Meromorphic Modular Forms: Theory & Applications

• Meromorphic modular forms are functions on the upper half-plane that obey a modular transformation law while allowing controlled poles at isolated points and cusps.
• They generalize holomorphic and weakly holomorphic modular forms by admitting Fourier expansions with finitely many negative terms and connecting to harmonic Maass forms.
• Their study yields practical insights in number theory, algebraic geometry, and physics, notably through cycle integrals, theta lifts, and explicit arithmetic invariants.

A meromorphic modular form is a function on the upper half-plane (or higher-dimensional symmetric domain) that satisfies a modular transformation law (possibly with a multiplier system) and is meromorphic, admitting only poles at isolated points and at the cusps. The space of such forms generalizes both holomorphic modular forms and weakly holomorphic modular forms by permitting controlled pole data in the upper half-plane. The structural and arithmetic properties, Fourier expansions, period relations, and connections to auxiliary objects such as harmonic Maass forms and special cycles are a major focus of current research, with implications for number theory, algebraic geometry, and mathematical physics.

1. Definitions and Basic Properties

Let $\Gamma \subset SL_2(\mathbb{Z})$ (or a congruence subgroup), and let $k \in \mathbb{Z}$, possibly a half-integer if a suitable multiplier is fixed.

A meromorphic modular form $f$ of weight $k$ and multiplier $\chi$ is a function $f : \mathbb{H} \to \mathbb{C}$ such that:

• Modularity: $f(\gamma z) = (cz + d)^k \chi(\gamma) f(z)$ for all $\gamma = \bigl(\begin{smallmatrix}a&b\c&d\end{smallmatrix}\bigr) \in \Gamma$ and all $z \in \mathbb{H}$;
• Meromorphicity: $f$ is holomorphic except for finitely many poles in $\mathbb{H}$ and possibly poles at the cusps;
• q-Expansion: At each cusp, $f$ admits a Fourier (or $q$-) expansion with only finitely many negative powers, i.e., $f(z) = \sum_{n \gg -\infty} a_n q^{n/h}$ for $q = e^{2\pi i z}$ and some integer $h$ (the cusp width).

For higher-dimensional analogues, e.g., Hilbert modular forms, $f$ is a function on $\mathbb{H}^g$ for $g$ a degree and must satisfy appropriate transformation and meromorphicity criteria (Alfes et al., 5 Jun 2024).

The principal part of $f$ at a pole $z_0 \in \mathbb{H}$ is the polar part of its Laurent expansion in local coordinates $(z-z_0)$.

Polar Harmonic Maass Forms

A polar harmonic Maass form of weight $k$ is a real-analytic $\Gamma$-invariant function $F$ on $\mathbb{H}$ (with finitely many poles) satisfying

• $\Delta_k F = 0$, where $\Delta_k$ is the weight-$k$ hyperbolic Laplacian;
• Principal parts at poles;
• At most exponential growth at the cusps.

The space of meromorphic modular forms embeds as the intersection of kernel spaces for certain differential operators within the polar harmonic Maass forms (Bringmann et al., 2015, Bringmann et al., 2016).

Cycle Integrals

Given a positive-definite or indefinite binary quadratic form $Q=[a,b,c]$ of discriminant $D$, the associated geodesic $C_Q$ in $\mathbb{H}$ is defined by $a|z|^2 + b\,\Re(z) + c = 0$. The cycle integral of $f$ along $C_Q$ is

$\int_{C_Q} f(z) Q(z,1)^{k-1} dz,$

interpreted (if necessary) as a principal value if $f$ is singular on $C_Q$. Traces and linear combinations of such cycle integrals are central arithmetic invariants (Alfes-Neumann et al., 2020, Schwagenscheidt, 2023).

2. Fourier Expansions and q-Series

Any meromorphic modular form $f$ at a cusp (e.g., $i\infty$) admits a Fourier expansion

$$f(z) = \sum_{n \gg -\infty} a(n) q^n,\qquad q=e^{2\pi i z}$$

with $a(n)$ in a number field or $\mathbb{Q}$ under standard hypotheses (e.g., when the pole data is defined over $\mathbb{Q}$).

For meromorphic modular forms with poles at CM points or other algebraic points, the local expansions (including Laurent and q-expansions) encode deep arithmetic information. The Laurent coefficients at a CM point $z_0$ can be computed: $$a_m = \frac{y_0^m}{m!} \operatorname{CT}\left(R_k^m f\right)(z_0),$$ where $R_k$ is the Maass–Shimura raising operator, $y_0 = \Im z_0$, and $\operatorname{CT}$ denotes the constant term in the local parameter $w$ (Bogo et al., 2023).

Special cases include the explicit expansions for the reciprocal of the Eisenstein series as in Ramanujan's work, now generalized to any negative- or zero-weight (quasi-)meromorphic cusp form via Poincaré series techniques, Bessel functions, and Kloosterman sums (Bringmann et al., 2016, Bringmann et al., 2016).

Poincaré Series Realization

Meromorphic modular forms with prescribed pole structure can be constructed using Poincaré series (with analytic continuation for convergence in the non-positive weight regime). For example, for a pole at $\tau$, the Petersson-type or generalized Poincaré series give

$$H_{m,\ell}(\tau,z) := (2\Im \tau)^{\ell+1}/\ell! \cdot \partial_\tau^\ell[H_m(\tau, z)],$$

with controlled poles and explicit $q$-expansions, including for higher-order poles (Bringmann et al., 2016).

3. Arithmetic Properties of Coefficients

Integrality and Divisibility

Meromorphic modular forms associated with positive definite binary quadratic forms, such as

$$f_{k,d,D}(z) = C_{k,d,D} \sum_{Q\in \mathcal{Q}_{dD}} \chi_D(Q)\, Q(z,1)^{-k},$$

have Fourier coefficients $a(n)$ satisfying surprisingly strong divisibility properties: $n^{k-1} \mid a(n)$ whenever the space of holomorphic cusp forms of weight $2k$ vanishes (Löbrich et al., 2020). In general, after subtracting appropriate cusp forms, divisibility by $n^{k-1}$ persists for Hecke-translates as well.

The sign pattern is determined by $(-1)^{k+n d D}$, ensuring explicit knowledge of the sign sequence.

Connections to Partition and $j$-Functions

Fourier coefficients of some meromorphic modular forms satisfy combinatorial relations to partition functions and the $j$-invariant, e.g.,

$$A(n) = \sum_{m\mid n} m\, p\bigl(\tfrac n m -1\bigr) \bigl(2 m^3 + 1\bigr),$$

where $p(n)$ is the partition function, and $A(n)$ arises from a specific linear combination of $f_{2,-23,r,1}$ forms (Löbrich et al., 2020).

p-adic and Atkin–Swinnerton-Dyer Congruences

For forms with poles at CM points, the Fourier coefficients display $p$-adic recurrence relations generalizing classical ASD congruences. For example, for $F_{k,C}(z) = E_k(z)/(j(z) - j(C))$ attached to an elliptic curve $C/\mathbb{Q}$,

$$a_{n p}(F_{k,C}) \equiv a_p(C)^{k-2} a_n(F_{k,C}) \pmod{p}$$

for primes $p$ of good reduction, with $a_p(C)$ the trace of Frobenius. In the supersingular case, supercongruences reflecting higher divisibility occur, linking divisibility depth to arithmetic properties of $C$ (Zhang, 27 Oct 2025, Allen et al., 7 Nov 2025).

More generally, for meromorphic modular forms on curves associated to noncongruence subgroups, ASD recurrences can be traced to the Frobenius action on de Rham or $\ell$-adic cohomology of the associated modular curve and its elliptic sheaf, and the residue data at the poles enters the polynomials governing the $p$-adic recurrence (Allen et al., 7 Nov 2025).

Magnetic Modular Forms

"Magnetic" modular forms are those whose Fourier coefficients $a_n$ satisfy $n^d \mid a_n$ up to bounded denominators ("depth $d$"). Known examples have algebraic residues at CM points and vanishing period cocycles. Conjecturally, closure under Hecke, Atkin–Lehner, and $\mathrm{SL}_2(\mathbb{Z})$ is expected, and explicit examples such as

$$C_4(\tau) = \frac{A(\tau)}{3(1+64A(\tau))}(4\phi_0^2-E_4)$$

with $A(\tau),\phi_0(\tau)$ products of $\eta$-functions, have verified $p$-divisibility properties (Bönisch et al., 5 Apr 2024).

4. Periods and Cycle Integrals: Rationality and Explicit Formulas

Cycle integrals and periods of meromorphic modular forms encode critical arithmetic information.

• Cycle Integrals: For $f$ as above, the cycle integral along the closed geodesic $C_A$ associated to $Q=[a,b,c]$ is

$\int_{C_A} f(z) Q(z,1)^{k-1} dz.$

Linear combinations of such integrals for $f_{k,P}$, over $A$ satisfying certain Hecke and period constraints, often yield rational numbers (Löbrich et al., 2019, Schwagenscheidt, 2023, Alfes-Neumann et al., 2020, Alfes-Neumann et al., 2018).

• Explicit Rationality: Under explicit vanishing of cusp form spaces or suitable Hecke relations, cycle integrals are rational or have bounded denominators, often via explicit finite formulas involving Legendre polynomials, zeta functions, and lattice sums. For binaries $P,Q$,

$$C(f_{k,P},A) = |d|^2 T_A (R^{2k-1} F_{1-k,A})(T_P),$$

where $T_P$ is a CM point and $F_{1-k,A}$ a locally harmonic Maass form (Löbrich et al., 2019).

• Relation to Harmonic Maass Forms: Cycle traces and periods can be expressed as Fourier coefficients and constant terms of harmonic Maass forms via theta lifts and Rankin–Cohen brackets: $$c_h(-D)\,\mathrm{Tr}(f_{k,A};D) = \frac{\pi\;2^{2k-3}|d|^{k/2}}{\Gamma(k)}\, \mathrm{CT}\left(f(\tau),[G(\tau),\Theta_{N^-}(\tau)]_{k-1}\right),$$ with $G$ a Maass form, $\Theta$ a theta series, and $[\cdot,\cdot]_{k-1}$ the $(k-1)$st Rankin–Cohen bracket (Alfes-Neumann et al., 2020).
• Splitting Theorems and Rationality of Periods: For periods $r_n(f)$, there are explicit splitting theorems for locally harmonic Maass forms, which yield rationality results for linear combinations of periods when the associated cusp forms satisfy vanishing relations (Löbrich et al., 2020).
• Hilbert Modular Generalization: Analogous results hold for meromorphic Hilbert modular forms, with traces of cycle integrals computable in terms of data attached to auxiliary harmonic Maass forms and Rankin–Cohen brackets, extending to the Doi–Naganuma correspondence framework (Alfes et al., 5 Jun 2024).

5. Theta Lifts, Maass Forms, and Modularity

Theta Lifts and Borcherds–Shimura Construction

Theta lifts relating harmonic Maass forms to meromorphic modular forms provide powerful structural and computational tools. For an even lattice $L$ of signature $(1,2)$, if $f(\tau)$ is a vector-valued harmonic Maass form, the regularized theta lift,

$$\Phi_{BS}(f;z) = \int^{\mathrm{reg}}_{SL_2(\mathbb{Z}) \backslash \mathbb{H}} \langle f(\tau),\Theta_L^{(0,k)}(\tau,z)\rangle\, d\mu(\tau),$$

yields a meromorphic form of weight $2k$ with explicit pole structure at CM points determined by the principal part of $f$ (Bogo et al., 2023, Schwagenscheidt, 2023).

6. Applications and Research Directions

Feynman Integrals and Physics

Meromorphic modular forms arise in analytic computation of Feynman integrals, where elliptic and modular geometry govern the underlying differential equations. For instance, the banana diagrams (multi-loop equal-mass integrals) involve iterated integrals of meromorphic modular forms corresponding to the monodromy group of the relevant second-order differential operators. The decomposition of the space of modular forms into total derivatives and a finite algebraic basis yields a finite "alphabet" for such iterated integrals (Broedel et al., 2021).

Magnetism, Cohomology, and Modularity

Studying "magnetic" modular forms and their period data links the arithmetic of modular forms to motivic and cohomological perspectives, especially through the vanishing of the period cocycle and closure properties under various modular operators (Bönisch et al., 5 Apr 2024, Allen et al., 7 Nov 2025).

Generalizations

— Extension to noncongruence subgroups, half-integer weights, vector-valued settings. — Rationality questions extend to the Hilbert modular case and Siegel domains (Alfes et al., 5 Jun 2024, Schwagenscheidt, 2023). — Open questions on analytic continuation and nonholomorphic parts, connections to special $L$-values and heights on Shimura varieties (Alfes-Neumann et al., 2020). — Intricate relations between the pole structure, residues at CM points, and modular period data.

7. Summary Table

Feature	Reference Examples	Arithmetic/Structural Implications
Fourier coefficients, divisibility	(Löbrich et al., 2020, Zhang, 27 Oct 2025, Allen et al., 7 Nov 2025)	$n^{k-1} \mid a(n)$, supercongruences, magnetic forms
Cycle integrals, periods	(Löbrich et al., 2019, Schwagenscheidt, 2023, Alfes-Neumann et al., 2020, Löbrich et al., 2020)	Explicit rationality, Rankin–Cohen expressions
Explicit $q$-expansions	(Bringmann et al., 2016, Bringmann et al., 2016, Elenberg et al., 2015)	Generalized Rademacher sums, Kloosterman/Bessel sums
Theta lifts and modularity	(Alfes-Neumann et al., 2020, Bogo et al., 2023, Schwagenscheidt, 2023)	Link to harmonic Maass forms, modular completions, Borcherds products

Meromorphic modular forms are thus a natural meeting ground for the analytic, arithmetic, and geometric aspects of modern automorphic theory. Their paper organizes and generalizes classical theorems of modular form theory, provides new links to physics and arithmetic geometry, and reveals previously hidden structures in the theory of special values, congruence relations, and the cohomology of modular curves.