symplectic group Sp8of rank four. The cohomology of the space of automorphic forms onGhas a natural subspace, which is spanned by classes represented by residues and derivatives of cuspidal Eisenstein series. It is called Eisenstein cohomology. In this paper we give a detailed description of the Eisenstein cohomologyHqEis(G,E) ofGin the case of regular coefficientsE. It is spanned only by holomorphic Eisenstein series. For non-regular coefficientsEwe really have to detect the poles of our Eisenstein series. SinceGis not quasi-split, we are out of the scope of the so-called ‘Langlands–Shahidi method’ (cf. F. Shahidi,On certainL-functions, Amer. J. Math.103(1981), 297–355; F. Shahidi,On the Ramanujan conjecture and finiteness of poles for certainL-functions, Ann. of Math. (2)127(1988), 547–584). We apply recent results of Grbac in order to find the double poles of Eisenstein series attached to the minimal parabolicP0ofG. Having collected this information, we determine the square-integrable Eisenstein cohomology supported byP0with respect to arbitrary coefficients and prove a vanishing result. This will exemplify a general theorem we prove in this paper on the distribution of maximally residual Eisenstein cohomology classes.

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