Atkin–Swinnerton-Dyer Congruences

• Atkin and Swinnerton-Dyer congruences are p-adic recurrence relations that define coefficients in modular forms, hypergeometric series, and combinatorial sequences.
• They employ two- and three-term recurrences modulo growing powers of p to generalize classical Hecke recursions across diverse arithmetic structures.
• Their study connects p-adic arithmetic, formal group laws, and Galois representations, with practical implications for partition functions and supercongruences.

Atkin and Swinnerton-Dyer Type Congruences

Atkin and Swinnerton-Dyer (ASD) type congruences are $p$-adic recurrence relations for coefficients of arithmetic or automorphic sequences, especially those arising from modular forms and truncated hypergeometric series. Such congruences are generalizations of classical Hecke recursions and were first observed in the 1970s for noncongruence modular forms, but now appear across several arithmetic and combinatorial contexts. Atkin and Swinnerton-Dyer congruences are typically recursive congruences between coefficients at arithmetic progressions indexed by $p$-powers, with modulus growing as a power of $p$, and are a key part of the structure theory for $p$-adic and mod $p$ modular forms, partition functions, and truncated hypergeometric series. Their study reveals deep relationships between modularity, $p$-adic arithmetic, Galois representations, and the theory of formal groups.

1. Definition and Characteristic Form of ASD Congruences

The prototypical ASD congruence arises for a sequence $\{a_n\}$ attached to a modular-like object (e.g., a modular form, partition function, or truncated hypergeometric sum). The general shape is a two-term or three-term recurrence, valid modulo a growing power of $p$, typically

$$a_{mp^s} \equiv A_p a_{mp^{s-1}} + B_p a_{mp^{s-2}} \pmod{p^{c s}}$$

for all $m, s \ge 1$, with $A_p$, $B_p$ depending on the modular form, the sequence, or additional arithmetic data (e.g., a character value, Hecke eigenvalue, or Legendre symbol). In the one-dimensional case (space of modular forms of dimension 1), $B_p$ is typically a (possibly trivial) character times $p^{k-1}$, $k$ the weight, which replicates the Hecke recursion for congruence forms but holds for noncongruence forms only modulo $p^{(k-1)s}$.

In higher-dimensional situations, one encounters $(2d+1)$-term recurrences, where $2d$ is the dimension of the associated Galois or cohomological representation (Li et al., 2013). For truncated hypergeometric series or combinatorial sequences, the recurrence may be explicitly two-term: $$S(p^\alpha n-1;m) \equiv \left(\frac{m(m-4)}{p}\right) S(p^{\alpha-1}n-1;m) \pmod{p^{2\alpha}}$$ as in the case of the ${}_1F_0\bigl(\tfrac12;;-\frac{4}{m}\bigr)$ family (Zhang et al., 2018).

For the partition function $p(n)$ and related objects, Atkin and Swinnerton-Dyer congruences take the form

$p(Q^3 \ell n + \beta) \equiv 0 \pmod{\ell}$

for suitable primes $Q$, $\ell \geq 5$, and $\beta$ in arithmetic progressions, often parameterized by quadratic residue symbols (Ahlgren et al., 2021, Ahlgren et al., 4 Apr 2025).

2. Theoretical Foundations: Modular Forms, Formal Groups, and Galois Representations

ASD congruences are fundamentally rooted in the arithmetic of modular forms and their generalizations to noncongruence subgroups and arithmetic or combinatorial sequences. For congruence subgroups and Hecke eigenforms, the three-term recurrences for Fourier coefficients are identities, but for noncongruence contexts there is typically no Hecke operator, and recurrences only persist modulo powers of $p$ (Li et al., 2013).

The underlying mechanism is often geometric or cohomological:

• Scholl’s construction attaches to spaces of weight $k$ cusp forms on $\Gamma$ a compatible family of $2d$-dimensional $\ell$-adic Galois representations. Local Frobenius polynomials at $p$ control the recurrence satisfied by the modular or arithmetic sequence, with the modulus reflecting the $p$-adic action on (log-)crystalline cohomology (Kazalicki et al., 2013, Li et al., 2013, Allen et al., 7 Nov 2025).
• For weakly holomorphic or meromorphic modular forms, Kazalicki–Scholl and Allen–Long–Saad extend the cohomological framework to include forms with poles, resulting in polynomial-length recurrences where components corresponding to residues at poles are explicitly encoded (Kazalicki et al., 2013, Allen et al., 7 Nov 2025).
• In the case of truncated hypergeometric series, the connection is via formal group laws: the series are coefficients in the formal logarithm of a 1-dimensional commutative formal group. The associated Witt–Frobenius functional equations yield ASD-type recursions (Kibelbek et al., 2012, Li et al., 2013).
• When the underlying motive has complex multiplication (CM), the strength of the congruence can double, resulting in so-called "supercongruences" (i.e., higher modulus than predicted by formal groups), a phenomenon arising from extra endomorphisms on the motive (Kibelbek et al., 2012).

3. Key Results for Truncated Hypergeometric Series

ASD-type congruences have been systematically established for truncated ${}_1F_0$ and related hypergeometric series. For $p$ odd prime, positive integer $n$, and $m \in \{1,2,3\}$,

$$S\left(p^\alpha n-1; m \right) \equiv \left( \frac{m(m-4)}{p} \right) S\left(p^{\alpha-1}n-1; m\right) \pmod{p^{2\alpha}}$$

where the sum is over $k=0$ to $p^\alpha n-1$ of $$\frac{(\frac12)_k}{k!} \left( -\frac{4}{m}\right)^k$$ and $(\tfrac{a}{p})$ is the Legendre symbol (Zhang et al., 2018). For $m=4$, the congruence simplifies: $$\sum_{k=0}^{p^\alpha n-1} (-1)^k \frac{(\frac12)_k}{k!} \equiv p \sum_{k=0}^{p^{\alpha-1} n-1} (-1)^k \frac{(\frac12)_k}{k!} \pmod{p^{2\alpha}}.$$ The proof strategy relies on: $p$-adic expansions of binomial coefficients, Lucas sequences, and base-$p$ combinatorial reductions to eliminate error terms modulo $p^{2\alpha}$, and induction on $\alpha$. The base case for $\alpha=1$ is governed by classical Gauss–hypergeometric evaluations mod $p$.

More generally, hypergeometric supercongruences and their ASD-type recurrences are linked to the arithmetic of associated algebraic varieties (elliptic curves, K3 surfaces), and their formal group laws. For truncated $_rF_{r-1}(1/2, ..., 1/2; 1, ..., 1; \lambda)$, one has

$$F_r(\lambda)_{N_{s+1}} \equiv y_p(\lambda) F_r(\lambda)_{N_s} \pmod{p^{s}}$$

for a $p$-adic unit root $y_p(\lambda)$, and, in certain CM cases, the congruence modulus is $p^{2s}$ (Kibelbek et al., 2012).

4. Atkin and Swinnerton-Dyer Congruences in Combinatorics

Atkin and Swinnerton-Dyer congruences permeate the arithmetic study of partition functions and related combinatorial sequences:

• For the partition function $p(n)$, Ramanujan's classical congruences generalize via Atkin's discovery that for all primes $\ell \geq 5$,

$p(Q^3 \ell n + \beta) \equiv 0 \pmod{\ell}$

for infinitely many $Q$ and appropriate $\beta$, with the set of such progressions governed by the square class of $1-24\beta \pmod{\ell}$ (Ahlgren et al., 2021). This is always possible for “Family I” and for at least $17/24$ of all primes for “Family II,” with the conditions made explicit using modular Galois representations.

• For generalized Frobenius partitions $c\phi_m(n)$ (the $m$-colored Frobenius partition function), similar Atkin-type congruences hold for all but a finite set of pairs $(m, \ell)$:

$$c\phi_m\left( \frac{\ell Q^2 n + m}{24} \right) \equiv 0 \pmod{\ell}$$

for all $n$ satisfying a quadratic residue condition modulo $Q$, and infinitely many auxiliary primes $Q \not = \ell$ (Ahlgren et al., 4 Apr 2025).

• The coefficients of powers of Euler products ($p_k(n)$) and partition-theoretic functions—core partitions, $\ell$-regular partitions, overpartitions—admit infinite families of ASD-type congruences, typically deduced using the action of Atkin $U$-operators, modular equations, and recurrence relations controlled by the arithmetic of eta-quotients and their modular equations (Du et al., 2018, Mestrige, 2020).

5. Cohomological, Modular, and Noncongruence Aspects

Scholl's theory extends ASD congruences to higher rank, noncongruence subgroups, and weakly holomorphic or meromorphic modular forms. For spaces with $\dim S_k(X) = d$, the recurrence takes the form

$$\sum_{j=0}^{2d} p^{(k-1)j} A_j a_{n/p^j} \equiv 0 \pmod{p^{(k-1)(1+v_p(n))}}$$

with $A_j$ arising from the Frobenius polynomial of the attached Galois representation (Li et al., 2013, Kazalicki et al., 2013, Allen et al., 7 Nov 2025). These congruences are valid for any finite-index subgroup, including noncongruence groups, with explicit examples showing new phenomena such as the non-existence of a $p$-adic Hecke eigenbasis in certain weight 3 Fermat-curve cases (Kazalicki et al., 2013).

For meromorphic forms, the ASD congruence is governed by a corresponding cohomology group incorporating residues at the loci of poles, leading to recurrences of degree $2d + (k-1)\#\mathfrak{u}$ for meromorphic forms with poles at $\mathfrak{u}$, and reduction to simpler (two-term) recurrences in the presence of CM (Allen et al., 7 Nov 2025).

6. Relation to Hecke Recursion, Formal Groups, and Supercongruences

ASD congruences provide a $p$-adic shadow of the exact Hecke recursions for congruence modular forms but encode congruence relations, not identities, due to the absence of the Hecke operator structure for noncongruence groups (Li et al., 2013). The congruence modulus is always a (growing) power of $p$, typically $p^{(k-1)s}$, reflecting the weight and $p$-adic geometry rather than the rational representation theory alone.

The formal group law approach demonstrates the interpolation between $p$-adic congruences for sequences associated with motives (e.g., elliptic curves, hypergeometric motives). The “strict logarithm” determines the recurrence mod $p^{s+1}$, with the modulus improving to $p^{2s}$ or higher in the supercongruence case when extra endomorphisms (CM) exist (Kibelbek et al., 2012). The key supercongruence phenomenon occurs, for example, when truncated hypergeometric sums align with the theory of certain modular forms (e.g., coefficients of $\eta(4z)^6$), and the modulus doubles due to CM (Kibelbek et al., 2012).

7. Open Directions and Generalizations

Multiple open problems remain:

• A complete classification of which noncongruence subgroups or arithmetic sequences admit three-term ASD bases for almost all $p$;
• The behaviour and structure of higher-dimensional Scholl representations, including reducibility, automorphy, and explicit description of their $p$-adic L-factors (Li et al., 2013);
• Extensions to higher-order truncated hypergeometric series ${}_rF_s$ or nonstandard parameters, including the derivation of three-term ASD-type recurrences in wider classes;
• Determining explicit congruence conditions on partition-theoretic and combinatorial functions (e.g., overpartitions, smallest parts functions) beyond the classical Hecke and Ramanujan congruences (Garvan, 2010, Andersen, 2012);
• Deeper understanding of “magnetic” meromorphic forms (in the sense of bounded denominators and strong $p$-adic recursions) and their arithmetic/dynamical implications (Allen et al., 7 Nov 2025).

The field continues to link $p$-adic analytic, modular, and Galois-theoretic techniques with explicit arithmetic for families of sequences of geometric or combinatorial origin, frequently leveraging cohomological, modular form, and formal group structures.