Ramanujan-style congruences of local origin

Automorphic Forms on Feit’s Hermitian Lattices

“…The modulus arises as a divisor of 3 12 − 1, and may be viewed as a divisor of ζ {3} (12)/π 12 , where ζ {3} (s) = (1 − 3 −s )ζ(s), the Riemann zeta function with the Euler factor at 3 omitted. Such congruences "of local origin" were anticipated by Harder in [23, §2.9], and proved in [16,Theorem 1.1] or [6,Theorem 1].…”

“…On the unramified part, Frob −1 3 acts by the U 3 eigenvalue, which is the coefficient of q 3 , and det ρ f is the product of the (k ′ − 1) power of the cyclotomic character and the Galois character associated to χ −3 . It follows that P (X) = 1 + 5022X + 43046721X 2 , noting that 43046721 = 3 16 [2]. Remark 6.3.…”

“…Mégarbané successfully employed a somewhat cruder version of their method [34,Lemme 4.3.2], finding optimal moduli for his congruences. Following him, for a second proof of the congruence 16 ≡ 11 (mod 11) we checked that the reductions mod 11 of v 16 and v 11 are scalar multiples of one another, from which the mod 11 congruence of Hecke eigenvalues follows easily. The greatest common divisor of (λ 16 (T (2) )−λ 11 (T (2) )) and (λ 16 (T ( √ 3) ) − λ 11 (T ( √ −3) )) is 2 3 · 3 3 · 11, but mod 2 and mod 3, v 16 is not a scalar multiple of v 11 , so we were unable to use his method to increase the modulus beyond 11.…”

“…Assuming these guesses, after cancellation we more-or-less recover various congruences involving elliptic modular forms of levels 1 or 3, including Ramanujan's congruence, Eisenstein congruences "of local origin" [16] and Ribet-Diamond level-raising congruences [12,38]. But we also obtain instances of conjectured congruences involving automorphic representations of U(2, 2), analogous to those of Harder, again with the moduli coming from critical L-values [15].…”

A generalization of Ramanujan's congruence to modular forms of prime level

“…This and later congruences mean that the difference between the Fourier coefficients at the cusp ∞ of the two sides belong to I (after clearing the denominator of the constant term), and we always normalize Hecke eigenforms to have the coefficient of q equal to 1. This congruence was recently generalized to newforms f of prime level by Billerey-Menares [3] and by Dummigan-Fretwell [13], for Fourier coefficients of index coprime to the level. In this paper we refine these results, by determining also the Atkin-Lehner eigenvalue of the newform involved, thus obtaining congruences for all coefficients.…”