WHERE THE SLOPES ARE

FERNANDO Q. GOUVÊA

Let N be a positive integer (the “level”) and k ≥ 2 be an integer (the “weight”). Let S_{k}(N, ) denote the finite-dimensional -vector space of cuspidal modular forms of weight k and level N defined over . Elements f ∈ M_{k}(N, ) 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, ) consisting of forms all of whose Fourier coefficients are in fact rational. We denote the -vector space spanned by this basis by S_{k}(N, ). Note that then we have

S_{k}(N, ) = S_{k}(N, ) ⊗ .

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

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). Hence, it is interesting to attempt to understand how these eigenvalues are distributed. This problem is the subject of Serre’s article [Ser97], which studies (fill in).

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For each prime number p, one can also consider the theory in a padic setting. One way of doing this is simply to choose an embedding of the algebraic closure of into the completion _{p} of an algebraic closure of _{p}, after which one can define

S_{k}(N, _{p}) = S_{k}(N, ) ⊗ _{p},

and similarly for S_{k}(N, F) where F is any extension of _{p}. In the p-adic 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.

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

This paper begins an investigation of the distribution of the slopes of the U operator for fixed level and varying weight. 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.

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, and Naomi Jochnowitz made significant suggestions and observations at several points.

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 = _{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 _{p}. Our computations will actually 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}.

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].

The Hecke operator T_{p} can be diagonalized on S_{k}(N, _{p}). Let f ∈ S_{k}(N, _{p}) be a normalized cuspidal eigenform, and let a_{p} be the eigenvalue of T_{p} acting on f. Finally, let f_{1} and f_{2} be the two images of f in S_{k}(Np, _{p}). Then 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 can be diagonalized, and 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, _{p}) is semisimple. In particular, we always have α′ ≠ α′′.

This has been proved for p ≠ 2 and k = 2 by Coleman and Edixhoven in [CE98]. Ulmer has shown in [Ulm95] that the case k = 3 follows from the conjecture of Birch and Swinnerton-Dyer for elliptic curves over function fields. We know of no other results about this conjecture.

It is easy to see that if the polynomial has a double root then we must have ord(a_{p}) = (k − 1)/2 (unless p = 2, in which case we would have ord(a_{p}) = (k + 1)/2). In our computations (which were restricted to the case N = 1), we never found a form for which ord(a_{p}) ≥ (k − 1)/2, which shows that Ulmer’s conjecture is true in every case we considered. This shows that the computations below are correct independent of Ulmer’s conjecture, and also provides evidence that the conjecture is indeed true in general.

For what follows, we assume Ulmer’s conjecture holds. In that case, we can determine the slopes of U acting on the oldforms in S_{k}(Np, _{p}) by determining the slopes of T_{p} acting on S_{k}(N, _{p}). 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})

Prime Largest weight computed  Prime Largest weight computed
2  226  43  140
3  202  47  134
5  178  53  134
7  166  59  142
11  154  61  130
13  154  67  130
17  152  71  130
19  146  73  130
23  142  79  136
29  142  83  130
31  142  89  130
37  142  97  130
41  142

Table 1. Maximum weight k considered for each prime p ≤ 100

for U acting on the oldforms in S_{k}(Np, _{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, _{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. The discussion above shows that this first half is the same as the slope sequence for T_{p} acting on S_{k}(N, _{p}), except that all slopes larger than (k − 1)/2 are replaced by (k − 1)/2. (As mentioned above, we found no example in which the latter case occurs.)

Since we want to understand the variation of the slopes as the weight k varies, it makes sense to normalize the slopes by dividing them by k − 1. Hence we define the normalized slope sequence

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

by

η_{i} = \frac{α_{i}}{k − 1}.

Prime p Weights k
59  16, 46, 76, 106, 136
79  38, 44, 118, 124
2411 12
15271 16
187441 16
3371 20
64709 20
Table 2. Known exceptions to η_{i} ≤ 1/(p + 1)

The normalized slope sequence is contained in the interval [0, 1] and is symmetric under η ⌊ 1 − η.

The main problem we consider is to understand the distribution of the normalized slopes in the interval [0, 1] when we fix the level N and let k ) . This problem can be expressed in measure-theoretic 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 normalized slope η_{i}: let d_{k} = dimM_{k}(N, _{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.

2. Computations

For our computations, we restricted to the case N = 1, which then means that one only gets non-trivial results for 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 as many small weights k as possible. 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 modular forms 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 normalized slopes 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. As one can see from Table 1, the largest weight we could handle decreased as p increased. The full output of the computations can be found on the

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

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 is diagonalizable 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. Fix a prime number p and a level N. Let (α_{i}) be the p-adic slope sequence of T_{p} acting on M_{k}(N, _{p}). Is it true that

α_{i} < \frac{k − 1}{2}

always?

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 normalized slopes for weight k and prime p are smaller than 1/(p + 1). In other words, the inequality

η_{i} ≤ \frac{1}{p + 1}

holds almost always for the lower half of the slope sequence; by symmetry, the upper half of the slope sequence almost always satisfies

1 − η_{i} ≥ \frac{p}{p + 1}.

Weight k Slope Sequence (lower half)
16  (1)
46  (0, 0, 1)
74  (1/2, 1/2, 1, 1, 1)
76  (0, 1, 1, 1, 1, 2)
104  (0, 0, 1/2, 1/2, 1, 1, 1, 1)
106  (0, 0, 0, 0, 1, 1, 1, 2)
132  (1/2, 1/2, 1, 1, 1, 2, 2, 2, 2, 2, 2)
134  (0, 1, 1, 1, 1, 3/2, 3/2, 2, 2, 2)
136  (0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3)

Table 3. Exceptional and fractional slopes for p = 59

Question. Fix a prime number p and a level N. Let (η_{i}) be the normalized p-adic slope sequence of T_{p} acting on M_{k}(N, _{p}). Is it true that

η_{i} ≤ \frac{1}{p + 1}

almost always as k ) ?

To be more explicit about “almost always,” we found exactly nine slopes (in the lower half of the slope sequence) which fail to satisfy the inequality above. They occur for p = 59, k = 16, 46, 76, 106, 136, and for p = 79, k = 38, 44, 118, 124. In every other case, the normalized slopes were smaller than 1/(p + 1). Other exceptions to the inequality, outside the range of this computation, can be read off from the results in [Gou97]; they correspond to forms that are non-ordinary with respect to large primes. The full list of primes and weights for which we know of a slope that does not satisfy the inequality is given in Table 2. For each (p, k) pair, we found that exactly one slope in the first half of the slope sequence violates the inequality.

We can make “almost always” precise by using the measure-theoretic formulation:

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

One way to think about these results is by analogy with the results in [Gou97]. There, we fixed a modular form and computed its slopes with respect to varying primes p, and found that the slope was almost always zero. In this case, we fix a prime and consider the full slope sequences for varying weights k, and we find that the slopes are almost always bounded by (k − 1)/(p + 1). The two exceptional sets are connected, of course. For example, a form of weight k which has slope equal to 1 at a prime p > k + 2 will be exceptional from both points of view.

We know of no theoretical reason^{1} to expect the slopes to satisfy this inequality. It seems to us, however, 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.

Question. Can one identify a specific property of the modular forms or the Galois representations corresponding to exceptional pairs (p, k)?

The list of exceptional cases is itself suggestive. Consider the case of p = 59. For k = 16, one finds a form of slope 1, which is therefore an exception to the inequality. There is no clear explanation for this first exceptional slope, but the fact that the other counterexamples occur at weights k = 46, 76, 106, 136 suggests a systematic pattern. A possible interpretation of this pattern via the theory of Θ-cycles developed in [Joc82] was suggested by Kevin Buzzard and will be discussed in the next section.

Finally, we note 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 all happen in one of two situations:

1Note, however, that Lawren Smithline has recently shown that the inequality does always hold when p = 3.

Weight k Normalized slope sequence (lower half)
16  (0.066)
46  (0, 0, 0.0222)
76  (0, 0.0133, 0.0133, 0.0133, 0.0133, 0.0266)
106  (0, 0, 0, 0, 0.00952, 0.00952, 0.00952, 0.0190)
136  (0, 0.00740, 0.00740, 0.00740, 0.00740, 0.00740, 0.0148,

0.0148, 0.0148, 0.0148, 0.0222) Table 4. Normalized exceptional slopes for p = 59

Weight k Slope sequence (lower half)
38  (0, 1)
44  (0, 0, 1)
116  (0, 1/2, 1/2, 1, 1, 1, 1, 1, 1)
118  (0, 0, 0, 1, 1, 1, 1, 1, 2)
122  (0, 0, 1/2, 1/2, 1, 1, 1, 1, 1)
124  (0, 0, 0, 1, 1, 1, 1, 1, 1, 2)

Table 5. Exceptional and fractional slopes for p = 79

1. A weight for which there exists exceptional form with slope equal to 2 is preceded by a weight for which the slope sequence contains two slopes equal to 1/2. (For p = 59, this happens in weights 74 and 76, then 104 and 106; for p = 79, weights 116 and 118, then 122 and 124.) See Tables 3 and 5.

2. A weight for which there exists exceptional form with slope equal to 3 is preceded by a weight for which the slope sequence contains two slopes equal to 3/2, and that weight is preceded by a weight whose slope sequence contains two slopes equal to 1/2. (For p = 59 and weights 132, 134, 136.) See Table 3.

This suggests a final question:

Question. How often are the slopes integral? Is there a connection between non-integral slopes and exceptional slopes?

Weight k Normalized slope sequence (lower half)
38  (0, 0.0270)
44  (0, 0, 0.232)
118  (0, 0, 0, 0.00854, 0.00854, 0.00854, 0.00854,

0.00854, 0.0170)

124  (0, 0, 0, 0.00813, 0.00813, 0.00813, 0.00813,

0.00813, 0.00813, 0.0162) Table 6. Normalized exceptional slopes for p = 79

3. Θ-cycles?

In this section we look a little more closely at the exceptional cases that occur for p = 59 and p = 79. The relevant slope sequences for p = 59 are listed in Tables 3 and 4. Note that in this case 1/(p+1) = 1/60 = 0.0166 …; we have written the normalized slopes as decimals with three significant digits in order to make it easier to compare the normalized slopes to 1/(p + 1). The exceptional slopes are printed in boldface. Similarly, Tables 5 and 6 contain slope sequences and normalized slope sequences for p = 79.

The weights that occur in this table strongly suggest a connection to the theory of Θ-cycles, as in [Joc82]. We recall the basic ideas. 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]). 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]).

When one considers modular forms modulo p, 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, we define its filtration w(f) to be the minimal weight for which there is a modular form whose q-expansion is congruent modulo p to the q-expansion of f. The theory of Θ-cycles describes what happens to the filtration under the Θ operator.

The basic facts are the following. Let f be a modular form of weight k on Γ_{0}(N). If f is not ordinary (i.e., U(f) ≡ 0 (mod p), which is the case in all the examples we will consider), then Θ^{p−1}f ≡ f (mod p), and hence w(Θ^{p−1}f) = w(f). This is why one speaks of a “Θ-cycle.” The filtration normally increases by p + 1 each time we compute Θ; specifically, if w(f) is not divisible by p, then w(Θf) = w(f) + p + 1. If, on the other hand, w(f) is divisible by p, the filtration goes down w(Θf) = w(f) + p + 1 − n(p − 1), where n is an integer. What values n can assume is completely determined in [Joc82]. The case that is relevant here is the one where w(f) = k, 4 ≤ k ≤ p − 1, and f is not ordinary. In this case, the cycle looks as follows. First,

w(f) = k, w(Θf) = k + p + 1, …, w(Θ^{p−k}f) = p^{2} − (k − 1)p.

Then, since this is divisible by p, the filtration falls, with n = p+2−k, so

w(Θ^{p−k+1} f) = p + 3 − k, …, w(Θ^{p−2}f) = (k − 2)p,

and finally w(Θ^{p−1}f) = k, closing the cycle.

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, 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 (a property which is “visible” modulo p), but there seems to no reason to predict anything further about their slope.

Consider now the exceptional slopes for p = 59. For the case of weight 16, the ocurrence of a large slope seems to be “accidental,” but its “propagation” to higher weights seems to be linked a Θ-cycle. Let f_{0} be the (unique) form of weight 16 and level 1; it’s 59-adic slope is 1, because its 59-th Fourier coefficient is divisible (once) by 59. For each i, let f_{i} denote the minimal weight modular form which is congruent modulo p to Θ^{i}f. Then we have:

• f_{1} is of weight 76, • f_{2} is of weight 136,

• …

• f_{43} is of weight 2596 = 22 × 59,

• f_{44} is of weight 46,

• f_{45} is of weight 106,

• …

• f_{57} is of weight 826 = 14 × 59,

• finally, f_{58} = f_{0}.

Note that the smaller slopes on this list are precisely the weights for which we found exceptional slopes.

Because the reductions modulo p of these forms are in the image of Θ, all of the liftings of forms in the Θ-cycle must have slope at least one, but the lifting theory suggests no reason to expect the slope to increase along the cycle. What we observe, however, is the following:

• The exceptional form of weight 76 is congruent modulo 59 (but not modulo 59^{2}) to Θf_{0} and has slope 2.

• The exceptional form of weight 136 is congruent modulo 59 (but not modulo 59^{2}) to Θ^{2}f_{0} and has slope 3.

• …

• The exceptional form of weight 46 is congruent modulo 59 (but not modulo 59^{2}) to Θ^{44}f_{0} and has slope 1.

• The exceptional form of weight 106 is congruent modulo 59 (but not modulo 59^{2}) to Θ^{45}f_{0} and has slope 2.

• …

In other words, the forms we get are only connected via Θ modulo 59, but their slopes increase by 1 as one might expect under Θ, except that at the point where the filtration falls back down to 46 the slope falls back down to 1. This suggests, for example, that there should be exceptional forms of weight 166 and 196, of slopes 3 and 4 respectively. We haven’t yet been able to extend our computations that far. If the pattern continues to hold for the full cycle, every one of the resulting forms will be exceptional (i.e., their normalized slopes will be bigger than 1/60), though their normalized slopes get closer and closer to 1/60 as the weight gets larger.

Similarly, we can examine the exceptional slopes for p = 79. The relevant slope sequences are listed in Tables 5 and 6. Note that in this

14  Fernando Q. Gouvêa

case 1/(p + 1) = 1/80 = 0.0125 …; we have written the normalized slopes as decimals with three significant digits in order to make it easier to compare the normalized slopes to 1/(p+1). The exceptional slopes are printed in boldface. Again, the exceptional slopes seem to fit into a Θ-cycle starting from a form of weight 48 (the weights in the cycle would be 48, 128, 208, …, 2528, 34, 114, 194, …, 3634,

48. Again, all we see is the lower end of the cycle, and again, if the pattern persists throughout the cycle every one of these forms would be exceptional.

If the existence of such “Θ-cycles” of slopes reflects a general phenomenon, then this would suggest that the set of possible slopes for a given p and varying k has far more structure than predicted, say, in [GM92]. This agrees with very recent observations of Kevin Buzzard regarding the case p = 2.

References

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

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

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

[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.

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

[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.

[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.

Where the Slopes Are  15

[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.

[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.

[Ulm95] D. L. Ulmer, A construction of local points on elliptic curves over modular curves, Internat. Math. Res. Notices (1995), 349–363.

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

E-mail address: fqgouvea@colby.edu