Dwork towers modulo p^{(k+1)r} for power series with a modular parametrization

Let F(t) = Σ a_n t^n be a power series with integer coefficients such that Θ(τ) = F(t(τ)) is a modular form of weight k, where t(τ) is a Hauptmodul with t = q + O(q²), q = e^(2πiτ). This is the situation of the generating functions of Σ C(2n,n)² t^n, the Franel numbers, the Apéry numbers for ζ(3), the Domb numbers, Σ C(2n,n)³ t^n and the other Apéry-like sequences. Write F_m(t) = Σ_{n<m} a_n t^n for the truncations and t^σ = t(q^p) for the excellent Frobenius lift. In other words, t^σ is the map q ↦ q^p in the canonical coordinate, written as a power series in t. Dwork's classical theory gives F(t)/F(t^p) ≡ F_{p^r}(t)/F_{p^(r−1)}(t^p) modulo p^r. F. Beukers (Supercongruences using modular forms, arXiv:2403.03301) proved that with the excellent lift one has F(t) ≡ F_p(t)·F(t^σ) modulo p² for second-order equations and modulo p³ for third-order ones; this is the case r = 1. He observed that the whole tower F(t)/F(t^σ) ≡ F_{p^r}(t)/F_{p^(r−1)}(t^σ) (mod p^(2r)), r ≥ 1, seems to hold whenever the case r = 1 does, and reported that an attempt to prove it in at least one case had failed. F. Beukers and M. Vlasenko (Dwork crystals III, Conjecture 7.5) state a closely related conjecture for constant-term families and write that they were unable to prove the congruence modulo p^(2s) for any example. As far as we know, no case with r ≥ 2 had been proved. Main results. Theorem 2.1. Assume the following: Θ has no zeros in the upper half-plane (for instance, Θ is an eta-product); t has its pole at a cusp c_∞, where Θ vanishes to order at most that of the pole, and Θ does not vanish at the other cusps; the level is N = 2^a·m with a ≤ 3 and m odd and squarefree; the group may be extended by Atkin–Lehner involutions when the character is trivial. Then, for every prime p not dividing N and every r ≥ 1, F(t)·F_{p^(r−1)}(t^σ) ≡ F_{p^r}(t)·F(t^σ) (mod p^((k+1)r) Z[[t]]). So the modulus is p^(2r) in weight 1, the case Beukers observed. In weight 2 it is the stronger modulus p^(3r), which does not seem to have been stated before, and in weights 3 and 4 it is p^(4r) and p^(5r). Theorem 2.2 (explicit form). The Hecke trace T_r of a natural modular function is a polynomial in t with integer coefficients, of degree at most p^r − p + (p − 1)r_0 < p^r, and F_{p^r}(t) ≡ T_r(t) − χ(p)·p^(k+1)·F_{p^(r−2)}(t) (mod p^((k+1)r)), where the last term is absent for r = 1. Examples covered by the theorem: modulus p^(2r): Σ C(2n,n)² t^n (level 4); the Franel numbers and Σ_j C(n,j)² C(2j,j) (level 6); Σ_j C(n,j) C(2j,j) C(2n−2j,n−j) (level 8); modulus p^(3r): the Apéry numbers for ζ(3) (level 6); Σ C(2n,n)³ t^n (level 4); the Domb numbers (level 6); modulus p^(4r): 2F1(1/3,1/3;1;−27t)³ (level 3); modulus p^(5r): 2F1(1/4,1/4;1;−64t)⁴ (level 2). Why this matters. 1. For a natural class of modular examples, it proves the Dwork-type tower modulo p^(2r) that Beukers observed and that Beukers and Vlasenko conjectured in a related form, for every r and at the level of power series. Each such congruence contains, for every prime p, infinitely many congruences between the coefficients a_n. 2. It explains why the excellent lift improves Dwork's precision p^r to p^((k+1)r). In the proof the gain is a single factor p^(k+1): p^k comes from the automorphy factor of weight k, and one more p from summing p conjugates. The exponent is the weight of Θ plus one. 3. For third-order equations (Apéry numbers for ζ(3), Domb numbers, Σ C(2n,n)³) it gives the modulus p^(3r). This is the function-level analogue of the classical supercongruences A(mp^r) ≡ A(mp^(r−1)) (mod p^(3r)) of Gessel, Beukers and Coster. For higher weights it gives p^(4r) and p^(5r). 4. The method is short and elementary. It uses double cosets of the Hecke operator T_p, orders of modular forms at cusps and the identity U_p V_p = id, and no crystalline cohomology. Method. In the canonical coordinate q the tower becomes a Dwork congruence for the plain lift q ↦ q^p. With P_r = F_{p^r}(t)/Θ it reads P_r(q) ≡ P_(r−1)(q^p) (mod p^((k+1)r)). The key object is G_r(τ) = Θ(τ)·P_(r−1)(pτ), a modular function for Γ_0(Np). Its trace to Γ_0(N) is computed exactly: Tr G_r = G_r + χ(p)·p^(k+1)·Θ·U_p(P_(r−1)). An analysis of the orders at the cusps shows that the trace is a polynomial in t of degree less than p^r; this is where the absence of zeros of Θ in the upper half-plane is used. By induction, U_p(P_(r−1)) ≡ U_p V_p (P_(r−2)) = P_(r−2), so the correction term is congruent to the polynomial p^(k+1)·F_{p^(r−2)}(t). Hence G_r is congruent to a polynomial of degree less than p^r. Comparing the first p^r coefficients shows that this polynomial is F_{p^r}(t). For r = 1 the argument recovers the eta-product case of Beukers' theorems. Verification. All examples were checked in exact integer arithmetic with PARI/GP (via cypari2). The script verify_towers.py runs in about 8 minutes, and its full output is in verify_towers.log (103 checks, all passed). It does the following. (0) For each example it computes F = Θ∘q(t) from the eta-quotients, checks that F has integer coefficients and compares it with the closed formula up to t^59. (a) Tower: the exponent attained is exactly (k+1)r. This is checked for the three smallest primes p in {3, 5, 7, 11} not dividing the level, and for r ≤ 3 with p^r ≤ 125. (b) Control: with the naive lift t ↦ t^p the exponent drops to Dwork's r (r + 1 in two cases with p = 3). (c) Trace: it is a polynomial exactly for the sign χ(p), and its degree equals p^r − p + (p − 1)r_0 in every case tested. The script verify_cubic.py checks the counterexample to the method described in the paper. ligozat.py computes the orders of the eta-products at the cusps. Limits and open questions. The method does not apply when Θ vanishes at an elliptic point instead of a cusp, as for Σ C(2n,n) C(3n,n) t^n. There the trace is not a polynomial, although the tower modulo p^(2r) still holds numerically. Other open questions are: the condition on the level, which can probably be weakened (Zagier's sequence B, at level 9, satisfies the conclusion numerically); the sharpness of the exponent (k+1)r, which is observed in all the examples and for which the paper gives an exact formula of the defect; fourth-order Calabi–Yau equations without a modular parametrization, such as the Hulek–Verrill numbers. Authorship. The author of this work is Roberto Blanco Gómez. The results were obtained with SyntheticMind, the mathematical reasoning engine that the author created, designed and programmed himself. The AI system Claude (Anthropic) collaborated as an assistant, under the author's direction, in finding and checking the argument, in the literature search, in the verification scripts and in the writing. The author has checked the arguments and computations and takes full responsibility for the content. Files: Blanco_Gomez_2026_Dwork_towers_EN.pdf: the paper. Blanco_Gomez_2026_Dwork_towers_ES.pdf: the Spanish version, with the same content. verify_towers.py: the main verification script (Python 3 with cypari2 / PARI/GP). verify_towers.log: its output. verify_cubic.py and verify_cubic.log: the example outside the hypotheses. ligozat.py: orders of the eta-products at the cusps (Python 3, standard library). sources.zip: LaTeX sources of both versions.