Ramanujan's Congruences and Modular Forms

• Ramanujan's Congruences are specific congruence properties observed in partition and tau functions, highlighting deep links between combinatorial identities and modular form theory.
• These congruences extend to a variety of modular forms—including Siegel, Hermitian, and prime-level forms—using techniques like generating functions and Hecke operators.
• Their study illuminates the arithmetic of Bernoulli numbers, L-values, and Galois representations, driving advances in p-adic analysis and automorphic forms.

Ramanujan's congruences are a foundational phenomenon in arithmetic geometry and the theory of modular forms, characterized by striking divisibility properties of functions such as the partition function and modular form coefficients on specific arithmetic progressions modulo primes or prime powers. These congruences, initially observed for the partition function $p(n)$ and the tau function $\tau(n)$, have evolved into a deep and general theory connecting the arithmetic of modular forms, generalized Bernoulli numbers, Hecke algebras, and automorphic forms in one and several variables.

1. Classical Congruences and Modular Forms

Ramanujan's original congruences state that: $$p(5n + 4) \equiv 0 \pmod{5},\quad p(7n + 5) \equiv 0 \pmod{7},\quad p(11n + 6) \equiv 0 \pmod{11},$$ where $p(n)$ is the partition function. Simultaneously, for the unique normalized cusp form $\Delta(z) = q\prod_{n=1}^\infty (1 - q^n)^{24}$, whose Fourier coefficients are $\tau(n)$, one has

$\tau(5n + 5) \equiv 0 \pmod{5},$

and, more generally, the deep congruence

$\tau(p) \equiv p^{11} + 1 \pmod{691}$

for all primes $p$, with $691$ appearing as an "exceptional prime" dividing the numerator of $\zeta(12)/\pi^{12}$ and thus the Bernoulli number $B_{12}$ (Bal et al., 30 Aug 2025, Parnoff et al., 5 Mar 2024). These results, obtained via generating function techniques, combinatorial recurrences, and modular form theory, are the archetype of "Ramanujan-type congruences."

2. Generalizations: Modular Forms of Several Variables and Higher Levels

The classical congruences admit systematic generalizations in the context of modular forms with several variables and modular forms of higher level.

• Siegel Modular Forms (Degree 2):

If a prime $p$ divides the Bernoulli number $B_{2k-2}$ (i.e., $(p,2k-2)$ is an irregular pair), Theorem 2.1 of (Kikuta et al., 2012) yields a Siegel modular form $f\in M_k(\mathrm{Sp}_2(\mathbb{Z}))$ such that

$G_k \equiv f \pmod{p},$

where $G_k = -\frac{4k(k-1)}{B_{2k-2}} E_k$ (normalized Eisenstein series).

• Hermitian Modular Forms:

For an imaginary quadratic field $K$ with Kronecker character $\chi_K$, if $p$ divides the $(k-1)$-th generalized Bernoulli number $B_{k-1,\chi_K}$ and $k < p-1$, then there is a nontrivial Hermitian cusp form $f$ with

$G_{k,k} \equiv f \pmod{p},$

with $G_{k,k} = -\frac{4k(k-1)}{B_{k-1,\chi_K}} E_{k,k}$ (Kikuta et al., 2012).

• Klingen-Eisenstein Series and Prime Ideals:

In the setting of Siegel modular forms of arbitrary degree $n$, for almost all prime ideals $\mathfrak{p}$, mod $\mathfrak{p}^m$ "cusp forms" (those for which "non-cusp" coefficient indices vanish modulo $\mathfrak{p}^m$) are congruent to true cusp forms, and the Klingen-Eisenstein series is congruent to a cusp form:

$[f]_r \equiv F \pmod{\mathfrak{p}^m}$

for suitable Hecke eigen cuspidal $f$ and $F$ in the same weight/degree, generalizing the $\sigma_{11}(n) \equiv \tau(n) \pmod{691}$ paradigm (Kikuta et al., 2014).

• Congruences for Newforms of Prime Level:

For modular forms on $\Gamma_0(N)$ of prime level, congruences of the type

$f \equiv E_k^{(\epsilon)} \pmod{I}$

hold between newforms with prescribed Atkin-Lehner eigenvalue $\epsilon$ and appropriately "twisted" Eisenstein series $$E_k^{(\epsilon)}(z) = E_k(z) + \epsilon p^{k/2} E_k(pz)$$, refining the level-raising congruences and broadening the context for prime-level forms (Gaba et al., 2016).

3. The Role of (Generalized) Bernoulli Numbers and $L$-values

The arithmetic of Ramanujan-type congruences is inextricably linked with the divisibility of Bernoulli numbers or their generalizations:

• Bernoulli Numbers:

The classical modulus $691$ arises because $691$ divides the numerator of $B_{12}$, governing the constant term in the Fourier expansion of Eisenstein series and unlocking the congruence between Eisenstein and cusp forms.

• Generalized Bernoulli Numbers and Dirichlet $L$-functions:

For congruence subgroups with nebentypus character $\chi$, congruence primes frequently appear as divisors of special values

$L(k,\chi) \sim B_{k,\chi}$

and the explicit formula for $L(2m,\chi)$ in terms of Bernoulli numbers and Gauss sums. For example, $67$ appears as a congruence prime in weight 6, level 5, with nebentypus, because $67 \mid 5 B_{6,\chi}$, prompting a congruence between an Eisenstein series and a cusp form in $M_6(\Gamma_0(5),\chi)$ (Parnoff et al., 5 Mar 2024).

4. Arithmetic Progressions, Hecke Operators, and Representation Theory

The congruence structure of modular form coefficients on arithmetic progressions is governed by Hecke operator theory and the arithmetic of Hecke algebras:

• Hecke Eigenvalues and Stability:

If a weakly holomorphic modular form $f$ satisfies $c(f; Mn + \beta) \equiv 0 \pmod{\ell}$ for all $n$, the vanishing propagates to certain "gap" progressions, and such Ramanujan-type congruences are determined by explicit congruence conditions on the eigenvalues $\lambda_p$ of the Hecke operators (Raum, 2020, Raum, 2021).

• Hecke Module Structure:

The "shallow" Hecke algebra (generated by operators outside the level and congruence prime) preserves the subspace of forms exhibiting these congruences, and the maximal such congruences are characterized by Steinberg representation theory in the local $p$-adic setting (Raum, 2020).

• Propagation to Square-Classes:

When an "explainable Ramanujan-type congruence" (i.e., arising from a Jacobi form/rank statistic) holds on $M\mathbb{Z} + \beta$, representation theory typically propagates the congruence to the entire square-class $M\mathbb{Z} + u^2\beta$ for all $u$ coprime to $M$ (Raum, 3 Apr 2024).

5. Combinatorial Invariants, Mock Modular Forms, and Ramanujan-Type Congruences

Not all instances of Ramanujan-type congruences are merely modular artifacts—some have genuine combinatorial explanations:

• Ranks and Cranks:

The existence of "rank" and "crank" statistics enables a combinatorial interpretation of congruences such as $p(5n+4) \equiv 0 \pmod{5}$. Generating functions for these statistics, often realized as (weakly holomorphic) Jacobi forms with prescribed cyclotomic divisibility at torsion points, explain the partition equidistribution modulo primes (Rolen et al., 2020, Raum, 3 Apr 2024, Amdeberhan et al., 26 May 2025).

• Mock Theta Functions and Generalized Borcherds Products:

Congruences for mock theta function coefficients (e.g., Ramanujan’s $f$ and $\omega$) are analyzed by realizing them as holomorphic parts of harmonic Maass forms, constructing generalized Borcherds products, and leveraging modular form congruences for the resulting logarithmic derivatives (Berg et al., 2013).

6. Explicit Generating Functions and Closed-Form Expressions

Ramanujan’s work connected congruences to explicit $q$-series identities involving infinite products:

• Partition Generating Functions Modulo $\ell$:

For primes $\ell \geq 5$, the mod $\ell$ generating function for the sequence $p(\ell n - \delta_\ell)$, $\delta_\ell = (\ell^2 -1)/24$, has an explicit description:

$$P_\ell(q) = \sum_{n \ge 0} p(\ell n - \delta_\ell) q^n \equiv c_\ell \frac{T_\ell(q)}{(q^\ell;q^\ell)_\infty} \pmod{\ell},$$

where $T_\ell(q)$ is a generating function for weighted Hecke traces of $\ell$-ramified values attached to weight~$\ell-1$ cusp forms (Bringmann et al., 6 Jun 2025).

• Bell Polynomials and Higher Congruences:

Identities expressing partition numbers in arithmetic progressions (like $p(7n+5)$) in terms of complete Bell polynomials further elucidate the combinatorial underpinnings of congruences (Leung, 2018).

7. Applications, Scarcity Phenomena, and Open Directions

Ramanujan-type congruences have wide-ranging implications for the arithmetic of modular forms, automorphic representations, and combinatorics:

• Scarcity and Dichotomy:

Only a limited class of congruences are "combinatorially explainable" via generalized ranks or cranks, typically occurring on square-classes of arithmetic progressions; the generic case (Hecke eigenvalue-driven) is much more abundant but often lacks a natural combinatorial interpretation (Raum, 2021, Raum, 3 Apr 2024).

• Generalization to Additive and Multiplicative Functions:

The theory extends via $p$-adic valuation analysis to multiplicative functions, stipulating that congruence properties in arithmetic progressions for such functions (e.g., divisor functions, $\tau(n)$) are dictated by local factors and prime power multiplicities (Craig et al., 2021).

• Role in Modern Theories:

Ramanujan-type congruences inform the paper of Galois representations, $p$-adic $L$-functions, Iwasawa invariants, and deep phenomena in the arithmetic of automorphic forms, as well as stimulating ongoing advances in supercongruence and $q$-series techniques (Doyon et al., 2021, Feng et al., 24 Jun 2025).

In total, Ramanujan’s congruences, and their modern generalizations, form a nexus joining modular representation theory, p-adic analysis, combinatorics, and algebraic geometry, with classical results such as the congruence modulo 691 serving both as striking arithmetic phenomena and as templates for deeper structural theorems in the arithmetic of modular forms.