Ramanujan-Type Congruences

• Ramanujan-type congruences are congruences between coefficients of modular forms and partition functions that reveal deep arithmetic connections via Bernoulli numbers and L-values.
• They extend classical results, generalizing Ramanujan’s partition congruences to sophisticated settings like Siegel and Hermitian modular forms with explicit numerical examples.
• The mechanism leverages divisibility properties of Bernoulli numbers to construct congruences between Eisenstein series and cusp forms, offering new insights into congruence primes.

Ramanujan-type congruences are congruences between the Fourier coefficients of modular forms (or related arithmetic functions) and appear throughout number theory, combinatorics, and the arithmetic theory of modular and automorphic forms. Their paper traces its origins to Ramanujan’s celebrated congruences for the partition function, but the concept generalizes naturally to a broad spectrum of modular forms and partition-related functions. Such congruences are now understood to reveal deep connections between arithmetic invariants (such as Bernoulli numbers and special values of $L$-functions), the structure of modular forms, and algebraic or combinatorial symmetries on arithmetic progressions.

1. Historical Context and Definition

Ramanujan-type congruences originally referred to arithmetic congruences of the form

$$p(5n + 4) \equiv 0 \pmod 5, \quad p(7n + 5) \equiv 0 \pmod 7, \quad p(11n + 6) \equiv 0 \pmod{11},$$

where $p(n)$ is the partition function counting the number of integer partitions of $n$. These discoveries by Ramanujan also include his observation that the Fourier coefficients of the normalized cusp form $\Delta(q) = q \prod_{n=1}^{\infty} (1 - q^n)^{24}$ (with coefficients $\tau(n)$) satisfy

$\tau(n) \equiv \sigma_{11}(n) \pmod{691},$

where $\sigma_{11}(n)$ denotes the divisor sum and 691 is a divisor of the 12th Bernoulli number. This congruence is between the coefficients of a cusp form and an Eisenstein series and is a fundamental example of a Ramanujan-type congruence.

General Definition:

A Ramanujan-type congruence is a congruence between the coefficients (or values) of modular forms, automorphic forms, or related partition functions on arithmetic progressions modulo a prime or a prime power, often reflecting deeper algebraic or geometric invariants.

2. Modular Forms of Several Variables: Siegel and Hermitian Cases

The classical phenomenon of Ramanujan-type congruences extends to the setting of modular forms in several variables, notably Siegel modular forms (of degree $n$) and Hermitian modular forms. The essential structure of these congruences in higher rank is as follows:

Siegel Modular Forms (Degree 2)

• For $M_k(\Gamma_2)$, the space of Siegel modular forms of weight $k$ for $\Gamma_2 = Sp_2(\mathbb{Z})$, there exists a normalized Eisenstein series $G_k$, and a subspace $S_k(\Gamma_2)$ of cusp forms.
• If $p$ divides the Bernoulli number $B_{2k-2}$ (i.e., $(p, 2k-2)$ is an irregular pair), then there exists a cusp form $f$ such that

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

where the normalization for $G_k$ is

$G_k := -\frac{4k(k-1)}{B_{2k-2}} E_k.$

This congruence shows that the Eisenstein series is congruent mod $p$ to a cusp form, generalizing the classical Ramanujan congruence for $\tau(n)$ (Kikuta et al., 2012).

Hermitian Modular Forms

• For $K$ an imaginary quadratic field of class number 1 and Hermitian modular group $U_2(\mathcal{O}_K)$, the Hermitian Eisenstein series $E_{k, K}$ is normalized as

$$G_{k,K} := -\frac{4k(k-1)}{B_k\, B_{k-1,\chi_K}} E_{k,K},$$

where $B_{k-1, \chi_K}$ is the generalized Bernoulli number for the Kronecker character $\chi_K$.

• If a prime $p$ divides $B_{k-1,\chi_K}$ and other technical conditions are met (notably avoiding divisors of smaller Bernoulli numbers and $k < p-1$), then there exists a nontrivial cusp form $f$ such that

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

Significance:

The arithmetic of (generalized) Bernoulli numbers governs the existence of such congruences, creating a bridge between the special values of $L$-functions and the modular/siegel/harmonic analysis of modular forms.

3. Mechanisms and Arithmetic Invariants

The key arithmetic input for these congruences is the divisibility of Bernoulli numbers or generalized Bernoulli numbers by certain primes:

• Classical case: Primes dividing Bernoulli numbers govern Eisenstein-to-cusp congruences in the modular forms of one variable.
• Siegel and Hermitian modular forms: The same principle applies, with generalized Bernoulli numbers replacing the classical ones, and the corresponding congruence includes the extra structure endowed by Fourier expansions indexed by matrices (with rank considerations).

This phenomenon is closely related to the arithmetic of congruence primes, the existence and properties of nontrivial cusp forms modulo $p$, and the algebraic and analytic structure of the Hecke algebra.

4. Explicit Examples and Numerical Realizations

Numerical work enumerates explicit congruences in the higher rank setting:

• Siegel case: For weight 10, with the explicit cusp form $X_{10}$ and normalization constants,

$G_{10} \equiv 11313\, X_{10} \pmod{43867},$

indicating that all Fourier coefficients of $G_{10}$ and $X_{10}$ agree modulo 43867 after appropriate scaling.

• Hermitian case: For discriminant $d_K = -3$ and prime 809,

$G_{10} \equiv 554\, F_{10} \pmod{809},$

and similarly for higher weights using the explicit computation of coefficients in the corresponding modular forms (Kikuta et al., 2012).

The scaling factors (such as 11313 or 554) arise from the normalization of Eisenstein series and specific cusp forms involved.

5. Theoretical and Structural Implications

The structure underlying these congruences elucidates fundamental aspects of modular form theory:

• Such congruences generalize the classical relationship between Eisenstein series and cuspidal eigenforms to higher dimension, showing that arithmetic invariants (Bernoulli-type numbers) play a pervasive role.
• The existence of these congruences implies the presence of nontrivial cusp forms modulo $p$, often when the arithmetic of the field or level admits sufficient structure (e.g., class number 1 in the Hermitian case).
• The analytic approach is harmoniously intertwined with the algebraic and arithmetic: the normalization denominators of the Eisenstein series and the role of the Swinnerton-Dyer result on modular forms modulo primes are essential in producing cusp forms congruent to Eisenstein series modulo $p$.
• These congruences provide insight for the paper of congruence primes—primes for which two modular forms (for instance, an Eisenstein and a cusp form) are congruent modulo $p$. This relates to the explicit realization of Galois representations attached to modular forms.

6. Context in Broader Number Theory

The findings generalize and extend core themes in arithmetic:

• The phenomenon connects the geometry and arithmetic of modular forms with the special values of $L$-functions encoded in Bernoulli numbers or their generalizations.
• The explicit congruences further equip researchers with concrete tools to identify or construct modular forms (cusp or Eisenstein) that are congruent mod $p$, facilitating deeper investigations into the mod $p$ reduction of automorphic representations, the nature of congruence primes, and the arithmetic of Shimura varieties.
• The techniques exemplify the interplay between the algebraic manipulation of modular forms (polynomial relations, rank considerations), the deep arithmetic invariants (Bernoulli numbers, congruence primes), and computationally explicit verification.

7. Influence and Future Directions

The generalization of Ramanujan-type congruences to Siegel and Hermitian modular forms, via explicit relationships governed by (generalized) Bernoulli numbers, has broad potential for ongoing research:

• The role of special values of $L$-functions continues to be illuminated, pointing towards ongoing developments in the arithmetic geometry of modular forms in several variables.
• The combinatorial and algebraic methods employed inspire further exploration into higher-rank modular forms, their congruence properties, and the associated Galois representations.
• Understanding and classifying primes that divide particular Bernoulli numbers becomes a guiding principle for predicting or explaining the existence of congruence relations.
• These results motivate new analytic and computational strategies for detecting congruences and for constructing explicit bases for spaces of modular forms with prescribed congruence properties.

In summary, Ramanujan-type congruences in modular forms of several variables reveal profound interactions between arithmetic invariants, modular symmetries, and the structure of automorphic forms, with explicit numerical and theoretical consequences that deepen our understanding of modern number theory.