Meromorphic Modular Forms Research Articles

S-duality, triangle groups and modular anomalies in N = 2 $$ \mathcal{N}=2 $$ SQCD

We study N = 2 superconformal theories with gauge group SU(N) and 2N fundamental flavours in a locus of the Coulomb branch with a Z_N symmetry. In this special vacuum, we calculate the prepotential, the dual periods and the period matrix using equivariant localization. When the flavours are massless, we find that the period matrix is completely specified by [N/2] effective couplings. On each of these, we show that the S-duality group acts as a generalized triangle group and that its hauptmodul can be used to write a non-perturbatively exact relation between each effective coupling and the bare one. For N = 2, 3, 4 and 6, the generalized triangle group is an arithmetic Hecke group which contains a subgroup that is also a congruence subgroup of the modular group PSL(2,Z). For these cases, we introduce mass deformations that respect the symmetries of the special vacuum and show that the constraints arising from S-duality make it possible to resum the instanton contributions to the period matrix in terms of meromorphic modular forms which solve modular anomaly equations.

On direct integration for mirror curves of genus two and an almost meromorphic Siegel modular form

This work considers aspects of almost holomorphic and meromorphic Siegel modular forms from the perspective of physics and mathematics. The first part is concerned with (refined) topological string theory and the direct integration of the holomorphic anomaly equations. Here, a central object to compute higher genus amplitudes, which serve as the generating functions of various enumerative invariants, is provided by the so-called propagator. We derive a universal expression for the propagator for geometries that have mirror curves of genus two which is given by the derivative of the logarithm of Igusa's cusp form of weight 10. In addition, we illustrate our findings by solving the refined topological string on the resolutions of the three toric orbifolds of order three, five and six. In the second part, we give explicit expressions for lowering and raising operators on Siegel modular forms, and define almost holomorphic Siegel modular forms based on them. Extending the theory of Fourier-Jacobi expansions to almost holomorphic Siegel modular forms and building up on recent work by Pitale, Saha, and Schmidt, we can show that there is no analogue of the almost holomorphic elliptic second Eisenstein series. In the case of genus 2, we provide an almost meromorphic substitute for it. This, in particular, leads us to a generalization of Ramanujan's differential equation for the second Eisenstein series. The two parts are intertwined by the observation that the meromorphic analogue of the almost holomorphic second Eisenstein series coincides with the physical propagator. In addition, the generalized Ramanujan identities match precisely the physical consistency conditions that need to be imposed on the propagator.

Regularized pairings of meromorphic modular forms and theta lifts

We show that the meromorphic modular forms recently considered by Bringmann and Kane can be obtained as images of regularized theta lifts of Poincaré series under weight raising operators. We use this fact in order to simplify the evaluation of the regularized pairing, also defined by Bringmann and Kane, of these functions with other meromorphic modular forms.

A Gross–Kohnen–Zagier type theorem for higher-codimensional Heegner cycles

We prove that Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool for generating the necessary relations is a Borcherds style theta lift with polynomials. We also show how this lift defines a new singular Shimura-type correspondence from weakly holomorphic modular forms of weight 1/2-m to meromorphic modular forms of weight 2m+2.

The complex Lorentzian Leech lattice and the bimonster (II)

LetDDbe the incidence graph of the projective plane overF3\mathbb {F}_3. The Artin group of the graphDDmaps onto the bimonster and a complex hyperbolic reflection groupΓ\Gammaacting on1313dimensional complex hyperbolic spaceYY. The generators of the Artin group are mapped to elements of order22(resp.33) in the bimonster (resp.Γ\Gamma). LetY∘⊆YY^{\circ } \subseteq Ybe the complement of the union of the mirrors ofΓ\Gamma. Daniel Allcock has conjectured that the orbifold fundamental group ofY∘/ΓY^{\circ }/\Gammasurjects onto the bimonster.In this article we study the reflection groupΓ\Gamma. Our main result shows that there is homomorphism from the Artin group ofDDto the orbifold fundamental group ofY∘/ΓY^{\circ }/\Gamma, obtained by sending the Artin generators to the generators of monodromy around the mirrors of the generating reflections inΓ\Gamma. This answers a question in Allcock’s article “A monstrous proposal” and takes a step towards the proof of Allcock’s conjecture. The finite groupPGL⁡(3,F3)⊆Aut(D)\operatorname {PGL}(3, \mathbb {F}_3) \subseteq \mathrm {Aut}(D)acts onYYand fixes a complex hyperbolic line pointwise. We show that the restriction ofΓ\Gamma-invariant meromorphic automorphic forms onYYto the complex hyperbolic line fixed byPGL⁡(3,F3)\operatorname {PGL}(3, \mathbb {F}_3)gives meromorphic modular forms of level1313.

S-Duality for surfaces with $$A_n$$ A n -type singularities

We show that the generating series of Euler characteristics of Hilbert schemes of points on any algebraic surface with at worst \(A_n\)-type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the Dedekind eta function. In particular it is a Fourier development of a meromorphic modular form with possibly half integer weight. The key ingredient is to apply the flop transformation formula of Donaldson–Thomas type invariants counting two dimensional torsion sheaves on threefolds proved in the author’s previous paper.

Meromorphic analogues of modular forms generating the kernel of Shimura’s lift

We study the meromorphic modular forms defined as sums of -k (k>1) powers of integral quadratic polynomials with negative discriminant. These functions can be viewed as meromorphic analogues of the holomorphic modular forms defined in the same way with positive discriminant, first investigated by Zagier in connection with the Doi-Naganuma map and then by Kohnen and Zagier in connection with the Shimura-Shintani lifts. We compute the Fourier coefficients of these meromorphic modular forms and we show that they split into the sum of a meromorphic modular form with computable algebraic Fourier coefficients and a holomorphic cusp form.

-INVARIANT AND MODULAR FORMS

We show that theAtiyah–Patodi–Singer reduced η-invariant of the twisted Dirac operator on a closed 4m − 1-dimensional spin manifold, with the twisted bundle being the Witten bundle appearing in the theory of elliptic genus, is a meromorphic modular form of weight 2m up to an integral q-series. We prove this result by combining our construction of certain modular characteristic forms associated to a generalized Witten bundle on spin c -manifolds with a deep topological theorem due to Hopkins.

Linear Mahler measures and double L-values of modular forms

We consider the Mahler measure of the polynomial 1+x1+x2+x3+x4, which is the first case not yet evaluated explicitly. A conjecture due to F. Rodriguez-Villegas represents this Mahler measure as a special value at the point 4 of the L-function of a modular form of weight 3. We prove that this Mahler measure is equal to a linear combination of double L-values of certain meromorphic modular forms of weight 4.

Congruences for Ramanujan’s f and ω functions via generalized Borcherds products

Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujan's mock theta functions f and {\omega} in detail.

BORCHERDS PRODUCTS FOR U(1,1)

In [Borcherds products on unitary groups, preprint (2012); arXiv:1210.2542] the author constructed a multiplicative Borcherds lift for indefinite unitary groups U (1, n). In the present paper the case of U(1,1) is examined in greater detail. The lifting in this case takes weakly holomorphic elliptic modular forms of weight zero as inputs and lifts them to meromorphic modular forms for U(1,1), on the usual complex upper half-plane ℍ. In this setting, the Weyl-chambers can be described very explicitly and the associated Weyl-vectors can also be calculated. This is carried out in detail for weakly holomorphic modular forms with q-expansions of the form [Formula: see text], n > 0, and for a constant function, which together span the input space, [Formula: see text]. The general case for the lifting of an arbitrary [Formula: see text] comes by as a corollary. The lifted functions take their zeros and poles along Heegner-divisors, which consist of CM-points in ℍ. We find that their CM-order can to some extent be prescribed.

Harmonic lifts of modular forms

It is shown that each complex conjugate of a meromorphic modular form for $\mathrm{SL}_{2}(\mathbb{Z})$ of any complex weight p occurs as the image of a harmonic modular form under the operator $2i y^{p} \partial_{\bar{z}}$ . These harmonic lifts occur in holomorphic families with the weight as the parameter.

The circle method and non-lacunarity of meromorphic modular forms

Abstract Serre proved that any holomorphic cusp form of weight one for Γ1(N) is lacunary while a holomorphic modular form for Γ1(N) of higher integer weight is lacunary if and only if it is a linear combination of cusp forms of CM-type (see [Publ. Math. I.H.E.S. 54 (1981), 323–401, Sections 7.6 and 7.7]). In this paper, we show that when a non-zero meromorphic modular form of arbitrary real weight for any finite index subgroup of the modular group SL 2 ( ℤ ) ${\operatorname{SL}}_2(\mathbb {Z})$ is lacunary, it is necessarily holomorphic on the upper-half plane, finite at the cusps and has non-negative weight.

ON FOURIER COEFFICIENTS OF SOME MEROMORPHIC MODULAR FORMS

We prove a congruence modulo a prime of Fourier coefficients of several meromorphic modular forms of low weights. We prove the result by establishing a generalization of a theorem of Garthwaite.

Regularized theta lifts for orthogonal groups over totally real fields

We define a regularized theta lift from SL2 to orthogonal groups over totally real fields. It takes harmonic ‘Whittaker forms’ to automorphic Green functions and weakly holomorphic Whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors, generalizing Borcherds' work on automorphic products. To prove our results, we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of Kudla and Millson.

A mod ℓ Atkin–Lehner theorem and applications

If f(z) is a weight \({k\in \frac{1}{2}\mathbb {Z}}\) meromorphic modular form on Γ0(N) satisfying $$f(z)=\sum_{n\geq n_0} a_ne^{2\pi i mnz}, $$ where \({m \nmid N,}\) then f is constant. If k ≠ 0, then f = 0. Atkin and Lehner [2] derived the theory of integer weight newforms from this fact. We use the geometric theory of modular forms to prove the analog of this fact for modular forms modulo l. We show that the same conclusion holds if gcd(N l,m) = 1 and the nebentypus character is trivial at l. We use this to study the parity of the partition function and the coefficients of Klein’s j-function.

How do black holes predict the sign of the Fourier coefficients of siegel modular forms?

Single centered supersymmetric black holes in four dimensions have spherically symmetric horizon and hence carry zero angular momentum. This leads to a specific sign of the helicity trace index associated with these black holes. Since the latter are given by the Fourier expansion coefficients of appropriate meromorphic modular forms of \({Sp(2,{\mathbb{Z}})}\) or its subgroup, we are led to a specific prediction for the signs of a subset of these Fourier coefficients which represent contributions from single centered black holes only. We explicitly test these predictions for the modular forms which compute the index of quarter BPS black holes in heterotic string theory on T 6, as well as in \({{\mathbb{Z}}_N}\) CHL models for N = 2, 3, 5, 7.

ON FOURIER COEFFICIENTS OF MODULAR FORMS

AbstractRecursive formulae satisfied by the Fourier coefficients of meromorphic modular forms on groups of genus zero have been investigated by several authors. Bruinieret al. [‘The arithmetic of the values of modular functions and the divisors of modular forms’,Compositio Math.140(3) (2004), 552–566] found recurrences for SL(2,ℤ); Ahlgren [‘The theta-operator and the divisors of modular forms on genus zero subgroups’,Math. Res. Lett.10(5–6) (2003), 787–798] investigated the groups Γ0(p); Atkinson [‘Divisors of modular forms on Γ0(4)’,J. Number Theory112(1) (2005), 189–204] considered Γ0(4), and S. Y. Choi [‘The values of modular functions and modular forms’,Canad. Math. Bull.49(4) (2006), 526–535] found the corresponding formulae for the groups Γ+0(p). In this paper we generalize these results and find recursive formulae for the Fourier coefficients of any meromorphic modular formfon any genus-zero group Γ commensurable with SL(2,ℤ) , including noncongruence groups and expansions at irregular cusps. The form of the recurrence relations is well suited for the computation of the Fourier coefficients of the functions and forms on the groups which occur in monstrous and generalized moonshine. The required initial data has, in many cases, been computed by Norton (private communication).

TWISTED BORCHERDS PRODUCTS ON HILBERT MODULAR SURFACES AND THE REGULARIZED THETA LIFT

We construct a lifting from weakly holomorphic modular forms of weight 0 for SL 2(ℤ) with integral Fourier coefficients to meromorphic Hilbert modular forms of weight 0 for the full Hilbert modular group of a real quadratic number field with an infinite product expansion and a divisor given by a linear combination of twisted Hirzebruch–Zagier divisors. The construction uses the singular theta lifting by considering a suitable twist of a Siegel theta function. We generalize the work by Bruinier and Yang who showed the existence of the lifting for prime discriminants using a different approach.

Poincaré series and the divisors of modular forms

Recently, Bruinier, Kohnen and Ono obtained an explicit description of the action of the theta-operator on meromorphic modular formsffonSL2(Z)SL_2(\mathbb {Z})in terms of the values of modular functions at points in the divisor offf. Using this result, they studied the exponents in the infinite product expansion of a modular form and recurrence relations for Fourier coefficients of a modular form. In this paper, we extend these results to meromorphic modular forms onΓ0(N)\Gamma _0(N)for an arbitrary positive integerN>1N>1.