Supercongruences in Hypergeometric Functions

Supercongruences are congruences between special values of truncated hypergeometric series that hold modulo high powers of primes. Originally motivated by Ramanujan’s remarkable formulae for 1/π, these congruences connect p-adic analysis, modular forms and arithmetic geometry. In essence, one studies truncated series of the form ∑_{k=0}^{p−1} A_k z^k where A_k are coefficients drawn from generalised hypergeometric functions, and seeks congruences of the type ∑_{k=0}^{p−1} A_k z^k ≡ B_p (mod p^r), with r>1 often much larger than in classical results. Such phenomena were first systematised by Dwork in the late 1960s and later conjectured in families by van Hamme. In recent years the field has broadened dramatically through the introduction of q-analogues, new transformation techniques and connections to cyclotomic polynomials, offering deeper insight into the nature of p-adic L-values, motives of algebraic varieties and q-deformations of modular forms. Concrete instances include congruences modulo p^3 or p^5 for truncated 2F1 and 3F2 series, and q-supercongruences reflecting additional structure over cyclotomic rings.

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Technical terms

Supercongruence: A congruence between truncated hypergeometric sums valid modulo a high power of a prime, exceeding what is expected from naïve combinatorial arguments.

Hypergeometric series: A power series ∑ A_k z^k where the coefficients A_k satisfy a ratio of products of linear functions in k, denoted {}_{n+1}F_n for generalised cases.

p-adic analysis: A branch of number theory studying properties of numbers and functions in the topology induced by the p-adic norm, enabling analytic continuation and congruence investigations.

q-analogue: A deformation of classical objects or identities, replacing integers or binomial coefficients by q-integers or basic hypergeometric terms, often recovering the original result as q→1.

Cyclotomic polynomial: An integer polynomial whose roots are the primitive nth roots of unity; congruences modulo its powers appear naturally in q-series contexts.

Creative microscoping: A method for establishing q-supercongruences by dissecting sums into more tractable pieces and combining transformation formulae in a stepwise, ‘microscopic’ fashion.

References

1. A p p -adic supercongruence conjecture of van Hamme. Proceedings of the American Mathematical Society (2008).
2. Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series. Constructive Approximation (2020).
3. Dwork-type supercongruences through a creative q-microscope. Journal of Combinatorial Theory Series A (2021).