0.1 Conjectures

Let M_{k} denote overconvergent eigenforms of weight k. Let f ∈ M_{k} be a finite slope eigenform of slope m. If k = −2n then θ^{2n+1}f is overconvergent of slope m + 2n + 1 and weight 2n + 2. If k is sufficiently close in weight space to −2n, then one might expect that θ^{2n+1}f is close to an overconvergent form, if one is lucky, of slope m + 2n + 1. This motivates the next definition.

Definition 0.1 Let f ∈ M_{k} be an overconvergent eigenform of slope m. We say that f is n-deficient if there exists no overconvergent form g with the following properties:

1. wt(g) = k + 2(2n + 1),

2. θ^{2n+1}f ≡ g mod p,

3. slope(g) = slope(f) + 2n + 1.

Remark. If f is a (classical) oldform of weight k, then f is n-deficient if and only if its twin form is.

We now introduce a space D_{n} that controls n-deficient forms of weight close to −2n.

Definition 0.2 Define D_{n} as follows:

D_{n} := lim_{i→∞}{f|f n-deficient, wt(f) ≡ −2n mod (p − 1)p^{i−1}}.

Here by “limit” we mean the following: An eigenform g ∈ D_{n} if and only if it is approximable by n-deficient forms of weight arbitrarily close to −2n.

It is an observable fact that for overconvergent forms of weight close to −2n, almost all forms are not n-deficient. Here is the conjecture:

Conjecture 1 There is an equality:

θ^{2n+1}D_{n} = infinite slope eigenforms in S_{2n+2}(Γ_{0}(p^{2}), \overline{ℚ}_{p})

Here “cusp form” is meant in the sense of p-adic modular forms, ie. a cusp forms at ∞, which includes forms such as twists of evil twins.

We contrast this with the conjecture (guess?) of Mazur:

Conjecture 2 Let f be a classical modular form on Γ_{0}(p^{n}) for some n. Then f is approximable by finite slope forms if and only if the \overline{ℚ}_{p} representation ρ_{f} is tamely potentially semistable.

Since X_{0}(p^{2}) has semistable reduction over ℚ_{p}(ζ_{p})^{+}, Conjecture 1 is consistent with Conjecture 2. In contrast, Robert (and William?) have proved that the weight 2 form arising from X_{0}(32) is not approximable, and suggested that the same is true of the form corresponding to X_{0}(27). Both these forms are not tamely potentially semistable.

0.2 Computational Evidence

Here are two examples, which are typical of the evidence so far. Let p = 7. There appear to be 0-deficient forms f_{i} of weight 2 · 6 · 7^{i−1} and slope (3 · 7^{i−1} − 1)/2 converging to g, where

θg = q + q^{2} − q^{4} − 3q^{8} + 4q^{11} + O(q^{16}) ∈ S_{2}(Γ_{0}(49))

Here g is the form associated to X_{0}(49), which has CM by ℤ[(1 + \sqrt{−7})/2]. One has f_{0} = Δ, and it appears that θf_{i} ≡ g mod 7^{i}. “Geometrically”, f_{i+1} appears to live in the same family as the twin form of f_{i} (at least, they both have the same slope and are congruent mod 7^{i}).

Here is another example, with n ≠ 0. Let p = 5, and n = 1. Thus we are looking for forms of weight congruent to −2 modulo 4 · 5^{i−1} for large i. The first example I tried was weight k = 4 · 5 · 5 − 2 = 98. The classical slopes are [2, 5, 7, 8, 9, 11, 15], whilst the slopes at weight 104 are [1, 3, 5, 8, 10, 11, 12, 14]. Thus the form of slope 15 is deficient. If h is this form, then one calculates that

θ^{3}h ≡ g = q + q^{2} + 7q^{3} − 7q^{4} + 7q^{6} + 6q^{7} + O(q^{8}) mod 5^{3}

Where g is an element of (the three dimensional space) S_{4}(Γ_{0}(25)). Again something geometric seems to be going on, this slope 15 form h is in the same family as the slope 15 twin form of Δ · E_{6}, which is 1-deficient of weight 4 · 5 − 2. (As expected, θ^{3}Δ · E_{6} ≡ g mod 5^{2}).