WHERE THE SLOPES ARE

FERNANDO Q. GOUVÊA

Let N be a positive integer (the “level”), let k ≥ 2 be an integer (the “weight”), and let S_{k}(N, ℂ) denote the finite-dimensional ℂ-vector space of cuspidal modular forms of weight k and trivial character on Γ_{0}(N) defined over ℂ. Elements f ∈ S_{k}(N, ℂ) can be specified by giving their Fourier expansions

f = a_{1}q + a_{2}q^{2} + ⋯ = ∑_{n=0}^{∞} a_{n}q^{n},

where q = e^{2πiz} and z is in the complex upper halfplane. This expansion is sometimes described as “the q-expansion at infinity” of the modular form f. There exists a natural basis of S_{k}(N, ℂ) consisting of forms all of whose Fourier coefficients are in fact rational. We denote the ℚ-vector space spanned by this basis by S_{k}(N, ℚ). Note that then we have

S_{k}(N, ℂ) = S_{k}(N, ℚ) ⊗ ℂ.

For each prime number p which does not divide N there is a linear operator T_{p} acting on S_{k}(N, ℂ), known as the p-th Hecke operator. (In fact, the T_{p} stabilize S_{k}(N, ℚ).) A modular form which is an eigenvector for all of these linear operators simultaneously is called an eigenform; the space S_{k}(N, ℂ) has a basis made up of eigenforms, and the Fourier coefficients of these eigenforms can be normalized (by requiring a_{1} = 1) to belong to a finite extension of ℚ.

The eigenvalues of the T_{p} operator encode significant arithmetic information about the modular form and various other objects which can be attached to it (for example, a Galois representation). In our setting, the eigenvalue of T_{p} acting on an eigenform f ∈ S_{k}(N, ℂ) is a totally real algebraic number whose absolute value (with respect to any embedding of \overline{ℚ} into ℂ) is between −2p^{(k−1)/2} and 2p^{(k−1)/2}. If we normalize the eigenvalues by dividing by p^{(k−1)/2}, the normalized eigenvalues are real numbers in the interval [−2, 2], and we can ask about their distribution in that interval. The Sato-Tate Conjecture, still very much an open problem, predicts the properties of that distribution for a fixed modular form f and varying p. We can also, however, fix the prime p and consider the distribution as k → ∞ of all the eigenvalues of T_{p} corresponding to eigenforms of weight k. This was done by Serre in [Ser97] and by Conrey, Duke, and Farmer in [CDF97].

Project sponsored in part by the National Security Agency under Grant number MDA904-98-1-0012. The United States Government is authorized to reproduce and distribute reprints.

The goal of this paper is to begin the study of an analogous question in the p-adic setting by presenting a wide range of numerical data. The unexpected regularities in the data suggest several interesting questions that deserve further investigation.

We fix a prime number p, then, and consider the situation in a p-adic setting. We choose an embedding of the algebraic closure of ℚ into the completion ℂ_{p} of an algebraic closure of ℚ_{p}, and then we define

S_{k}(N, ℂ_{p}) = S_{k}(N, ℚ) ⊗ ℂ_{p},

and similarly for S_{k}(N, F) where F is any extension of ℚ_{p}. In the padic context, it turns out that the right operator to consider is not T_{p} but rather the Atkin-Lehner U operator, which can be described by its action on q-expansions:

U (∑ a_{n}q^{n}) = ∑ a_{np}q^{n}.

If p does not divide N, this operator does not stabilize the space S_{k}(N, F), but it does stabilize the larger space S_{k}(Np, F), and once again we can consider eigenforms and the corresponding eigenvalues of U.

Assume p ∤ N, and let f ∈ S_{k}(Np, ℂ_{p}) be an eigenform for U, so that U(f) = λf. The p-adic valuation of the eigenvalue λ turns out to play a crucial role in the p-adic theory. We shall call this valuation the slope of the eigenform f:

Definition. Given an U-eigenform f of level Np, weight k and eigenvalue λ, we define the slope of f by

slope(f) = ord_{p}(λ).

The name “slope” comes from the p-adic theory of Newton polygons: the slopes of the eigenforms in S_{k}(Np, ℂ_{p}) are determined by the slopes of the Newton polygon of the characteristic polynomial of the U operator acting on this space.

We are interested in the distribution of the slopes of the U operator for fixed level and varying weight. (Thus, we are writing the eigenvalues as a p-adic unit times a power of p, and then we are ignoring the unit part.) All of our results are numerical, but we feel they are of sufficient interest and that they raise significant questions that need to be addressed on a theoretical level.

Though the questions we ask are supported by a substantial amount of numerical data, we are a little hesitant to label them as conjectures, basically because all our data is for small values of p and of k. On the other hand, we emphasize that the statements labeled as “Question” in what follows are indeed supported by a considerable amount of data.

I am grateful to several people for their contributions to this work. The main question discussed in this paper was raised by Dipendra Prasad in conversation with the author. The computations were done with the gp program using a modified version of a script written by Robert Coleman. Finally, Barry Mazur, Kevin Buzzard, Jean-Pierre Serre, David Farmer, Brian Conrey, Siman Wong, and Naomi Jochnowitz made significant suggestions and observations at several points.

I am grateful to NSA for a grant which offered support for the first phase of this work. Most of the computations were done one a large SGI machine at the Paul J. Schupf Scientific Computing Center at Colby College; I would like to thank the college and Mr. Schupf for making this resource available.

1. Setting up the Problem

Let p be a prime number, k ≥ 2 an even integer, and N a positive integer not divisible by p. Let ord_{p} be the p-adic valuation mapping, normalized by ord_{p}(p) = 1. For any field F of characteristic zero, we write S_{k}(N, F) to denote the F -vector space of cuspidal modular forms of weight k for Γ_{0}(N) (with trivial character) whose Fourier coefficients all belong to F. We will essentially be concerned only with F = ℚ_{p}, since the Newton polygon (and therefore the slopes) can be computed already in this context, though the eigenforms themselves may only be defined over some extension of ℚ_{p}. Our computations will be restricted to the case N = 1 (and hence k ≥ 12), but it seems reasonable to set up the problem for general level.

There are two natural inclusions of S_{k}(N, F) into S_{k}(Np, F); on q-expansions the first is the identity mapping and the second is the Atkin-Lehner V operator, which sends q to q^{p}. The subspace spanned by the images of both maps is called the space of oldforms in S_{k}(Np, F); it has a natural complement called the space of newforms.

The Atkin-Lehner U operator maps S_{k}(Np, F) to itself, acting on q-expansions by

U(∑ a_{n}q^{n}) = ∑ a_{np}q^{n}.

This action stabilizes the space of newforms and also the space of oldforms. It follows from the Atkin-Lehner theory of change of level (see [AL70]) that the action of U on newforms can be diagonalized (possibly after extending the base field), and that all the eigenvalues are equal to ±p^{(k−2)/2}, and hence have slope equal to (k − 2)/2. Thus, as far as the slopes are concerned, the interesting questions have to do with the action of U on the oldforms. This is best understood by relating it to the action of the Hecke operator T_{p} on forms of level N; this yields the theory of “twin eigenforms” discussed in [GM92], which we recall briefly.

The Hecke operator T_{p} can be diagonalized on S_{k}(N, ℂ_{p}). Let f ∈ S_{k}(N, ℂ_{p}) be a normalized cuspidal eigenform, and let a_{p} be the eigenvalue of T_{p} acting on f. Finally, let f_{1}, f_{2} ∈ S_{k}(Np, ℂ_{p}) be the two images of f under the maps described above. The U operator stabilizes the two-dimensional space generated by f_{1} and f_{2}, and its characteristic polynomial is x^{2} −a_{p}x+p^{k−1}. If this polynomial has two distinct roots, the action of U on this two-dimensional subspace can be diagonalized, and the eigenvalues will be precisely the two roots of the characteristic polynomial. Thus, the slopes of the two resulting eigenforms can be easily determined:

• If ord_{p}(a_{p}) < (k − 1)/2, the two eigenvalues have p-adic valuation equal to ord_{p}(a_{p}) and k − 1 − ord_{p}(a_{p}).

• If ord_{p}(a_{p}) ≥ (k − 1)/2, then both eigenvalues have p-adic valuation (k − 1)/2.

It has been conjectured by Ulmer that the polynomial x^{2}−a_{p}x+p^{k−1} always has two distinct roots. Specifically:

Conjecture (Ulmer). The action of U_{p} on S_{k}(Γ_{0}(Np), ℚ_{p}) is semisimple. In particular, the polynomial x^{2} − a_{p}x + p^{k−1} always has distinct roots.

Coleman and Edixhoven have shown that this is true for k = 2 and that for general k it follows from the Tate Conjecture (see [CE98]).

It is easy to see that if the polynomial has a double root then we must have a_{p} = ±2p^{(k−1)/2}. For N = 1, it is possible to show that this cannot happen.

Theorem 1. If N = 1, then a_{p} ≠ ±2p^{(k−1)/2}, and therefore the polynomial x^{2} − a_{p}x + p^{k−1} always has distinct roots.

Proof. (Conrey and Farmer)Let f(x) be the characteristic polynomial of T_{p} acting on S_{k}(1, ℚ). Suppose that one of the roots of f(x) is equal to ±2p^{(k−1)/2}. Then, since f(x) has rational coefficients and k is even, ∓2p^{(k−1)/2} must also be a root of f(x). Hence, f(x) must be divisible by x^{2} − 4p^{k−1}.

Consider first the case p ≠ 3. Then one knows from [CFW00], that

f(x) ≡ \begin{cases}(x − 2)^{d} (mod 3) if p ≡ 1 (mod 3) \\ x^{d} (mod 3) if p ≡ 2 (mod 3) \end{cases}

It is easy to see that either factorization is inconsistent with divisibility by x^{2} − 4p^{k−1}.

Finally, if p = 3, we know, from [Hat79], that

f(x) ≡ (x − 4)^{d} (mod 8),

which again is inconsistent with divisibility by x^{2} − 4p^{k−1}. Hence, no root of f(x) can be equal to ±2p^{(k−1)/2}.

We will always have N = 1 in the computations below, and thus won’t need to worry about double roots. In general, whenever the analogue of Theorem 1 holds, we can indeed read off the slopes of U acting on the oldforms in S_{k}(Np, ℚ_{p}) by determining the slopes of T_{p} acting on S_{k}(N, ℚ_{p}). Specifically, for each slope obtained in level N one obtains a pair of slopes α′ and α′′ in level Np, satisfying

• 0 ≤ α′ ≤ α′′ ≤ k − 1

• α′ + α′′ = k − 1 with α′ < α′′ unless they are both equal to (k − 1)/2. We define the slope sequence for level N, weight k, and prime p to be the ordered list of slopes

(α_{1}, α_{2}, …, (k − 1) − α_{2}, (k − 1) − α_{1})

for U acting on the oldforms in S_{k}(Np, ℚ_{p}), where we repeat slopes that occur with multiplicity. The number of elements in this sequence is equal to twice the dimension of S_{k}(N, ℚ_{p}). Since the slope sequence is symmetric under α ↔ (k −1)−α, we will usually specify it by giving only the first half of the slope sequence. Whenever ord_{p}(a_{p}) < (p−1)/2, it follows from the discussion above that this first half is the same as the slope sequence for T_{p} acting on S_{k}(N, ℚ_{p}).

Since we know that the slopes are in the interval [0, k − 1] (and we want to vary k), it makes sense to normalize the slopes by dividing them by k − 1.

Definition. Suppose f is either a T_{p}-eigenform of level N or a Ueigenform f of level Np. Let k be the weight of f and let a_{p}(f) be the eigenvalue (of T_{p} or of U). We define the supersingularity of f by

ss(f) = \frac{ord_{p}(a_{p}(f))}{k − 1}.

Let f ∈ S_{k}(N, ℂ_{p}) is an eigenform for T_{p} and (assuming Ulmer’s Conjecture is true) f′, f′′ are the two old u-eigenforms corresponding to it as above. Then, provided that ss(f) ≤ 1/2, we have

ss(f′) = ss(f)

and

ss(f′′) = 1 − ss(f),

and both numbers are in the interval [0, 1]. Thus, the sequence of supersingularities corresponding to old eigenforms of weight k and level Np is a normalized version of the slope sequence, and can be computed via the supersingularities of forms of level N, provided these are small enough.

(One can think of ss as a function on the eigencurve studied by Coleman and Mazur in [CM98]. It will be an continuous function on the eigencurve, except along the k = 1 locus. Notice, however, that classical eigenforms of weight 1 will always have slope zero; defining ss(f) = 0 for such forms gives a continuous extension of ss to classical forms of weight 1. No such continuous extension is possible at points corresponding to non-ordinary forms of weight 1.)

We define the supersingularity sequence in weight k

(η_{1}, η_{2}, …, 1 − η_{2}, 1 − η_{1})

by

η_{i} = ss(f_{i}) = \frac{slope(f_{i})}{k − 1}

as f_{i} runs through the old eigenforms of weight k on Γ_{0}(Np). The supersingularity sequence is contained in the interval [0, 1] and is symmetric under η ↔ 1 − η.

One of the main problems we want to consider is to understand the distribution of the supersingularities in the interval [0, 1] when we fix the level N and let k → ∞. This problem can be expressed in measuretheoretic terms, as in [Ser97]: considering N and p as fixed, for each k we define a probability measure μ_{k} on the interval [0, 1] by putting a point mass at each supersingularity η_{i}: let d_{k} = dim S_{k}(N, ℚ_{p}), and set

μ_{k} = \frac{1}{2d_{k}} ∑_{i=1}^{d_{k}} (δ_{η_{i}} + δ_{1−η_{i}}) ,

where δ_{x} is the Dirac measure at x. The question then is whether the measures μ_{k} tend to a limit as k → ∞, and if so to determine that limit measure. One can also consider several variants of this idea. For example, we might study the measure given by the first half of the slope sequence only (or, equivalently if we always have ord_{p}(a_{p}) < (k − 1)/2, by the slope sequence in level N).

2. Computations

For our computations, we restricted to the case N = 1, which then means that one only gets non-trivial results for even weights k ≥ 12. For each prime number p ≤ 100, we computed the Newton polygon of T_{p} acting on forms of weight k and level 1 for weights k ≤ 500. Since in every case the slopes were less than (k − 1)/2, the slopes we obtained are exactly the first half of the slope sequence for the U operator acting on oldforms of level p, as described above.

The method used for computation was straightforward: the space of cuspforms of weight k and level 1 has a basis consisting of forms E_{4}^{a}E_{6}^{b}Δ, where E_{4} and E_{6} are the Eisenstein series of weight 4 and 6 respectively, Δ is the unique cuspform of weight 12, and 4a + 6b + 12 = k. Using this explicit basis we determined the characteristic polynomial of T_{p} and computed its Newton slopes, then produced supersingularities by dividing by k − 1. The computation was done with the GP calculator [BBCO]; the basic GP functions we needed were based on a script originally written by Robert Coleman. The main constraint on the computation was the memory required for computing the characteristic polynomial: larger k meant working with a larger basis, and larger p meant that we needed to use more terms from the q-expansion of the modular forms. The full output of the computations can be found on the web at

    http://www.colby.edu/personal/fqgouvea/slopes/

3. The slopes are smaller than expected

As already mentioned above, in every case we found that every slope in the Newton polygon of T_{p} acting on forms of level N was smaller than (k − 1)/2, from which it follows that U acts semisimply on the space of oldforms and that the slope sequence for T_{p} is indeed the same as the first half of the slope sequence of U acting on oldforms of level Np.

Question 1. Fix a prime number p. Let S_{k}(1, ℂ_{p}) be the space of cuspforms of weight k and level 1. Let f ∈ S_{k}(1, ℂ_{p}) be an eigenform, and let a_{o}(f) be the eigenvalue of T_{p} acting on f. Is it true that

ord_{p}(a_{p}(f)) < \frac{k − 1}{2}

always?

Notice that a positive answer to this question implies that a_{p}(f) ≠ 0, yielding a vast generalization of a famous conjecture of Lehmer about the case k = 12. This observation also shows that any generalization of this statement to higher level must be worded so as to exclude forms with complex multiplication and must also be stated for weight k sufficiently large.

As pointed out above, the fact that this inequality does hold in every case we computed means that we do not need to worry about Ulmer’s conjecture and that for the weights and primes under consideration the slope sequence for T_{p} in level 1 is identical to the lower half of the slope sequence for U on oldforms of level p.

4. The slopes are much smaller than expected

In fact, one sees much more. Even a cursory observation of the tables suggests that the slopes are much smaller than one might expect. In fact, we found that in almost every case the supersingularities for weight k and prime p are smaller than 1/(p + 1). In other words, the inequality

ss(f) ≤ \frac{1}{p + 1}

holds almost always for forms of level N. It follows that the sequence of supersingularities for weight k is almost always contained in [0, \frac{1}{p+1}] ∪ [\frac{p}{p+1}, 1].

We need to be precise about what we mean by “almost always.” What seems to emerge from the data is this: for almost all primes, the inequality is true for every eigenform. For a few exceptional primes, there is a thin set of weights for which we find supersingularities that are just a bit bigger than 1/(p + 1).

Question 2. Is it true that for almost all primes p the inequality

ss(f) ≤ \frac{1}{p + 1}

holds for every eigenform f ∈ S_{k}(1, ℂ_{p})? Equivalently, is it true that for almost all primes p we have

ss(f) ∈ [0, \frac{1}{p + 1} ∪ [\frac{p}{p + 1}, 1]

for every eigenform f ∈ S_{k}(p, ℂ_{p})?

Prime p Weights k
59  16, 46, 76, 106, 136, 166, 196, 226, 256

286, 316, 346, 376, 406, 436, 466, 496

79  38, 44, 118, 124, 198, 204, 278, 284

358, 364, 438, 444

2411 12
15271 16
187441 16
3371 20
64709 20
27310421 26

Table 1. Known exceptions to η_{i} ≤ 1/(p + 1)

To be more explicit about “almost all primes,” in our computations exceptions to this inequality occurred only for p = 59 and p = 79; for each of these primes, the inequality fails to hold for the highest-slope form in certain weights. See Table 1 for the list of weights at which exceptional slopes appear; we discuss this list of weights further below. Other exceptions to the inequality, outside the range of this computation, can be read off from the results in [Gou97]; they correspond to forms of weights k = 12, 16, 20 that are non-ordinary with respect to large primes. The final entry in the table comes from a computation by Atkin. The full list of primes and weights for which we know of a slope that does not satisfy the inequality is given in Table 1. For each (p, k) pair, we found that exactly one slope in the first half of the slope sequence violates the inequality.

The structure of the table suggests that if there are any exceptional forms for a prime p, then there will be exceptional forms of relatively small weight, and the remaining exceptional forms will be in some sense related to these. This suggests the question

Question 3. Let p be a prime. Is it true that if the inequality

ss(f) ≤ \frac{1}{p + 1}

holds when f ∈ S_{k}(1, ℂ_{p}) and k ≤ p + 1, then it holds for every f ∈ S_{k}(1, ℂ_{p}) for all k?

We will postpone the discussion of the specific list of weights for which an exceptional form exists to a later section, and consider what these inequalities imply about the distribution of the supersingularities as k grows.

Assuming an affirmative answer to the questions above, we know that for almost all primes the measure μ_{k} is supported on [0, \frac{1}{p+1}] ∪ [\frac{p}{p+1}, 1]. This is not true for the other primes. However, if we focus on the exceptional slopes and compute the corresponding supersingularities, we see that in our examples ss(f) seems to get closer to 1/(p + 1) as k grows. Let p = 59, for example; the sequence of supersingularities corresponding to the exceptional slopes in Table 1 is

0.066, 0.022, 0.026, 0.019, 0.022, 0.018, 0.020, 0.017, 0.019,
0.017, 0.019, 0.017, 0.018, 0.017, 0.018, 0.017, 0.018

Here, of course, 1/(p + 1) = 1/60 = 0.01666 …, and the exceptional values of the supersingularity seem to be (slowly) approaching this value as the weight grows.

Encouraged by this, we can try to make “almost always” precise by using the measure-theoretic formulation:

Question 4. Is it true that the sequence of measures {μ_{k}} converges, as k → ∞, to the uniform measure on the set [0, \frac{1}{p+1}] ∪ [\frac{p}{p+1}, 1]?

We have checked this by computing the first five moments of μ_{k} for our range of (p, k). The results strongly suggest an affirmative answer when there are no exceptional forms. When exceptional forms exist, the convergence is much slower, but it still seems reasonable that the limit measure will be as above.

It is natural, at this point, to ask what, if anything, has actually been proved in this direction. We know of no general results which suggest that “exceptional” slopes are rare. For specific primes, there are some hints. For p = 3, Lawren Smithline has shown in [Smi00] and [Smi] that for every form of level 1 and weight k = 2 · 3^{a} we have

slope(f) < \frac{k}{4}.

For these values of k this is in fact equivalent to our inequality

slope(f) ≤ \frac{k − 1}{4}.

if we assume that all the slopes are integral (as we point out below, the slopes seem to be almost always integral).

For p = 2, Kevin Buzzard has formulated a conjectural description of all the slopes that implies, in particular, that the inequality

slope(f) ≤ \frac{k − 1}{3}

always holds for forms of level 1, i.e., that there are no exceptional forms for p = 2. (Buzzard’s conjectural description also implies that all the 2-adic slopes for level 1 are integral.)

These hints suggest that something is going on. It seems to us that the fact that the inequality is so often true demands some explanation. In particular, one would like to know whether there is something special about the cases where it fails. It is natural to wonder whether one can identify a specific property of the modular forms or the Galois representations corresponding to exceptional pairs (p, k). See the discussion below on the possible connection to the Θ operator for more on this.

5. The slopes are almost always integers

Perhaps one of the more surprising outcomes of the computations is the fact that almost all the slopes we obtained are integers. Of all the observations we make, this is the one that is most likely to be merely an effect of the fact that we work only with small primes. The location of the exceptions, however, suggests that something else may be going on. Specifically, non-integral slopes occur in our computations only for p = 59 and p = 79, the same primes for which exceptional slopes occur. Furthermore, the fractional slopes we observe are connected to the exceptional slopes, in the following remarkable way:

1. A weight k for which there exists exceptional form with slope equal to 2 is preceded by a weight k − 2 for which the slope sequence contains two slopes equal to 1/2. For p = 59, this happens for the pairs of weights (74, 76) and (104, 106); for p = 79, weights (116, 118), (122, 124).

2. A weight k for which there exists exceptional form with slope equal to 3 is preceded by a weight k − 2 for which the slope sequence contains two slopes equal to 3/2, and that weight is preceded by a weight k − 4 whose slope sequence contains two slopes equal to 1/2. This happens for p = 59 and weights (132, 134, 136) and (162, 164, 166); for p = 79, weights (194, 196, 198) and (200, 202, 204).

3. A weight k for which there exists exceptional form with slope equal to 4 is preceded by a weight k − 2 for which the slope sequence contains two slopes equal to 5/2, that weight is preceded by a weight k−4 whose slope sequence contains two slopes equal to 3/2, and that weight is preceded by a weight k−6 whose slope sequence contains two slopes equal to 1/2. This happens for p = 59 and weights (190, 192, 194, 196) and (220, 222, 224, 226); for p = 79 and weights (272, 274, 276, 278) and (278, 280, 282, 284).

4. And so on. An exceptional form of slope n is and weight k is accompanied by a “trail” of pairs of forms of slope

\frac{2n − 3}{2}, \frac{2n − 5}{2}, \frac{2n − 7}{2}, …, \frac{1}{2}

and weight

k − 2, k − 4, k − 6, …, k − 2(n − 1)

(one pair for each weight). These patterns can overlap without interfering. For example, there

are exceptional forms of slope 6 for p = 79 and weights 438 and 444. Each has its trail of weights for which fractional slopes exist. Because of the exceptional form at weight 438, there are forms of slope 1/2 at weight 428 and at each subsequent weight, up to forms of slope 9/2 at weight 436. Because of the exceptional form at weight 444, there are forms of slope 1/2 at weight 434 and at each subsequent weight, up to forms of slope 9/2 at weight 442. Hence, for example, at weight 436 we have both a pair of forms of slope 3/2 and a pair of forms of slope 9/2.

For the complete list of slope sequences which contain either exceptional or non-integral slopes, see

http://www.colby.edu/personal/fqgouvea/slopes/exceptional.html

This suggests the question:

Question 5. How often are the slopes integral? Is it true that nonintegral slopes occur only when p is a prime for which there exist exceptional slopes?

All of the non-integral slopes we see are in fact half-integral, as the description above shows.

6. The shadow of the Θ operator

For this section, we limit ourselves to the two primes for which we have found exceptional and non-integral slopes, p = 59 and p = 79. As we mentioned above, the list of weights for which exceptional slopes occur seems to have some structure. In this section, we explore a possible connection between the list of exceptional weights and the theory of the Θ operator.

We recall the basic facts about Θ. As above, we restrict to the case of level N = 1. The Θ operator is the operator that acts on q-expansions as q \frac{d}{dq}, so that

Θ (∑ a_{n}q^{n}) = ∑ na_{n}q^{n}.

(If we think of modular forms as functions on the complex upper halfplane, then q = e^{2πiz} and Θ is just \frac{d}{dz}.) As is well known (see [Gou97]), if f is a modular form then Θf is not a modular form, though it is “almost” modular in some sense (the p-adic story is a little different: see [Kat73, Gou88, CGJ95]). If we go ahead and formally compute the Hecke operators on Θf, we see that if f is an eigenform then so is Θf, and that slope(Θf) = 1 + slope(f).

On the other hand, Θ does define an operator on modular forms modulo p, in which case it maps forms of weight k to forms of weight k + p + 1 (see [Joc82, Kat77]). The formula for the change in the slope no longer makes sense, of course. When one considers modular forms modulo p, one can only distinguish between forms of slope zero and forms of positive slope. What we can say, then, is that if f is a modular form modulo p then Θf will be a modular form modulo p and will always have positive slope.

Finally, recall that it is possible for forms whose weight differs by a multiple of p − 1 to have identical q-expansions. (Basically, this is because the q-expansion of the Eisenstein series E_{p−1} is congruent to 1 modulo p.) Thus, if f is a modular form modulo p, one can ask what is the minimal weight k for which f lifts to a form of weight k in characteristic zero. Considering how the Θ operator affects this minimal weight leads to the theory of Θ-cycles, discussed in [Joc82]. We only need a small portion of the theory. Suppose f is a form of positive slope and weight k, 4 ≤ k ≤ p − 1. Consider the forms f_{i} = Θ^{i}f. It is clear that f_{p−1} = f (in the sense that they have the same q-expansion; remember that all of this is happening modulo p), so the f_{i} form a cycle. It is natural to ask whether there are any other values of i for which the minimal weight is small (i.e., less than p − 1). It turns out that this happens only for i = p − k + 1, in which case the form f_{i} is the reduction modulo p of a form of weight p + 3 − k.

The upshot, for us, is simply this: given a form of positive slope and weight k < p − 1, there must also exist a form of positive slope and weight p + 3 − k. This explains part of the data above: for p = 59, the existence of a form of weight 16 and slope 1 forces the existence of a form of weight 46 and positive slope; for p = 79 the existence of a form of weight 38 and slope 1 forces the existence of a form of weight 44. The theory does not predict that these forms should have slope 1, but it turns out that they do. (This is not really surprising, since the slopes tend to be integers and they also tend to be “as small as possible.”)

Notice that this whole theory refers only to modular forms modulo p. If f is an eigenform, then all of the Θ^{i}f are eigenforms modulo p; by the Deligne-Serre Lemma (see [DS74, Lemma 6.11] or [AS86, Prop. 1.2.2]), they lift to eigenforms in characteristic zero (but recall that the Θ^{i}f are not themselves modular forms, so the lifts will only be congruent to them). We know these eigenforms will have positive slope (which, as we noted, is “visible” modulo p as the fact that U(f) ≡ 0 (mod p)), but there seems to no reason to predict anything further about their slope.

Nevertheless, it seems that the Θ operator has a “shadow” in characteristic zero. To see this, consider the exceptional slopes for p = 59. For the case of weight 16, the occurrence of a large slope seems to be “accidental,” but its “propagation” to higher weights seems to be linked to the Θ operator. Let f_{0} be the (unique) form of weight 16 and level 1; its 59-adic slope is 1, because its 59-th Fourier coefficient is divisible (once) by 59. If we reduce f_{0} modulo 59, then its image under Θ can be lifted to an eigenform of weight 76. This is exactly the exceptional form of weight 76, and it has slope 2. The same pattern continues as we iterate Θ, both starting with the form of weight 16 and starting with the form of weight 46.

Thus, it seems that, at least in the case of forms with exceptional slope, if we start from a form f of slope a and weight k, the Θ operator does somehow “produce” a form of weight k + p + 1 and slope a + 1 (though of course it will only be congruent to Θf modulo p, so that the congruence is not strong enough to explain the relation between the slopes). If this is correct, it means that for p = 59 there will be two infinite sequences of forms of exceptional slope, one in weights 16 + 60i and the other in weights 46 + 60i. In each sequence, the slope is equal to i + 1 and each exceptional form is congruent modulo 59 to the image under Θ of the previous form in the sequence. For small enough weights, we have checked that this is indeed what happens.

The same analysis explains the two sequences of exceptional forms modulo 79, of weights 38 + 80i and 44 + 80i. Again, the forms in each sequence seem to represent a shadow in characteristic zero of the Θ operator on forms modulo p.

Question 6. Let f be an eigenform of level 1, weight k, and exceptional slope a. Is it true that for every integer i > 0 there exists an eigenform f_{i} of level 1, weight k + i(p + 1), and slope a + i, such that f_{i} ≡ Θ^{i}f (mod p)?

One can easily compute what happens to the supersingularity along such a family. Under the same assumptions as in the question above, suppose

ss(f) = \frac{a}{k − 1} = \frac{1}{p + 1} + e,

where, since we are assuming f is exceptional, we have e > 0. Assuming the existence of the forms f_{i}, set

ss(f_{i}) = \frac{1}{p + 1} + e_{i}.

Then one easily computes that

e_{i} = \frac{e}{1 + i\frac{p+1}{k−1}} .

In particular, we have e_{i} > 0 for all i, so that all of the f_{i} are exceptional. Note also that e_{i} → 0 as i → ∞, as suggested in question 4.

It might be useful to point out that the data show that if we start with a non-exceptional f it need not be true that a “shadow of Θf” will exist. One can see this by simply looking at the slope sequences at weight k and weight k + p + 1 and noting that the presence of a in the first sequence does not necessarily imply that a + 1 appears in the second. So we could formulate a broader question:

Question 7. Let f ∈ S_{k}(1, ℂ_{p}) be an eigenform of slope a. Under what conditions does there exist an eigenform f_{1} ∈ S_{k+p+1}(1, ℂ_{p}) which is of slope a + 1 and is congruent to Θf?

(Note that there always does exist an eigenform which is congruent to Θf, so the crucial question here concerns the behavior of the slopes.)

At the level of the Galois representations attached to modular forms modulo p, the Θ operator corresponds to a Tate twist. Thus, it follows from the discussion above that for p = 59 or p = 79 all the forms with exceptionally large slope are attached to Galois representations modulo p which seem to be connected by Tate twists.

In fact, the situation is even stranger. Let us take p = 59 start once again with the unique cuspform f of weight 16; it has slope 1. As pointed out above, the reduction modulo 59 of Θf is an eigenform of weight 76; its lift to characteristic zero is an eigenform of slope 2, which is therefore exceptional. We could also consider, however, the form E_{58}f, whose reduction modulo 59 is an eigenform of weight 16 + 58 = 74 whose q-expansion modulo 59 is identical to that of our initial form. It too must lift to an eigenform in characteristic zero. In fact, there are two such lifts, and both of them have slope 1/2. The two lifts are defined over a ramified quadratic extension of ℚ_{59}, and they are Galois-conjugate and congruent to each other.

One more step will make the overall pattern clear. From Θf in weight 76 we can go the Θ^{2}f in weight 136; its lift to characteristic zero has slope 3 and is therefore exceptional. We can also consider E_{58}Θf, which is an eigenform modulo 58 of weight 134. It has two lifts to characteristic zero, both of slope 3/2. Or we could look at E_{58}^{2}f, which is an eigenform modulo p of weight 132. It has two lifts to characteristic zero, both of slope 1/2.

Once again, we have only been able to check this pattern for small weights. What it suggests, however, is that every form on our list whose slope is either exceptional or non-integral corresponds, modulo p, to a Galois representation which is a Tate twist of the representation corresponding to the “initial” form of weight k < p + 1. This reinforces the feeling that there is a connection between forms whose slopes are unusually large and forms whose slopes are non-integral, and that all these forms correspond to Galois representations with unusual^{1} properties. Why this should be the case seems completely mysterious.

Question 8. Is there a representation-theoretic characterization of eigenforms that are of exceptional or non-integral slope?

7. The slopes are too constant

Finally, we would like to observe that our computations strongly support Kevin Buzzard’s observation (arising from his computations for p = 2) that the slopes of oldforms seem to be far more constant as the weight varies than one would expect. In this regard, recall that in [GM92] we conjectured that if we looked at two (sufficiently large) weights k_{1} and k_{2} such that k_{1} ≡ k_{2} (mod p)^{n}(p − 1), then the slope sequences for these two weights should be identical up to slope n. For n = 0, this is a theorem of Hida. In general, Coleman and Wan have shown that it is true if we strengthen the hypothesis to k_{1} ≡ k_{2} (mod p^{M(n)}(p − 1)), where M(n) is a quadratic function of n.

What one actually sees in the data, however, is much stronger. Consider, for example, the case where p = 5, n = 2, k_{i} = 112 + 100i. Our conjecture would predict that the portion of the slope sequences that has slopes less than or equal to 2 would be the same for two such weights. Table 2 gives the (lower halves) of the slope sequences for i = 0, 1, 2, 3. What we see is that the entire (lower half of the) slope sequence for weight k_{i} reappears in weight k_{i+1}. This suggests that something immensely stronger than the conjectures in [GM92] should be true, at least for the slopes of oldforms. Further examples of this behavior can easily be extracted from the data.

1The 59-adic representation attached to the unique form of weight 16 is known to have unusual properties, but we are unaware of anything unusual about the 79-adic representation attached to the form of weight 38.

Weight k Slope Sequence (lower half)
112  (1, 5, 5, 5, 10, 11, 14, 15, 16)
212  (1, 5, 5, 5, 10, 11, 14, 15, 16,

20, 21, 24, 25, 27, 30, 31, 34)

312  (1, 5, 5, 5, 10, 11, 14, 15, 16,

20, 21, 24, 25, 27, 30, 31, 34, 36, 37, 40, 41, 45, 46, 47, 50, 51)

412  (1, 5, 5, 5, 10, 11, 14, 15, 16,

20, 21, 24, 25, 27, 30, 31, 34, 36, 37, 40, 41, 45, 46, 47, 50, 51, 55, 55, 55, 59, 60, 63, 64, 65)

Table 2. Slope sequences for p = 5

8. Conclusions

Our computational results suggest several surprising regularities in the behavior of the slopes of p-oldforms for fixed p and varying k. It is quite possible that there are still more observations to make. For example, if it is true that the slopes for weight k are integers between 0 and (k − 1)/(p + 1), with what multiplicities do these integers occur? The data for small primes suggests that here too the behavior is quite regular. Can one come up with a precise conjecture? Such a conjecture would be closely related to the above conjectures about the distribution of the supersingularities in the interval [0, 1]. One could also consider the behavior of the slope for fixed k and varying p; in this case, the appropriate normalization seems to be to multiply the p-supersingularity by p + 1, so that the normalized supersingularities will be in [0, 1]. Finally, one could ask whether one can use the data to obtain predictions of what the non-classical slopes (i.e., the slopes corresponding to overconvergent p-adic modular forms of weight k) should be.

References

[AL70] A. O. L. Atkin and J. Lehner, Hecke operators on Γ_{0}(m), Math. Ann. 185 (1970), 134–160.

[AS86] A. Ash and G. H. Stevens, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, Journ. Reine Angew. Math. 365 (1986), 192–220.

[BBCO] C. Batut, D. Bernardi, H. Cohen, and M. Olivier, GP–PARI, Available

        by anonymous ftp from megrez.math.u-bordeaux.fr.

18  Fernando Q. Gouvêa

[CDF97] J. B. Conrey, W. Duke, and D. W. Farmer, The distribution of the eigenvalues of Hecke operators, Acta Arith. 78 (1997), 405–409.

[CE98] R. F. Coleman and B. Edixhoven, On the semi-simplicity of the U_{p}- operator on modular forms, Math. Ann. 310 (1998), 119–127.

[CFW00] J. B. Conrey, D. W. Farmer, and P. J. Wallace, Factoring hecke polynomials modulo a prime, Pacific J. Math. 196 (2000), 123–129.

[CGJ95] R. F. Coleman, F. Q. Gouvêa, and N. Jochnowitz, E_{2}, Θ, and overconvergence, Internat. Math. Res. Notices (1995), no. 1, 23–41.

[CM98] R. Coleman and B. Mazur, The eigencurve, (A. J. Scholl and R. L. Taylor, eds.), London Mathematical Society Lecture Note Series, vol. 254, Cambridge University Press, 1998, pp. 1–113.

[DS74] P. Deligne and J.-P. Serre, Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530, In J.-P. Serre, Œuvres, volume III.

[GM92] F. Q. Gouvêa and B. Mazur, Families of modular eigenforms, Math. Comp. 58 (1992), 793–806.

[Gou88] F. Q. Gouvêa, Arithmetic of p-adic modular forms, Lecture Notes in Mathematics, vol. 1304, Springer-Verlag, Berlin, Heidelberg, New York, 1988.

[Gou97] F. Q. Gouvêa, Non-ordinary primes: a story, Experiment. Math. 6 (1997), 195–205.

[Hat79] K. Hatada, Eigenvalues of Hecke operators on SL_{2}(𝟋), Math. Ann. 239 (1979), 75–96.

[Joc82] N. Jochnowitz, Congruences between systems of eigenvalues of modular forms, Trans. Amer. Math. Soc. 270 (1982), 269–285.

[Kat73] N. M. Katz, p-adic properties of modular schemes and modular forms, In Kuijk and Serre [KS73].

[Kat77] N. M. Katz, A result on modular forms in characteristic p, In Serre and Zagier [SZ77].

[KS73] W. Kuijk and J.-P. Serre (eds.), Modular functions of one variable III, Lecture Notes in Mathematics, vol. 350, Springer-Verlag, Berlin, Heidelberg, New York, 1973.

[Ser97] J.-P. Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke T_{p}, Journal of the American Mathematical Society 10 (1997), 75–102.

[Smi] L. Smithline, Bounding slopes of p-adic modular forms, preprint, August 2000.

[Smi00] L. Smithline, Exploring slopes of p-adic modular forms, Ph.D. thesis, University of California at Berkeley, 2000.

[SZ77] J.-P. Serre and D. B. Zagier (eds.), Modular functions of one variable V, Lecture Notes in Mathematics, vol. 601, Springer-Verlag, Berlin, Heidelberg, New York, 1977.

Department of Mathematics, Colby College, Waterville, ME 04901, U. S. A.

E-mail address: fqgouvea@colby.edu