Explicit computations of the slopes of the U_{5} operator acting on overconvergent modular forms

L. J. P. Kilford

August 31, 2004

Abstract

Let χ be a primitive Dirichlet character of conductor 25.

In this note, we will show that the slopes of the U_{5} operator acting on slopes of 5-adic overconvergent modular forms with character χ are given by either

\frac{1}{4} · ⌊ \frac{8i}{5} ⌋ or \frac{1}{4} · ⌊ \frac{8i + 4}{5} ⌋.

1 Introduction

We first define the slope of a (normalised) cuspidal eigenform.

Definition 1. Let f be a normalised cuspidal modular eigenform with q-expansion at ∞ given by ∑_{n=1}^{∞} a_{n}q^{n}. The slope of f is defined to be the 5-valuation of a_{5}; we normalise the 5-valuation of 5 to be 1.

As a consequence of the main result of this paper, we will prove the following theorem about classical modular forms.

Theorem 2. The cyclotomic polynomial Φ_{20}(x) factors over Q_{5} into the following factors

f_{1} : x^{4} + (2 + O(5))x^{3} + (4 + O(5))x^{2} + (3 + O(5))x + (1 + O(5)),  (1)
f_{2} : x^{4} + (3 + O(5))x^{3} + (4 + O(5))x^{2} + (2 + O(5))x + (1 + O(5)).  (2)

We define Q_{5}(f_{i}) to be the splitting field of f_{i} over Q_{5}.

Let χ be an odd primitive Dirichlet character of conductor 25 and let τ be an odd primitive Dirichlet character of conductor 5.

Let k be a positive integer. We fix an embedding of the field of definition of χ into Q_{5}(\sqrt[4]{5}).

Then the slopes of the U_{5} operator acting on S_{k}(Γ_{0}(25), χτ^{k−1}) are

\frac{1}{4} · ⌊\frac{8i+1}{5}⌋ if χ(6) ∈ Q_{5}(f_{1}),
\frac{1}{4} · ⌊\frac{8i+4}{5}⌋ if χ(6) ∈ Q_{5}(f_{2}).

2 Some previous work

This paper uses methods introduced by Emerton in his PhD thesis [7], which deals with the action of the U_{2} operator. It also uses methods developed by Smithline in his thesis [13], which were then also used in the author’s paper [9] and in the paper of the author with Buzzard [2].

In [13], the following theorem is proved about 5-adic modular forms:

Theorem 3 (Smithline [13]). We order the slopes of U_{5} by size, beginning with the smallest.

The sum of the first x nonzero slopes of the U_{5} operator acting on 5-adic overconvergent modular forms of weight 0 is at least x^{2}.

In Buzzard-Kilford, the following theorem was proved about the 2-adic slopes of U_{2} acting on certain spaces of modular forms.

Theorem 4 (Buzzard-Kilford, Theorem B). Let k be an integer and let θ be a character of conductor 2^{n} such that θ(−1) = (−1)^{k}.

If |5^{k}·θ(5)−1|_{2} > 1/8, then the slopes of the overconvergent cuspidal modular forms of weight k and character θ are {t, 2t, 3t, …}, where t = v(5^{k} · θ(5) − 1), and each slope occurs with multiplicity 1.

3 Defining 5-adic overconvergent modular forms

We now present the definition of the 5-adic overconvergent modular forms, first by defining overconvergent modular forms of weight 0, and then by deriving the definition for forms with weight and character.

This section follows Section 3 of [9] in its layout and direction; more details on the specific steps can be found there.

Following Katz [8], section 2.1, we recall that, for C an elliptic curve over an F_{5}-algebra R, there is a mod 5 modular form A(C) called the Hasse invariant, which has q-expansion over F_{5} equal to 1.

We consider the Eisenstein series of weight 4 and tame level 1 defined over Z, with q-expansion

E_{4}(q) := 1 + 240 ∑_{n=1}^{∞} ( ∑_{0<d|n} d^{3}) · q^{n}.

We see that E_{4} is a lifting of A(C) to characteristic 0, as the reduction of E_{4} to characteristic 5 has the same q-expansion as A(C), and therefore E_{4} mod 5 and A(C) are both modular forms of level 1 and weight 4 defined over F_{5}, with the same q-expansion. Note also that if C is an elliptic curve defined over Z_{5} then the valuation v_{5}(E_{4}(C)) can be shown to be well-defined.

It is interesting to note that one can use the same Eisenstein series, E_{4}, in this part of the definition for 2-adic, 3-adic and 5-adic overconvergent modular forms (as a lifting of the 4^{th}, 2^{nd} and 1^{st} power of the Hasse invariant, respectively).

We now let m be a positive integer which is at least 2. Using arguments exactly similar to those in [9], we define the affinoid subdomain Z_{0}(5^{m}) of X_{0}(5^{m}) to be the connected component containing the cusp ∞ of the set of points t = (C, P) in X_{0}(5^{m}) which have v_{5}(E_{4}(t)) = 0.

We now define strict affinoid neighbourhoods of Z_{0}(5^{m}).

Definition 5 (Coleman [5], Section B2). We think of X_{0}(5^{m}) as a rigid space over Q_{5}, and we let t ∈ X_{0}(5^{m})(\overline{Q}_{5}) be a point, corresponding either to an elliptic curve defined over a finite extension of Q_{5}, or to a cusp. Let w be a rational number, such that 0 < w < 5^{2−m}/6.

We define Z_{0}(5^{m})(w) to be the connected component of the affinoid

{t ∈ X_{0}(5^{m}) : v_{5}(E_{4}(t)) ≤ w}

which contains the cusp ∞.

Given this definition, we can now define 5-adic overconvergent modular forms.

Definition 6 (Coleman, [4], page 397). Let w be a rational number, such that 0 < w < 5^{2−m}/6. Let O be the structure sheaf of Z_{0}(5^{m})(w). We call sections of O on Z_{0}(5^{m})(w) w-overconvergent 5-adic modular forms of weight 0 and level Γ_{0}(5^{m}). If a section f of O is a w-overconvergent modular form, then we say that f is an overconvergent 5-adic modular form.

Let K be a complete subfield of C_{5}, and define Z_{0}(5^{m})(w)_{/K} to be the affinoid over K induced from Z_{0}(5^{m})(w) by base change from Q_{5}. The space

M_{0}(5^{m}, w; K) := O(Z_{0}(5^{m})(w)_{/K})

of w-overconvergent modular forms of weight 0 and level Γ_{0}(5^{m}) is a K-Banach space.

We now let χ be a primitive Dirichlet character of conductor 5^{m} and let k be an integer such that χ(−1) = (−1)^{k}. Let E_{k,χ}^{∗} be the normalised Eisenstein series of weight k and character χ with nonzero constant term.

The space of w-overconvergent 2-adic modular forms of weight k and character θ is given by

M_{k,θ}(2^{m}, w; K) := E_{k,θ}^{∗} · M_{0}(2^{m}, w; K).

This is a Banach space over K.

There are Hecke operators U_{5} and T_{p} (where p ∤ 5) acting on the space of modular forms M_{k,θ}(2^{m}, w; K); these are defined on the q-expansions of the overconvergent modular forms in exactly the same way as they are defined on the q-expansions of classical modular forms.

Using results of Coleman, we have the following theorem about the independence of the characteristic power series of U_{5} acting on M_{k,θ}(2^{m}, w; K):

Theorem 7 (Coleman [5], Theorem B3.2). Let w be a real number such that 0 < w < min(2^{2−m}/3, 1/4), let k be an integer and let θ be a character such that θ(−1) = (−1)^{k}.

The characteristic polynomial of U_{2} acting on overconvergent 2-adic modular forms of weight k and character θ is independent of the choice of w.

We will now rewrite the definition of Z_{0}(25)(w) in terms of a carefully chosen modular function of level 25, in order to prove the following theorem:

Theorem 8. Let w_{0} = 1/12. The space of w_{0}-overconvergent modular forms of weight 0 and level 25, with coefficients in Q_{5}(\sqrt[4]{5}), is a Tate algebra in one variable over Q_{5}(\sqrt[4]{5}).

Proof. We have given a valuation on the points t of the rigid space X_{0}(5^{m}), based on the lifting of the Hasse invariant by the Eisenstein series E_{4}. We recall that the modular j-invariant is defined to be j := E_{4}^{3}/Δ. Therefore, we see that, if the elliptic curve corresponding to t has good reduction, then Δ(t) has valuation 0, and therefore that

v_{5}(t) = v_{5}(E_{4}(t)) = \frac{1}{3}v_{5}((E_{4})(t)^{3}) = \frac{1}{3}v_{5}(j(t)).

We now recall that the modular curve X_{0}(25) has genus 0. This means that there is a modular function t_{25} which is a uniformiser on X_{0}(25); following the notation of McMurdy [10], we define this modular function to be

t_{25} := \frac{η_{1}}{η_{25}} .

We could write t_{25} as a rational function in j directly, but as the resulting rational function is very complicated, we will instead also work with the uniformiser t_{5} of X_{0}(5), defined as

t_{5} := (\frac{η_{1}}{η_{5}} )^{6} .

By explicit calculation, one can verify the following identities of modular functions:

j = \frac{(t_{5}^{2} + 250t_{5} + 3125)^{3}}{t_{5}^{5}} and t_{5} = \frac{t_{25}^{5}}{t_{25}^{4} + 5t_{25}^{3} + 15t_{25}^{2} + 25t_{25} + 25}.  (3)

We note also that

j(∞) = t_{5}(∞) = t_{25}(∞) = ∞;

this follows because the q-expansion of all of these functions begins q^{−1} + · · ·.

Because t_{5}(∞) = t_{25}(∞) = ∞, the connected components of Z_{0}(5) and Z_{0}(25) which contain the cusp ∞ are of the form v_{5}(t_{5}) < D_{1} and v_{5}(t_{25}) < D_{2}, for some rational numbers D_{1} and D_{2}.

By considering the Newton polygons of the numerators and denominators of the rational functions in (3), we see that if v_{5}(t_{25}) < 1/2, then v_{5}(t_{25}) = v_{5}(t_{5}) = v_{5}(j). This means that we have shown that

Z_{0}(25)(w) = {x ∈ X_{0}(25) : v_{5}(t_{25}(x)) ≤ 3w} for 0 < w < 1/6.

Now, we choose w = 1/12, and therefore we obtain

Z_{0}(25)(1/12) = {x ∈ X_{0}(25) : v_{5}(t_{25}(x)) ≤ 1/4} .

Let us define W := \sqrt[4]{5}/t_{25}. We can rewrite the definition of Z_{0}(25)(1/12) again in terms of W to get

Z_{0}(25)(1/12) = {x ∈ X_{0}(25) : v_{5}(W (x)) ≥ 0} .

Finally, we recall that the rigid functions on the closed disc over Q_{5} with centre 0 and radius 1 are defined to be power series of the form

∑_{n∈N} a_{n}z^{n} : a_{n} ∈ Q_{5}, a_{n} → 0.

Therefore, the 1/3-overconvergent modular forms of level Γ_{0}(25) and weight 0 are

Q_{5}(\sqrt[4]{5})⟨W ⟩,

which is what we wanted to show.

We have written down this space of overconvergent modular forms as an explicit Banach space. This means that we can write down its Banach basis: the set {W, W^{2}, W^{3}, …} forms a Banach basis for the overconvergent modular forms of weight 0 and level Γ_{0}(25). This Banach base is composed of weight 0 modular functions — we want to be able to consider the action of the U_{5} operator on overconvergent modular forms with non-zero weight k and character χ (here, as elsewhere in this note, χ has conductor 25 and θ(−1) = (−1)^{k}). Using an observation from the work of Coleman [5], we will be able to move between weight 0 and weight k and character χ via multiplication by a suitable quotient of modular forms.

Let F be an overconvergent modular form of weight k and character θ which has nonzero constant term, and let z be an overconvergent modular function of weight 0. In particular, we note that F may have negative weight. From the discussion in Coleman [5, page 450] we see that the pullback Ũ_{5} of the U_{5} operator acting on overconvergent modular forms of weight-character (k, θ) to weight 0 is 1/F · U_{5}(z · F).

Now by equation 3.3 of [6] we have that U_{5}(z · V (F)) = U_{5}(z) · F. We therefore consider the modular form H = V (G), and substitute H for F in the formula we have just derived for U_{2}(z · V (F)), to obtain:

Ũ_{5}(z · V (G)) = \frac{1}{V (G)} · U_{5}(z · V (G)) = \frac{G}{V (G)} · U_{5}(z).

We can also use this line of reasoning to see that

1/F · U_{5}(z · F) = U_{5} (z · \frac{F}{V (F)}) .

This allows us to now define the (twisted) U_{5} operator.

Definition 9 (The twisted U_{5} operator). Let k be an integer and let χ be an odd character of conductor 25. Let τ be a character of conductor 5 such that χ(−1) = (−1)^{k}.

The twisted U_{5} operator acting on forms of weight-character (1 + kt, χ · τ^{k}) is defined to be the following operator:

U_{5} (W^{i} · \frac{E_{1,χ}^{∗}}{V (E_{1,χ}^{∗})}) · (\frac{E_{k,τ}^{∗}}{V (E_{k,τ}^{∗})})^{t} .  (4)

We can now consider the action of this twisted U_{5} operator on these spaces of overconvergent modular forms.

Definition 10 (The matrix of the twisted U_{5} operator). Let k be an integer and let χ be an odd character of conductor 25. Let τ be a character of conductor 5 such that χ(−1) = (−1)^{k}.

Let M = (m_{i,j}) be the infinite compact matrix of the twisted U_{5} operator acting on overconvergent modular forms of weight-character (1 + kt, χ · τ^{k}), where m_{i,j} is defined to be the coefficient of W^{i} in the W -expansion of the operator defined in equation (4).

Here we will make the observation that the entries of our matrix M are functions of χ, τ, and t.

We know that U_{5} is a compact operator, so we can show that the trace, determinant and characteristic power series of M are all well-defined. We will use a theorem of Serre to prove our theorem on the slopes of U_{5} acting on M.

Theorem 11 (Serre [12], Proposition 7). 1. Let M_{n} be an n × n matrix defined over a finite extension of Q_{2}. Let det(1 − tM_{n}) = ∑_{i=0}^{n} c_{i}t^{i}. Let M_{m} be the matrix formed by the first m rows and columns of M_{n}.

Let s(i) be the formula for the i^{t}h slope; in our specific case, this will mean that either

s(i) = \frac{1}{4} · ⌊\frac{8i + 1}{5}⌋ or s(i) = \frac{1}{4} · ⌊\frac{8i + 4}{5}⌋.

Assume that there exists a constant r ∈ Q^{×} such that

(a) For all positive integers m such that 1 ≤ m ≤ n, the valuation of det(M_{m}) is r · ∑_{i=1}^{m} s(i).

(b) The valuation of elements in column j is at least r · s(i).

Then we have that, for all positive integers m such that 1 ≤ m ≤ n, v_{2}(c_{m}) = r · ∑_{i=1}^{m} s(i).

2. Let M_{∞} be a compact infinite matrix (that is, the matrix of a compact operator). If M_{m} is a series of finite matrices which tend to M_{∞}, then the finite characteristic power series det(1−tM_{m}) converge coefficientwise to det(1 − tM_{∞}), as m → ∞.

We now quote a result of Coleman that tells us that overconvergent modular forms of small slope are in fact classical modular forms:

Theorem 12 (Coleman [4], Theorem 1.1). Let k be a non-negative integer. Every 5-adic overconvergent modular eigenform of weight k with slope strictly less than k − 1 is a classical modular form.

We now state the main theorem of this paper, which tells us exactly what the slopes of the U_{5} operator acting on slopes of modular forms of level 25.

Theorem 13. We recall and use the notation of Theorem 2.

Let χ be an odd primitive Dirichlet character of conductor 25 and let τ be an odd primitive Dirichlet character of conductor 5.

Let k be a positive integer. We fix an embedding of the field of definition of χ into Q_{5}(\sqrt[4]{5}).

The slopes of overconvergent modular forms of weight k and character χτ^{k−1} are given by

\frac{1}{4} · ⌊\frac{8i}{5}⌋ if χ(6) ∈ Q_{5}(f_{1})
\frac{1}{4} · ⌊\frac{8i+4}{5}⌋ if χ(6) ∈ Q_{5}(f_{2}).

We can prove Theorem 2, assuming Theorem 13, by recalling the following theorem from Cohen-Oesterlé:

Theorem 14 (Cohen-Oesterlé [3], Théorème 1). Let χ be a primitive Dirichlet character of conductor 25 and let k be a positive integer greater than 1. The following formula holds:

d(k, χ) := dim S_{k}(Γ_{0}(25), χ) = \frac{5k − 7}{2} + ε · (χ(8) + χ(17)),

where ε is 0 for odd k, −1/4 if k ≡ 2 mod 4, and 1/4 if k ≡ 0 mod 4.

Proof of Theorem 2. The classical theorem will follow, because when we substitute d(k, χ) into the formula s(i) for the i^{th} slope, we see that the maximum value of s(d(k, χ)) is k − 1. An argument of Buzzard shows that slopes greater than k − 1 cannot be classical (see [2], the proof of the Corollary to Theorem B, which references the proof of Theorem 4.6.17(1) of [11]), so therefore as there are at most k − 1 slopes which are smaller than or equal to k − 1, we see that all of these small slopes are the slopes of classical eigenforms.

4 Observations

There are some interesting new features which appear when p = 5 that do not appear when p = 2 or p = 3.

Firstly, there is a computational issue. In previous work (such as [2] or [9]), the computations in magma [1] could be carried out either over the rational numbers or over the field Q_{p}. However, for p = 5 the calculation must be carried out over Q_{5}(\sqrt[4]{5}).

Secondly, and more importantly, there are now two different possibilities for the slopes, which depend on which character is chosen. These two possibilities correspond to the two factors of the cyclotomic polynomial Φ_{20}(x) over Q_{5}. This is a departure from the situation in [2] and [9], where the slopes are independent of choice of character.

Thirdly, the slopes are no longer in one arithmetic progression, as they are in the previously studied cases. Instead, there are five arithmetic progressions which interlace together; these all have a common difference between terms (which is 2). (One could, of course, view the arithmetic progression 1, 2, 3, … as being made up of the two arithmetic progressions 1, 3, 5, … and 2, 4, 6 … but this point of view is only reasonable after one has considered the action of U_{5} on forms of level 25, where the slopes form several arithmetic progressions).

Fourthly, part of the complexity in the calculations in [9] was the fact that (in the notation of that paper) the modular function U_{2}(z^{2i+1}) was identically zero. This meant that the first “matrix of the U_{2} operator” that was defined had identically zero determinant, which meant that some algebra had to be done to get a matrix to which Theorem 11 could be applied to. In the current note, this does not happen because the matrix M of the U_{5} operator does not have any identically zero columns.

Finally, the strategy of Section 5 was chosen because the modular functions involved were unusually simple, thus making the calculations more tractable (the corresponding functions for (say) weight 2 were much less pleasant).

5 The technical part; proof of Theorem 13

As the actual proof of Theorem 13 is somewhat technical, we will first outline a plan to show how the proof works.

Plan for the proof of Theorem 13. In this section, we will show that we can apply Theorem 11, which will prove Theorem 13. First we fix an arbitrary positive integer n, an integer k and a primitive Dirichlet character θ of conductor 25 such that θ(−1) = −1.

We will begin with the matrix M_{n}; the matrix formed by the first n rows and n columns of M, the matrix of the twisted U_{5} operator acting on forms of weight-character (1, θ) defined in Definition 9. The proof will then proceed in the following way:

1. Define the matrix D(β(i)) to be the diagonal matrix with β(j) in the jth row and the jth column. We define the matrix O_{n} := D(\sqrt{5}^{−j}) · O_{n} · D(\sqrt{5}^{j}).

2. We then show that the valuation of elements in the jth column of O_{n} are s(j); this verifies condition (b) of Theorem 11, with r = s(j).

3. We finally show that O_{n} has determinant of valuation ∑_{i=1}^{n} s(i), by considering the matrix P_{n} := D(5^{−s(j)}) · O_{n}. By showing that P_{n} has determinant of valuation 0, it can be seen that the valuation of the determinant of O_{n} is the valuation of the determinant of D(5^{s(j)}), which is ∑_{i=1}^{n} s(i). This will verify condition (a) of Theorem 11, with determinant of valuation ∑_{i=1}^{n} s(i).

4. Finally, we will show that, after multiplication by the multiplier (as defined in Definition 10)

(\frac{E_{1,τ}^{∗}}{V (E_{1,τ}^{∗})})^{t} ,

the matrix of the twisted U_{5} operator acting on forms of weight 1 + t and character θ · τ^{t−1} still satisfies properties (a) and (b) of Theorem 11.

At each step of this plan, we must show that the characteristic polynomial of the new matrix defined is the same as that of M_{n}. In the last step, we will show that P_{n} has unit determinant by reducing it modulo a prime ideal above 5 and showing that this reduction has determinant 1. This means that we must prove that P_{n} has coefficients which are integers in Q_{5}(\sqrt[4]{5}).

Proof of Theorem 13. In this section, we will use the modular function T instead of W; we define T as follows:

T := \frac{1}{t_{25}} .

This will make it easier to perform the calculations.

By computation, it can be shown that the first five columns of the matrix M_{n} (in weight 1) are polynomials in T of degree 5i; we will just give the valuations of the coefficients of these, as the actual coefficients are elements of Q_{5}(\sqrt[4]{5}) and thus take up a lot of space.

If we have chosen χ(6) to be a root of (1), then these are the valuations:

U_{5}(T · E_{1,χ}^{∗}) :  [\frac{1}{2}, \frac{3}{2}, \frac{9}{4}, \frac{13}{4}, 4]
U_{5}(T^{2} · E_{1,χ}^{∗}) :  [\frac{1}{2}, 1, 2, \frac{11}{4}, 4, \frac{9}{2}, \frac{11}{2}, \frac{25}{4}, \frac{29}{4}, 8]
U_{5}(T^{3} · E_{1,χ}^{∗}) :  [\frac{1}{4}, 1, \frac{5}{4}, \frac{9}{4}, 3, 5, 5, 6, \frac{27}{4}, 8, \frac{17}{2}, \frac{19}{2}, \frac{41}{4}, \frac{45}{4}, 12]
U_{5}(T^{4} · E_{1,χ}^{∗}) :  [\frac{1}{4}, \frac{3}{4}, \frac{5}{4}, \frac{7}{4}, 3, \frac{7}{2}, \frac{9}{2}, \frac{21}{4}, \frac{25}{4}, 7, \frac{17}{2}, 9, 10, \frac{43}{4}, 12, \frac{25}{2}, \frac{27}{2}, \frac{57}{4}, \frac{61}{4}, 16]
U_{5}(T^{5} · E_{1,χ}^{∗}) : [0, 1, 1, 2, 2, \frac{7}{2}, 4, 5, \frac{23}{4}, 7, \frac{15}{2}, \frac{17}{2}, \frac{19}{2}, \frac{21}{2}, 11, \frac{25}{2}, 13, 14, \frac{59}{4}, 16, \frac{33}{2}, \frac{35}{2}, \frac{73}{4}, \frac{77}{4}, 20]

If, on the other hand, we have chosen χ(6) to be a root of (2), then these are the valuations:

U_{5}(T · E_{1,χ}^{∗}) :  [\frac{1}{4}, \frac{5}{4}, 2, 3, 4]
U_{5}(T^{2} · E_{1,χ}^{∗}) :  [\frac{1}{4}, \frac{3}{4}, \frac{7}{4}, \frac{5}{2}, 4, \frac{17}{4}, \frac{21}{4}, 6, 7, 8]
U_{5}(T^{3} · E_{1,χ}^{∗}) :  [0, \frac{3}{4}, 1, 2, 3, \frac{17}{4}, \frac{19}{4}, \frac{23}{4}, \frac{13}{2}, 8, \frac{33}{4}, \frac{37}{4}, 10, 11, 12]
U_{5}(T^{4} · E_{1,χ}^{∗}) :  [0, \frac{1}{2}, 1, \frac{3}{2}, 3, \frac{13}{4}, \frac{17}{4}, 5, 6, 7, \frac{17}{2}, \frac{35}{4}, \frac{39}{4}, \frac{21}{2}, 12, \frac{49}{4}, \frac{53}{4}, 14, 15, 16]
U_{5}(T^{5} · E_{1,χ}^{∗}) : [0, 1, 1, 2, 2, \frac{7}{2}, \frac{15}{4}, \frac{19}{4}, \frac{11}{2}, 7, \frac{29}{4}, \frac{33}{4}, \frac{37}{4}, \frac{41}{4}, 11, \frac{49}{4}, \frac{51}{4}, \frac{55}{4}, \frac{29}{2}, 16, \frac{65}{4}, \frac{69}{4}, 18, 19, 20] .

The valuations of the T -coefficients of U_{5}(T^{5}) are independent of the choice of χ(6) (because T has q-coefficients in Q_{5}) and are as follows:

U_{5}(T^{5}) : [0, \frac{1}{4}, \frac{1}{4}, \frac{1}{2}, \frac{1}{2}, \frac{5}{4}, \frac{3}{2}, \frac{3}{2}, \frac{7}{4}, \frac{7}{4}, \frac{5}{2}, \frac{5}{2}, \frac{5}{2}, \frac{11}{4}, \frac{11}{4}, \frac{7}{2}, 4, \frac{15}{4}, 4, 4, \frac{9}{2}, \frac{19}{4}, \frac{19}{4}, 5, \frac{19}{4}] .

We also note that the following identity of modular functions holds:

\frac{E_{1,τ}^{∗}}{V (E_{1,τ}^{∗})} = 1 − \frac{5(T + (2 + 2I)T^{2})}{1 + (2 + I)T + (2 + I)T^{2}} .  (5)

5.1 Checking the valuations of the elements in the j^{th} column of O_{n}

We will show that the valuations of elements in the jth column of the matrix O_{n} have valuation at least s(j) by writing the operator which determines the (5a+b)^{th} column, U_{5}(T^{5a+b}·E_{1,χ}^{∗}/V (E_{1,χ}^{∗})) in terms of the operators U_{5}(T^{b}· E_{1,χ}^{∗}/V (E_{1,χ}^{∗})) and U_{5}(T^{5}).

Definition 15. We will write

\overline{U_{5}(T · E_{1,χ}^{∗}/V (E_{1,χ}^{∗}))}

to mean that we have changed the basis of the matrix M_{n} of the twisted U_{5} operator by conjugating it with the matrices D(5^{s(j)}) and D(5^{−s(j)}).

We define this matrix to be O_{n}, as defined in the Plan given at the beginning of this section.

Let π be a fixed fourth root of 5. By consulting the tables of valuations above, we see that (ignoring unit factors in the coefficients) the following congruences hold:

\overline{U_{5}(T · E_{1,χ}^{∗}/V (E_{1,χ}^{∗}))} ≡ π^{4s(1)}T mod π^{4s(1)+1}  (6)
\overline{U_{5}(T^{2} · E_{1,χ}^{∗}/V (E_{1,χ}^{∗}))} ≡ π^{4s(2)}(T + T^{2}) mod π^{4s(2)+1}  (7)
\overline{U_{5}(T^{3} · E_{1,χ}^{∗}/V (E_{1,χ}^{∗}))} ≡ π^{4s(3)}(T + T^{3}) mod π^{4s(3)+1}  (8)
\overline{U_{5}(T^{4} · E_{1,χ}^{∗}/V (E_{1,χ}^{∗}))} ≡ π^{4s(4)}(T + T^{2} + T^{3} + T^{4}) mod π^{4s(4)+1} (9)
\overline{U_{5}(T^{5} · E_{1,χ}^{∗}/V (E_{1,χ}^{∗}))} ≡ π^{4s(5)}(T + T^{3} + T^{5}) mod π^{4s(5)+1}  (10)
\overline{U_{5}(T^{5})} ≡ π^{8}(T + T^{3} + T^{5}) mod π^{9}.  (11)

Now we see that

\overline{U_{5} (T^{5a+b} · \frac{E_{1,χ}^{∗}}{V (E_{1,χ}^{∗})})} ≡ \overline{U_{5} (T^{b} · \frac{E_{1,χ}^{∗}}{V (E_{1,χ}^{∗})})} · \overline{U_{5}(T^{5})}^{a};  (12)

this is because T^{5} is congruent to V (T) modulo 5, and from the definition of the U_{5} and V operators, we see that \overline{U_{5}(T^{5a+b} · X)} (where X is a modular function) is congruent to T^{a} · \overline{U_{5}(T^{b} · X)} modulo 5. Finally, one can check explicitly that the q-expansions of T and \overline{U_{5}(T^{5})} are congruent modulo 5.

Now we can show that the valuations of the T -coefficients of \overline{U(T^{5a+b} · E_{1,χ}^{∗}/V_{1,χ}^{∗})} are at least s(5a + b), as is required for condition (b) of Theorem 11; from the argument above we see that

\overline{U_{5}(T^{5a+b} · X)} ≡ \overline{U_{5}(T^{5})}^{a} · \overline{U_{5}(T^{b} · X)} mod 5.

This means that the following equality holds:

\overline{U_{5}(T^{5a+b} · X)} = \overline{U_{5}(T^{5})}^{a} · \overline{U_{5}(T^{b} · X)} + 5^{ε}f(T),  (13)

for some function f(T) with integral coefficients in some extension of Z_{5}.

Now we know from the discussion above that the valuation of T -coefficients of \overline{U_{5}(T^{5})} is at least 2, and that the valuation of T -coefficients of \overline{U_{5}(T^{b} · X)} is at least s(b), so therefore the valuation of the T -coefficients of \overline{U_{5}(T^{5a+b} · X)} is at least s(5a + b), as is required.

5.2 Defining P_{n} and showing that it has unit determinant

We now postmultiply the matrix O_{n} by D(5^{−s(i)}) and define this product to be P_{n}. We will now show that P_{n} has determinant of valuation 0, and therefore that the valuation of the determinant of O_{n} is ∑_{i=1}^{n} s(i).

We now reduce the entries of the matrix P_{n} modulo a prime above 5; we call this matrix P′_{n}. Now, to show that P_{n} has unit determinant it will suffice to show that the columns of P′_{n} are linearly independent.

From the congruences shown in (6)–(11) and the argument at (12) and (13), we see that the elements of the (5a + b)^{th} column of P′_{n} are given by the coefficients T in the T -expansion of

\overline{U_{5} (T^{b} · \frac{E_{1,χ}^{∗}}{V (E_{1,χ}^{∗})})} · \overline{U_{5}(T)}^{a},

and that the highest coefficient of this that does not vanish after reduction modulo the prime ideal above 5 is exactly T^{5a+b}.

We can now see immediately that the matrix P′_{n} has determinant a unit, because there are units on the diagonal, and no elements below the diagonal. Therefore the determinant of P_{n} is a 5-adic unit, as required.

This means that the determinant of O_{n} and also the determinant of M_{n} both have valuation ∑_{i=1}^{n} s(i). This means that M_{n} satisfies condition (a) of Theorem 11 and therefore that we can apply this theorem to the matrix M to show that the slopes of the U_{5} operator are given by s(i).

5.3 Generalising all this to other weights

We note that this part of the proof has shown that the matrices M_{n} of the twisted U_{5} operator acting on overconvergent modular forms of weight 1 have determinants with 5-valuation ∑_{i=1}^{m} s(i), and that the valuations of elements in the j^{th} column are at least s(i). We will now prove that the matrix of the twisted U_{5} operator acting on weights of the form 1 + t, where t is an integer, also satisfies these two properties; this will be enough to prove Theorem 13.

It should be noted that the strange-seeming conditions involving χ and τ in Theorem 2 and Theorem 13 arise because we first proved the theorems for weight 1 and will then use the multiplier (E_{1,τ}^{∗} /V (E_{1,τ}^{∗}))^{k−1} to get to weight k.

We now check that the multiplier given in (5) has large enough valuations that after the change-of-basis, its T -coefficients still have non-negative valuation.

Under the embedding we have chosen of Q_{5}(χ) into the extension field of Q_{5}, both 3−I and 2+I have normalised valuation 1. This means that the valuation of the coefficient of T^{2} in the multiplier has valuation 1, so therefore under the change of basis by conjugation by the two diagonal matrices, we see that the valuation of the coefficients of the multiplier will be non-negative. This means that we are still able to reduce modulo a prime ideal above 5, so our analysis will carry through.

We now define matrices O_{n}, P_{n} and P′_{n} in the same way as above, but now using the operator

U_{5} (T^{5a+b} · \frac{E_{1,χ}^{∗}}{V (E_{1,χ}^{∗})}) · (\frac{E_{1,τ}^{∗}}{V (E_{1,τ}^{∗})})^{t} .

Because we have checked that the valuations of the weight 1 and level 5 multiplier (5) are compatible with those of the original twisted U_{5} operator in weight 1, we merely need to check that the columns of the matrix P′_{n} are still linearly independent.

We notice that, after the change of basis and reduction modulo the prime ideal, the weight 1 multiplier is of the form 1 + ⋯; in other words, it is a unit. So the columns of P′_{n} in weight 1 + t are linearly independent, because the columns of the matrix P′_{n} in weight 1 are linearly independent, and we have multiplied each of these columns by a unit.

This means that P_{n} has unit determinant, and therefore that O_{n} and M_{n} have determinant of valuation ∑_{i=1}^{n} s(i). Therefore we can apply Theorem 11 and hence we have proved Theorem 13.

6 Acknowledgements

This note was written during my stay at the Max-Planck-Institut in Bonn, Germany in the summer of 2004. The author would like to thank the Institut for its hospitality.

The results of this paper were mentioned in a short note circulated at the Banff International Research Station workshop “p-adic variation of motives”, which was held in December 2003. The author would also like to thank the BIRS for its hospitality.

Some of the computer calculations were executed by the computer algebra package magma [1] running on the machine crackpipe, which was bought by Kevin Buzzard with a grant from the Central Research Fund of the University of London. I would like to thank the CRF for their support.

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