Eisenstein Congruence Divisibility

• Eisenstein congruence divisibility is the arithmetic phenomenon where a prime divides (generalized) Bernoulli numbers, forcing the vanishing of low-rank Fourier coefficients in Eisenstein series.
• It employs explicit modular form constructions that use polynomial projection methods to establish congruences between normalized Eisenstein series and nontrivial cusp forms.
• Numerical examples in Siegel and Hermitian settings illustrate that when a prime divides the relevant Bernoulli number, key Fourier coefficients vanish modulo p, confirming the existence of congruent cusp forms.

Eisenstein congruence divisibility refers to the phenomenon wherein the divisibility of certain arithmetic invariants—most critically, generalized Bernoulli numbers or associated $L$-values—by a prime $p$ forces congruences modulo $p$ between Eisenstein series and cusp forms in the context of modular forms of several variables. This interplay reveals deep structural links between the arithmetic of modular invariants and the existence or construction of non-trivial modular or automorphic forms congruent to Eisenstein series, extending the ideas found in Ramanujan's classical congruence for modular forms to Siegel and Hermitian modular settings.

1. Framework of Eisenstein Series and Congruence

In both the Siegel and Hermitian cases, the theory focuses on comparing normalized Eisenstein series $G_k$ (or $G_{k,K}$ in the Hermitian setting) with suitably constructed cusp forms $f$, explicitly analyzing congruences between their Fourier coefficients modulo a prime $p$. The congruence is made precise by considering the expansions

$G_{k}(Z) = \sum_{T} a_{G_{k}}(T)q^T,$

with $T$ ranging over appropriate symmetric (or Hermitian) matrices, and comparing $a_{G_k}(T)\bmod p$ against the corresponding coefficients of a cusp form.

A key normalization in the Siegel case is given by

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

where $B_{2k-2}$ is the standard Bernoulli number, and in the Hermitian case

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

where $B_{k-1,\chi_K}$ is the $(k-1)$th generalized Bernoulli number associated to the Kronecker character of the imaginary quadratic field $K$.

2. Role of (Generalized) Bernoulli Numbers in Forcing Vanishing

The core divisibility condition is that a prime $p$ divides $B_{2k-2}$ (for Siegel forms) or $B_{k-1,\chi_K}$ (for Hermitian forms). Under these circumstances, the following vanishing modular congruences are established for Fourier coefficients:

• In the Siegel case, if $p\mid B_{2k-2}$, then $a_{G_k}(0_2)\equiv 0\ (\text{mod}\; p)$, and $a_{G_k}(T)\equiv 0\ (\text{mod}\; p)$ for all $T$ with $\operatorname{rank}(T)\leq 1$.
• In the Hermitian case, subject to further technical conditions (e.g., $p\not | B_{3,\chi_K}$, $B_{5,\chi_K}$ and $k<p-1$), $p\mid B_{k-1,\chi_K}$ implies $a_{G_{k,K}}(H)\equiv 0\ (\text{mod}\; p)$ for all $H$ with $\det H = 0$.

In both settings, the vanishing of low-rank Fourier coefficients modulo $p$ ensures that the Eisenstein series loses its non-cuspidal terms, thus behaving modulo $p$ as a genuine cusp form.

3. Existence and Construction of Congruent Cusp Forms

Given the vanishing of lower-rank terms of the normalized Eisenstein series modulo $p$, a polynomial or projection argument (e.g., Lemma 3.1 in the Siegel case and Lemma 3.4 in the Hermitian case) establishes the existence of an explicit non-trivial cusp form $f$ such that

$$G_k \equiv f \quad (\text{mod}\; p) \quad\text{and}\quad G_{k,K} \equiv f \quad (\text{mod}\; p).$$

This congruence holds for all Fourier coefficients, and $f$ is constructed so that in each case, there exists at least one maximal rank coefficient that does not vanish modulo $p$, ensuring $f\neq 0$.

Typical explicit formulas (in the Hermitian case) for the constant term are

$$a_{G_{k,K}}(0_2) = -\frac{B_k}{2k}B_{k-1,\chi_K}^{-1},$$

showing that divisibility $p\mid B_{k-1,\chi_K}$ forces $a_{G_{k,K}}(0_2)\equiv 0$.

4. Numerical Examples Illustrating Divisibility Phenomena

The general results are supported by concrete computations. For instance:

• In the degree 2 Siegel setting with $k=10$, $p=43867$ divides $B_8$, yielding

$$G_{10} \equiv 11313\, X_{10} \quad (\text{mod}\; 43867)$$

for $X_{10}$ the Igusa cusp form.

• In the Hermitian context with $K$ of discriminant $-3$ or $-4$, and $k=10$, $p=809$ divides $B_{9,\chi_K}$ and thus

$$G_{10,K} \equiv 554\, F_{10} \quad (\text{mod}\; 809)$$

with $F_{10}$ a Maass-lifted cusp form.

These examples demonstrate explicit instances of Eisenstein congruence divisibility, with the modulus $p$ always being a prime divisor of the relevant (generalized) Bernoulli number.

5. Structural Summary and Theoretical Implications

The main structural mechanism underlying Eisenstein congruence divisibility may be summarized as follows:

• The divisibility condition $p|B_{2k-2}$ or $p|B_{k-1,\chi_K}$ acts as an arithmetic obstruction, forcing vanishing of constant and low-rank coefficients in normalized Eisenstein series.
• This vanishing, in conjunction with suitably constructed polynomial relations among modular forms, produces modular forms which are cusp forms modulo $p$, congruent to the Eisenstein series.
• There is always at least one nonzero full-rank coefficient modulo $p$, guaranteeing the constructed cusp form is nontrivial.

This phenomenon connects the arithmetic of special values (Bernoulli numbers/generalized Bernoulli numbers) with modular form congruence, generalizing well-known facts (such as Ramanujan's congruence modulo 691 for the Eisenstein series $E_{12}$ on $\mathrm{SL}_2(\mathbb{Z})$) to the setting of modular forms in several variables and more general types (Siegel, Hermitian).

6. Relevance for the Arithmetic of Modular Forms

Eisenstein congruence divisibility exposes the interplay between special values of $L$-functions (through Bernoulli-type numbers), congruence primes, and the structure of spaces of modular forms. It answers, in concrete terms, when certain cusp forms exist modulo $p$ “because” $p$ is a nontrivial divisor of a relevant special value, and precisely which congruence phenomenon is enforced. In particular, the divisibility of generalized Bernoulli numbers signals the presence of hidden congruence relationships beyond the classical ones, and these can be explicitly realized in the Fourier expansions and algebraic structure of modular forms.