Slopes of 3-adic overconvergent modular forms

If r = 12 and (uij) is the matrix of the U operator in the above basis, then the numbers uij satisfy a recurrence formula: there is a p × p matrix M such that uij = ∑p r,s=1Mrsui−r,j−s. Furthermore, M is skew-upper-triangular and constant on off diagonals; and the coefficients uij satisfy uij = jiuji. The case p = 2 is extensively studied in [BC05]. Here the recurrence relation is simple enough that one can explicitly write down uij as a hypergeometric term in (i, j) and verify that it satisfies the correct initial conditions and recurrence. One can similarly spot a hypergeometric formula for the terms of the unique matrices A,D,B such that ADB = U and A is lower triangular, B is upper triangular, A and B have diagonal entries 1 and D is diagonal; that these formulae give the correct entries for U can be checked by evaluating the corresponding sum using Dougall’s 7F6 identity. From the formulae for A and B when p = 2, one can easily check that A and B are congruent to the identity matrix mod p; it follows that the slopes of U are equal to the valuations of the terms of D. One can also use this to prove the Gouvea-Mazur spectral expansion conjecture (that the finite slope eigenfunctions span the space), as shown in [Loe07]. For p = 3 things are more complicated. In [Loe07, Conjecture 3.1], it was noted that the entries of the matrix D appear to satisfy