Computing overconvergent forms for small primes

Abstract

We construct explicit bases for spaces of overconvergent p-adic modular forms when p = 2, 3 and study their interaction with the Atkin operator. This results in an extension of Lauder's algorithms for overconvergent modular forms. We illustrate these algorithms with computations of slope sequences of some 2-adic eigencurves and the construction of Chow-Heegner points on elliptic curves via special values of Rankin triple product L-functions.

Slopes of Modular Forms and the Ghost Conjecture

“…Buzzard's conjecture is concerned with the (classical) slopes in classical weights, so after observing Fact 3.1 we did not perform a large scale computation comparing ghost slopes against true slopes in classical weights. (Though we did spot check that the ghost conjecture is consistent with the examples computed in [20,Section 4] [20] (see [25] for small primes) takes in any integer weight k and outputs P k (t) mod p M for a specified integer M . One can take k = 0, for example, and compare the ghost series to the characteristic series P 0 (t).…”

Overconvergent modular forms and their explicit arithmetic

“…Said differently, the numbers A w,z u,v (j) are the entries of the infinite matrix of U p • G j with respect to our chosen orthonormal basis for M † k (r). The following lemma estimates their p-adic valuations and is an easy extension of Wan [Wan98, Lemma 3.1]; see [Von15]. Lemma 4.2.…”

Computing $\protect \mathcal{L}$-invariants via the Greenberg–Stevens formula

“…For p ≥ 5, the algorithm of Lauder presented in [17] computes P † (k, t) mod p m for given k and m and runs in polynomial time with respect to p, N and m, and linear time in log(k). This algorithm was extended by Vonk in [21] to include the primes p = 2, 3. These algorithms give us the input for our subsequent computations.…”

“…the polynomial Q ε p,4 (X) ∈ Q p [X] is given by a linear factor with the L-invariant of the unique newform in S 4 (Γ 0 (6)) p−new as a zero. Modulo 2 21 , respectively 3 21 , we have…”

“…The space S 2 (Γ 0 (77)) 7−new,ε is two-dimensional for ε = +1 and three-dimensional for ε = −1. The coefficients of the following polynomials have been reduced modulo 7 21 .…”