Slopes of overconvergent 2-adic modular forms.

Kevin Buzzard  Frank Calegari

January 13, 2003

1 Introduction.

Let p be a prime, and let N be a positive integer coprime to p. Let M_{k}(Γ_{1}(N); Q_{p}) denote the weight k modular forms of level Γ_{1}(N) defined over Q_{p}. In recent years, work of Coleman and others (for example [5],[6],[7],[8],[9]) has shown that a very profitable way of studying this finite-dimensional Q_{p}-vector space is to choose a small positive rational number r and then to embed M_{k}(Γ_{1}(N); Q_{p}) into a (typically infinite-dimensional) p-adic Banach space M_{k}(Γ_{1}(N); Q_{p}; p^{−r}) of p^{−r}-overconvergent p-adic modular forms, that is, sections of ω^{⊗k} on the affinoid subdomain of X_{1}(N) obtained by removing certain open discs of radius p^{−r} above each supersingular point in characteristic p (at least if N ≥ 5; see the appendix for how to deal with the cases N ≤ 4). The space M_{k}(Γ_{1}(N); Q_{p}; p^{−r}), for 0 < r < p/(p + 1), comes equipped with canonical continuous Hecke operators, and one of them, namely the operator U := U_{p}, has the property of being compact; in particular U has a spectral theory. Coleman exploited this theory in [5] to prove weak versions of conjectures of Gouvêa and Mazur on families of modular forms. One of us (K.B.) has made, in many cases, considerably more precise conjectures ([3]) than those of Gouvêa and Mazur, predicting the slopes of U, that is, the valuations of all the non-zero eigenvalues of U. These conjectures are very explicit, and display a hitherto unexpected regularity. However, they have the disadvantage of being rather inelegant. See also the forthcoming PhD thesis [13] of Graham Herrick, who has, perhaps, more conceptual conjectures about these slopes. We present here a very concrete conjecture in the case N = 1 and p = 2, which presumably agrees with the conjectures in [3] but which has the advantage of being much easier to understand and compute. Let S_{k} := S_{k}(Γ_{0}(1), Q) denote the level 1 cusp forms of weight k. If F (X) is a polynomial with rational coefficients then by its 2-adic Newton polygon we mean its Newton polygon when considered as a polynomial with 2-adic coefficients.

Conjecture 1 Let k ≥ 12 be even, and let m = dim S_{k}. Then the 2-adic Newton Polygon of det(1 − XT_{2}) on S_{k} equals the 2-adic Newton Polygon of

1 + ∑_{n=1}^{m} X^{n} ∏_{j=1}^{n} \frac{2^{2j}(k − 8j)!(k − 8j − 3)!(k − 12j − 2)}{(k − 12j)!(k − 6j − 1)!}.

This conjecture can be verified numerically, and we have verified it for all k ≤ 2048. Control theorems of Coleman ([6]) imply that complete knowledge of all slopes of classical cusp forms at level 2 for all weights is equivalent to complete knowledge of all slopes of finite overconvergent tame level 1 cusp forms for all integer weights. In fact, it is a little tedious but completely elementary to show that we may reformulate Conjecture 1 as follows. Let S_{k} := S_{k}(Γ_{0}(1); Q_{2}; 2^{−1/2}) denote the 2^{−1/2}-overconvergent forms of weight k and tame level 1.

Conjecture 2 Let k ≤ 0 be an integer. Then the Newton Polygon of det(1 − XU) on S_{k} is the Newton Polygon of

1 + ∑_{n=1}^{∞} X^{n} ∏_{j=1}^{n} \frac{2^{2j}(−k + 2 + 12j)!(−k + 6j)!}{(−k + 2 + 8j)!(−k − 2 + 8j)!(−k − 12j)}.

One can also find a form of this conjecture that makes sense if k > 0, for example if one reformulates the factorials as Gamma-functions and then is careful to make precise what is happening at poles. We leave this reformulation to the reader. As evidence for this conjecture, we have the following:

Theorem 1 Conjecture 2 is true when k = 0.

If x is a non-zero rational number, then by its slope we mean its 2-adic valuation v_{2}(x). It is easy to check that for n ≥ 0 an integer we have v_{2} ((2n)!) = n + v_{2}(n!), and it follows from this that

v_{2} (\frac{2^{2j}(12j + 2)!(6j)!}{(8j + 2)!(8j − 2)!(12j)}) = 1 + 2v_{2} (\frac{(3j)!}{j!}) .

Similarly one can check that if D denotes the infinite diagonal matrix whose (j, j)th entry, j ≥ 1, is given by

d_{j,j} = \frac{2^{4j+1}(3j)!^{2}j!^{2}}{3 · (2j)!^{4}} ,

then v_{2}(d_{j,j}) = 1 + 2v_{2} (\frac{(3j)!}{j!}) . Hence Theorem 1 above is equivalent to

Theorem 2 The Newton polygons of det(1 − XU) on S_{0} and det(1 − XD) coincide.

This is the form of the theorem that we shall actually prove. Note that the sequence v_{2}((3j)!/j!) is strictly increasing; this follows from the fact

\frac{(3j + 3)!/(j + 1)!}{(3j)!/j!} = 3(3j + 2)(3j + 1)

is even for all j. We deduce Corollary 1 Let |λ_{1}| ≥ |λ_{2}| ≥ |λ_{3}| ≥ … be the non-zero eigenvalues (with multiplicities) of U on S_{0}. Then the slope of λ_{n} is given by the following formula:

v_{2}(λ_{n}) = 1 + 2 · v_{2} (\frac{(3n)!}{(n)!}) .

In particular, the slopes are all distinct, and are all positive odd integers. We also have

Corollary 2 Let f = q + … ∈ S_{0} ˆ⊗C_{2} be a normalised finite slope overconvergent eigenform. Then the coefficients of f are all in Q_{2}.

Proof. We use only that the slopes of the non-zero eigenvalues of U are distinct. If λ denotes the eigenvalue of U on f then 1 − λ^{−1}U is not invertible on S_{0} ˆ⊗C_{2} and hence by Proposition 11 of [18] we see that λ^{−1} is a zero of the characteristic power series P (T) of U acting on S_{0}. Note that P (T) ∈ Q_{2}[[T]]. Choosing some big affinoid disc containing λ^{−1} and applying the Weierstrass preparation theorem to P (T) shows that λ^{−1} is a root of a polynomial with coefficients in Q_{2}. Hence λ ∈ \overline{Q}_{2}. Now all the Galois conjugates of λ have the same valuation and are also roots of the characteristic power series of U; hence, by Corollary 1, λ ∈ Q_{2}. Finally by Proposition 12 of [18] the subspace of S_{0} where U acts as multiplication by λ is one-dimensional over Q_{2} and so all the eigenvalues of all the other Hecke operators are also in Q_{2}. □

Remark. This Corollary provides some evidence towards Question 4.3 of [3]. See also Corollary 1.2 of [17].

Note that for p = 2 and N = 1, the map θ defines an isomorphism θ : S_{0} → S_{2} such that Uθ = 2θU. Thus, the slopes in weight two are precisely each of the slopes in weight zero, plus one. We have also proved Conjecture 2 for k = −12 using similar methods, although the combinatorics are too painful to write here, and the arguments do not seem to generalise to all k. Lawren Smithline was perhaps the first person to observe that there was some structure in the slopes of overconvergent modular forms of small level; his results ([19]) were primarily for the prime p = 3 but some of the techniques used in this paper for studying the explicit matrix representing U were inspired by his ideas. As far as we know, the first people to get explicit results pinning down all overconvergent slopes at a given weight were L. Kilford ([17]) and D. Jacobs ([14]) but their results differ in two respects from ours: firstly, they consider points nearer the boundary of weight space, and secondly the slopes at the weights they consider have a much simpler pattern—they form an arithmetic progression. We would like to thank Robert Coleman and Matthew Emerton for several helpful discussions. This genesis of this paper was a project at the 2001 Arizona Winter School.

2 Weight Zero

The curve X_{0}(2) has genus 0. A natural choice of uniformiser is given by the following function (Hauptmodul):

f(τ) = Δ(2τ)/Δ(τ)

In the sequel, we shall write this function simply as f. There is a product formula for f:

f = q ∏_{n=1}^{∞}(1+q^{n})^{24} = q ∏_{n=1}^{∞} \frac{1}{(1 − q^{2n−1})^{24}} = q+24q^{2}+300q^{3}+2624q^{4}+18126q^{5}+105504q^{6}+…

which follows immediately from the usual product formula for Δ. For p = 2, f is overconvergent of weight 0 and level 1. Furthermore, if g = 2^{6}f then the set {1, g, g^{2}, g^{3}, …} is a Banach basis for the space M_{0} := M_{0}(SL_{2}(Z); Q_{2}; 2^{−1/2}). It seems a little difficult to extract these concrete statements from the literature and so we sketch a proof of this in the appendix, and note that these ideas should easily be adaptable to cover other cases where one might want to do explicit computations. The reader who is happy to accept that the formal Banach space with basis {1, g, g^{2}, g^{3}, …} is some kind of p-adic space of modular forms might well want to avoid these technical details. To determine the spectral theory of U on M_{0}, we shall explicitly compute a matrix for U acting on M_{0}. These calculations are much in the spirit of classical congruences for coefficients of modular functions such as j, see for example Watson [20], or Atkin and O’Brien [1]. In this optic, the a fortiori presence of a spectral theory greatly simplifies matters. Concretely then, our task is to compute U(f^{k}) as a power series in f, for all k ≥ 0.

Lemma 1 The following identities are satisfied:

U(f) = 24f + 2^{11}f^{2}.

f(\frac{τ}{2})f(\frac{τ + 1}{2}) = −f(τ).

Proof. The operator U preserves the space of functions on X_{0}(2). Furthermore U(f), considered as a map X_{0}(2) → P^{1}, has degree at most 2, and hence U(f) − 24f − 2^{11}f^{2}, if non-zero, is a function X_{0}(2) → P^{1} of degree at most 4. Hence one can check that this function is identically zero by computing the first few terms of its q-expansion and verifying that they are zero. The second identity follows similarly, or directly from the product formulas:

f(τ)f(τ + \frac{1}{2}) = q ∏_{n=1}^{∞} \frac{1}{(1 − q^{2n−1})^{24}} × (−q) ∏_{n=1}^{∞}(1 + (−q)^{n})^{24}

= −q^{2} ∏_{n=1}^{∞} \frac{(1 − q^{2n−1})^{24}(1 + q^{2n})^{24}}{(1 − q^{2n−1})^{24}} = −q^{2} ∏_{n=1}^{∞}(1 + q^{2n})^{24} = −f(2τ).

□

Using this lemma, we may inductively determine U(f^{k}) for all positive k. To do this, we observe (by definition) that

2 · U(f^{k}) = f(\frac{τ}{2})^{k} + f(\frac{τ + 1}{2})^{k}.

Thus, if X_{k} := U(f^{k}), then multiplying out, one sees that the X_{k} satisfy the recurrence relation:

X_{k+2} − (f(\frac{τ}{2}) + f(\frac{τ + 1}{2})) · X_{k+1} + f(\frac{τ}{2})f(\frac{τ + 1}{2}) · X_{k} = 0 (k ≥ 0).

Moreover, From Lemma 1, we may evaluate the coefficients of this recurrence to conclude that X_{0} = 1, X_{1} = 24f + 2^{11}f^{2} and, for k ≥ 2,

X_{k} = U(f^{k}) = (48f + 2^{12}f^{2})U(f^{k−1}) + f · U(f^{k−2}).

In particular, we note that U(f^{k}) is a polynomial in f with integer coefficients and of degree at most 2k. These results are in Emerton’s thesis and apparently are originally due to Kolberg.

Definition 1 Define integers s_{i,j}, i, j ∈ Z_{≥0}, by

U(f^{j}) = ∑_{i=0}^{∞} s_{i,j}f^{i}.

Note that s_{i,j} = 0 for i > 2j. We also note that s_{i,j} = 0 for j > 2i, by comparing the coefficients of q^{i} in the definition of s_{i,j}. We have s_{0,0} = 1, s_{1,1} = 24 and s_{2,1} = 2^{11}, s_{1,2} = 1, and s_{i,j} = 0 in all other cases with 0 ≤ i ≤ 1 or 0 ≤ j ≤ 1.

Lemma 2 The integers s_{i,j} satisfy the recurrence relation:

s_{i,j} = 48s_{i−1,j−1} + 2^{12}s_{i−2,j−1} + s_{i−1,j−2} (i, j ≥ 2).

Moreover, for i, j ≥ 1 and i ≤ 2j, j ≤ 2i we have an equality:

s_{i,j} = \frac{(i + j − 1)!3j · 2^{8i−4j−1}}{(2i − j)!(2j − i)!} .

Proof. The recurrence for s_{i,j} follows directly from the recurrence for U(f^{k}). The explicit formula also satisfies this same recurrence and moreover equals s_{i,j} for s_{i,1}, s_{i,2}, s_{1,j} and s_{2,j}. This suffices to prove the second equality. □

The constant function 1 is an eigenform for U with eigenvalue 1. Thus to determine the spectral theory of U on M_{0} it suffices to work with the cuspidal subspace:

S_{0} := S_{0}(SL_{2}(Z); Q_{2}; 2^{−1/2})

of q-expansions with zero constant term. In the p-adic setting, cuspidal generalises the notion of vanishing at the single cusp ∞; thus certain Eisenstein series (such as the twin form of the usual Eisenstein series E_{2k}) are considered cuspidal. Lemma 2 provides us with an explicit description of the action of U on S_{0}. This allows us to gain fine control over the spectrum of U on S_{0}. Before we begin the proof of Theorem 2, we recall some elementary facts about continuous endomorphisms of Banach spaces. If M is a Banach space over Q_{p}, and {e_{1}, e_{2}, e_{3}, …} is a countable subset of M, then we say that {e_{1}, e_{2}, e_{3}, …} is an orthonormal Banach basis for M if

• |e_{i}| = 1 for all i,

• every m ∈ M can be written uniquely as m = ∑_{i≥1} a_{i}e_{i} for a sequence a_{i} ∈ Q_{p} such that a_{i} → 0 as i → ∞, and

• if m, a_{i} are as above, then |m| = max_{i} |a_{i}|.

If M is a Banach space and {e_{1}, …} is an orthonormal Banach basis for M, and if ϕ : M → M is a continuous Q_{p}-linear map, then we define the matrix of ϕ to be the collection (c_{i,j})_{i,j≥1} such that ϕ(e_{j}) = ∑_{j} c_{i,j}e_{i}. The collection (c_{i,j}) has the following two properties:

(i) For all j, lim_{i→∞} c_{i,j} = 0 (ii) There exists some C ∈ R such that |c_{i,j}| ≤ C for all i, j. Conversely, given a collection (c_{i,j})_{i,j≥1} satisfying (i) and (ii) above, there is a unique continuous linear map ϕ : M → M with matrix (c_{i,j}). Composition of linear maps corresponds to multiplication of matrices using the usual formula, which one easily checks to converge because of (i) and (ii) above. Set r = 1/2, let M_{0} denote the 2-adic Banach space of 2^{−r}-overconvergent 2-adic modular forms of weight 0 and tame level 1, equipped with the supremum norm, and let S_{0} denote its cuspidal subspace. We prove in the appendix that a Banach basis for M_{0} is {1, g, g^{2}, g^{3}, …} with g = 2^{6}f; we consider M_{0} as being equipped with this basis once and for all. Moreover, S_{0} has a natural basis given by {g, g^{2}, g^{3}, …}. We can write the matrix of the operator U on S_{0} as (u_{i,j})_{i,j≥1}, where

u_{i,j} = 2^{6j−6i}s_{i,j} = \frac{(i + j − 1)!3j · 2^{2i+2j−1}}{(2i − j)!(2j − i)!}

(and where we interpret this as being zero if i > 2j or j > 2i). Let A = (a_{i,j}), B = (b_{i,j}), D = (d_{i,j}) (i, j ≥ 1), be respectively the lower triangular, upper triangular and diagonal matrices defined as follows:

a_{i,j} = \frac{2^{2i−2j}i!^{2}(2j)!^{2}(2j + i − 1)!}{(2i)!(i − j)!j!(i + j)!(2j − i)!(3j − 1)!}, 2j ≥ i ≥ j, 0 otherwise.

b_{i,j} = \frac{j}{i} · \frac{2^{2j−2i}j!^{2}(2i)!^{2}(2i + j − 1)!}{(2j)!(j − i)!i!(j + i)!(2i − j)!(3i − 1)!}, 2i ≥ j ≥ i, 0 otherwise.

and

d_{i,i} = \frac{2^{4i+1}(3i)!^{2}i!^{2}}{3 · (2i)!^{4}} .

Note the symmetry in these formulas. One has i · b_{i,j} = j · a_{j,i}, and a_{i,i} = b_{i,i} = 1.

Lemma 3 The matrices defined by A, B and D satisfy (i) above, and all have coefficients in Z_{2} so also satisfy (ii) above. Moreover A ≡ B ≡ Id mod 2.

Proof. That A, B and D satisfy (i) is clear because if i > 2j then a_{i,j} = b_{i,j} = d_{i,j} = 0. Thus it suffices to prove that A ≡ B ≡ Id mod 2 and that D has coefficients in Z_{2}. Recall that v_{2} ((2n)!) = n + v_{2}(n!) and from this we see that v_{2}(d_{i,i}) = 1 + 2v_{2} ((3i)!/i!) > 0, so D has entries in Z_{2}. Next we deal with A. Because a_{i,i} = 1 for all i, it suffices to prove that the valuation of a_{i,j} is positive for 2j ≥ i > j ≥ 1. We write:

a_{i,j} = 6ij (\frac{(2j)!}{2^{j}j!})^{2} (\frac{2^{i}i!}{(2i)!})^{2} (\frac{(2i − 1)!}{(i + j)!}) (\frac{(2j + i − 1)!}{(3j)!}) \binom{j}{i − j}.

Again using that v_{2}((2n)!) = n + v_{2}(n!), we see that for i > j the right hand side is clearly 6 times a product of terms in Z_{2}, and so lies in 2Z_{2}. For B the argument is similar: since b_{i,i} = 1 for all i, it suffices to prove that the valuation of b_{i,j} is positive for 2i ≥ j > i. We write:

b_{i,j} = \frac{j}{i} · a_{j,i} = 6j^{2} (\frac{(2i)!}{2^{i}i!})^{2} (\frac{2^{j}j!}{(2j)!})^{2} (\frac{(2j − 1)!}{(i + j)!}) (\frac{(2i + j − 1)!}{(3i)!}) \binom{i}{j − i},

and again observe that the right hand side is 6 multiplied by a product of terms all of which lie in Z_{2}. □

Hence the matrices A, B, D all define continuous endomorphisms of S_{0}, which we also call A, B and D.

Lemma 4 We have ADB = U.

Proof. It suffices to show that

u_{i,j} = ∑_{k} a_{i,k}d_{k,k}b_{k,j}.

The right hand side of this equation becomes, after expanding out and simplifying,

u_{i,j} · \frac{4 · i!^{2}j!^{2}(2i − j)!(2j − i)!}{(2i)!(2j)!(i + j − 1)!} ∑_{k} \frac{(2k + i − 1)!k(2k + j − 1)!}{(i − k)!(i + k)!(j − k)!(j + k)!(2k − i)!(2k − j)!}.

Hence it suffices to prove that

\frac{(2i)!(2j)!(i + j − 1)!}{4 · i!^{2}j!^{2}(2i − j)!(2j − i)!} = ∑_{k} \frac{(2k + i − 1)!k(2k + j − 1)!}{(i − k)!(i + k)!(j − k)!(j + k)!(2k − i)!(2k − j)!}.

This identity, however, follows from classical results; for example, we derive it from a three term specialization of Dougall’s _{7}F_{6} summation formula (see [12], (7.2.3)). First note that by symmetry, we may assume that i ≤ j. To force the summation to start at k = 0, we let k = i − n. Let (a)_{n} = (a)(a + 1) ⋯ (a + n − 1). After repeated application of the formal relations

(a)_{n} := \frac{(a + n − 1)!}{(a − 1)!},  (−a)_{n} = \frac{a!(−1)^{n}}{(a − n)!},  (−a)_{n}(−a + 1/2)_{n} = \frac{(2a)!}{2^{2n}(2a − 2n)!}

to transform our sum into hypergeometric form, the required identity becomes the following:

7F_{6} ( \begin{matrix}(−i)/2, (1 − i)/2, (j − 2i)/2, (j − 2i + 1)/2, 1 − i, −2i, −i − j \\ (1 − 3i)/2, (2 − 3i)/2, (−2i + 1 − j)/2, (−2i + 2 − j)/2, −i, j − i + 1 \end{matrix} ; 1)

= \frac{3 · (2i)!^{2}(2j)!(i + j − 1)!(j − i)!(j + i)!}{4 · i!j!^{2}(2j − i)!(3i)!(2i + j − 1)!}.

The smallest integer in the numerator is i/2 or (i − 1)/2, and we consider each case separately. Dougall’s summation formula expresses this hypergeometric sum as a mélange of rising factorials that eventually simplify to the required answer. □

An alternative method for proving our identity would be via the automated “creative telescoping” of Zeilberger. See for example [21], where Zeilberger proves Dougall’s summation formula in one (rather long) line; the very short proof there specialises to a proof of the identity we require (note however that the statement of the theorem in [21] contains a typographical error: the (−1 − a − b − c − d)_{n} term in the denominator should be (−1 − a + b + c + d)_{n} and the proof should be modified similarly).

Lemma 5 The Newton Polygon of U := ADB is the same as the Newton Polygon of D.

Proof. We use only the fact that A and B are both congruent to the identity modulo 2, and that D is integral, diagonal, and compact. Because B is congruent to the identity mod 2 it has an inverse. Note by §5, Corollaire 2 of [18], ADB has the same Newton Polygon as B(ADB)B^{−1} = BAD, so it suffices to prove that CD has the same Newton Polygon as D, for any matrix C congruent to the identity modulo 2. If X = (x_{i,j})_{i,j≥1} is a matrix, and r_{1}, r_{2}, …, r_{n} are distinct positive integers, then by the n × n principal minor of X associated to these integers we mean the determinant of the n by n matrix formed from the r_{i}th rows and columns of X, 1 ≤ i ≤ n. If X is the matrix associated to a compact morphism, then the Newton Polygon of X is the lower convex hull of the points (n, Σ_{n}) ∈ R^{2}, where Σ_{n} is the valuation of the sum of all n × n principal minors of M. Firstly, note that if d is any n×n principal minor of the diagonal matrix D, then (n, v_{2}(d)) lies on or above the Newton Polygon of D. Secondly, note that for each (n, Σ_{n}) that lies at a vertex of the Newton Polygon of D, there is a unique n × n minor with maximal valuation. Both of these facts are easily verifiable using the fact that D is diagonal. Next note that if r_{1}, r_{2}, …, r_{n} are distinct positive integers then the principal minors of D and CD associated with these integers have the same valuation, because the n×n minor of CD associated to these integers is just the product of the minor associated to C (which is a unit) and the minor associated to D. Hence all principal minors of D and CD have the same valuation and now it is easy to check that this forces the Newton polygons of C and D to be the same.

□

Theorem 2 follows immediately from the above lemma.

3 Extensions and Generalizations

3.1 What is special about the function f?

There are many ways to parameterise a (p-adic) disc, but the choice of f that we made led to the simple formulae which enabled us to prove results about slopes. One possible reason why this f was a good choice is that the basis defined by powers of f behaves well with respect to a certain pairing, which we define below. Recall our function g defining an isomorphism of X_{0}(2) with the projective line. Let w denote the Atkin-Lehner involution on X_{0}(2).

Lemma 6 For n ∈ Z we have w^{∗}g^{n} = g^{−n}.

Proof. It suffices to prove the lemma for n = 1. Since Δ(−1/τ) = τ^{12}Δ(τ), we see that

w^{∗}g = g(−1/(2τ)) = \frac{2^{6}Δ(−1/τ)}{Δ(−1/(2τ))} = \frac{2^{6}τ^{12}Δ(τ)}{2^{12}τ^{12}Δ(2τ)} = \frac{1}{g}.

□

Let X denote the rigid affinoid annulus in X_{0}(2) defined by X = {x ∈ X_{0}(2) : |g(x)| = 1}. This is the width zero annulus “in the middle” of the supersingular annulus in 2^{−6} < |g| < 2^{6} in X_{0}(2), and the Atkin-Lehner involution w induces an involution X → X. If η is a holomorphic one-form on X then we can write η = (∑_{n} a_{n}g^{n})dg and we define ∫_{∞} η := a_{−1}. Similarly we can write η = (∑_{n} b_{n}(1/g)^{n}) d(1/g) and we define ∫_{0} η := b_{−1}. An easy check shows that ∫_{0} η + ∫_{∞} η = 0.

Definition 2 Let ⟨ , ⟩ denote the following bilinear form on M_{0}:

⟨α, β⟩ = ∫_{∞} w^{∗}α · dβ.

Lemma 7 The bilinear form ⟨ , ⟩ is symmetric.

Proof. Since w^{∗} swaps the two cusps, we see that

⟨α, β⟩ = ∫_{0} α · d(w^{∗}β)

Since ∫_{0} + ∫_{∞} = 0 we see that

⟨α, β⟩ = − ∫_{∞} α · d(w^{∗}β)

In particular,

⟨α, β⟩ − ⟨β, α⟩ = ∫_{∞} w^{∗}α · dβ + β · d(w^{∗}α) = ∫_{∞} d((w^{∗}α)β) = 0.
□

Lemma 8 The basis {g^{k}}_{k=0}^{∞} is an orthogonal basis for M_{0}, with respect to ⟨ , ⟩.

Proof. By lemma 6 we have w^{∗}g^{k} = g^{−k}. Hence

⟨g^{m}, g^{n}⟩ = ∫_{∞} g^{−m} · ng^{n−1}dg = ∫_{∞} ng^{n−m} · \frac{dg}{g} = n · δ_{m,n}.

□

Thus powers of g behave nicely with respect to this pairing. On the other hand, this pairing behaves nicely with respect to U:

Theorem 3 U is self-adjoint with respect to ⟨ , ⟩.

Proof. It is enough to show this for any pair g^{j}, g^{i}. We see that

⟨Ug^{j}, g^{i}⟩ = ∑_{k=0}^{∞} u_{k,j}⟨g^{k}, g^{i}⟩ = u_{i,j}⟨g^{i}, g^{i}⟩.

Now from the above calculation and our explicit evaluation of u_{i,j}, this is equal to

\frac{2^{2i+2j−1}3ij(i + j − 1)!}{(2i − j)!(2j − i)!} = ⟨g^{j}, Ug^{i}⟩,

since the penultimate expression is symmetric. □ It is natural to ask whether this theorem is a special case of a more general phenomenon.

3.2 Weights other than Zero

Our results in this section are unfortunately much more incomplete. We may relate the action of U in weight 0 to the action of U in weight k by “Coleman’s trick”, namely, a judicious application of the identity U(gV (h)) = hU(g). In our case we again obtain explicit formulae for matrix entries of U acting on S_{k}. Take for example the case k = −12m with m ≥ 0 an integer. Define

h_{k} = \frac{Δ(2τ)^{m}}{Δ(τ)^{2m}} = Δ(2τ)^{−m}f^{2m}

By Coleman’s trick we now observe that

U(h_{k}f^{j}) = U(Δ(2τ)^{−m}f^{j+2m}) = Δ(τ)^{−m}U(f^{j+2m}) = h_{k}f^{−m}U(f^{j+2m})

In particular, with respect to the basis {h_{k}g, h_{k}g^{2}, h_{k}g^{3}, …}, U in weight −12m can be given explicitly by the matrix (2^{−6m}u_{i+m,j+2m})_{i,j≥1}. Since for m ∈ ℕ, −12m is dense in 4Z_{2}, if one could prove Conjecture 2 in the case k = −12m then by a continuity argument one could prove it for all k congruent to 0 mod 4, and then by using the theta operator one should be able to deal with the case k ≡ 2 mod 4 as well. However, although we have a factorization of the form U = ADB, the matrices A and B (and BA) are unfortunately not integral, and new methods seem to be required.

3.3 Primes other than p = 2

Our methods rely strongly on the fact that X_{0}(p) has genus zero. Presumably our results can be extended to other primes with this property, with perhaps a corresponding increase in combinatorial difficulty. On the other hand, our techniques fail, or at least must be greatly modified, when the genus of X_{0}(p) is greater than zero, even though similar results seem to be true. For example, the following conjecture appears to be compatible with the conjectures in [3]:

Conjecture 3 Let |λ_{1}| ≥ |λ_{2}| ≥ |λ_{3}| ≥ … be the non-zero eigenvalues (with multiplicities) of U on S_{0}(SL_{2}(Z); Q_{11}; 11^{−r}) for some rational r with 11/12 > r > 0. Then the slope (that is, the 11-adic valuation) of λ_{n} is given by the following formula:

v_{11}(λ_{n}) = v_{11} (\frac{[\frac{6n+1}{5}]! [\frac{6n+4}{5}]!}{[\frac{n}{5}]! [\frac{n}{5}]!}) + ∑_{k=1}^{4} [\frac{n + k}{5} ]

where [x] denotes the largest integer less than or equal to x.

4 Appendix: Overconvergent forms at small level.

For this appendix we work with a general level and a general prime p. For want of a reference, we explain how to extend the theory of overconvergent p-adic modular forms to cases where the level structure is too coarse for the resulting moduli problem to be rigid. The motivation is that there is a growing theory of “explicit” computations with padic modular forms, where typically both the level and the prime are small—for example, this paper. On the other hand, at several points in theoretical papers on the subject the hypothesis is made that the level in which one is working is at least 5, or that p ≥ 3, for convenience (lifting the Hasse invariant, for example). One problem at small level is that there is, as far as we know, no reference for the theory of “rigid analytic stacks”, so we proceed by the usual low-level ad hoc methods. Here is an overview of the idea: take the level structure one is interested in, and a sufficiently small auxiliary Galois level structure with Galois group G (for example a full level M structure for some large prime M). The product level structure is sufficiently fine to imply the existence of a compact curve X representing a moduli problem on generalised elliptic curves, and X is equipped with an action of G. One can form the quotient curve X/G, which is not in general the solution to a moduli problem, but which is a coarse moduli space. One cannot always form an appropriate sheaf ω on these quotient curves, because ω does not always descend from X (problems at elliptic points, for example), but one can still define modular forms as G-invariant sections of tensor powers of ω on X. Next one has to check that the rigid-analytic subspaces that one is interested in are all G-invariant, which comes down to checking that they have an intrinsic definition that only depends on the underlying elliptic curve and not the level structure. We now formulate everything rigorously. Let U_{N} denote the compact open subgroup of GL_{2}(Ẑ) consisting of matrices congruent to the identity modulo N. If U is an arbitrary compact open subgroup of GL_{2}(Ẑ) then we define the level of U to be the smallest positive integer N such that U_{N} ⊆ U. Let Γ be a compact open subgroup of GL_{2}(Ẑ) and let p be a prime that does not divide the level of Γ. We shall give the main definitions in the theory of overconvergent p-adic modular forms for Γ. We say that a level structure Γ is “sufficiently small” if it satisfies the following two conditions:

(i) The identity element is the only element of Γ which is of finite order and is conjugate in GL_{2}(A_{f}) to an element of GL_{2}(Q), and

(ii) If C denotes the surjective Ẑ-module homomorphisms (Ẑ)^{2} → Ẑ, equipped with its natural right action of GL_{2}(Ẑ), then −1 operates without fixed points on C/Γ.

These properties are used in the following way. Property (i) implies that there will be no elliptic points in the associated moduli space, or, more precisely, that the associated moduli problem on elliptic curves is rigid and hence representable ([16], appendix to §4). Property (ii) implies that there will be no irregular cusps on the compactification of the representing object (see [16], section 10.13 and Proposition 10.13.4). We remark that Γ = U_{M} for any M ≥ 3 satisfies properties (i) and (ii). However, perhaps the most common modular curves that one sees are the curves X_{0}(N), corresponding to the compact open subgroup Γ of matrices in GL_{2}(Ẑ) which are upper triangular mod N, and property (i) always fails for these Γ, since −Id ∈ Γ.

Let us assume initially that Γ is sufficiently small. Because Γ satisfies (i), the associated moduli problem on elliptic curves is representable over Z_{p}, by a smooth affine curve Y (Γ), equipped with a universal elliptic curve π : E(Γ) → Y (Γ). Set ω := π_{∗}Ω_{E(Γ)/Y (Γ)}^{1}. Then ω is an invertible sheaf on Y (Γ). Because Γ satisfies (ii), the natural compactification X(Γ) of Y (Γ) is also the solution to a moduli problem (that of parameterising generalised elliptic curves with level structure—see [11], III.6, although we shall not use this in what follows). Next note that the sheaf ω extends in a natural way to an invertible sheaf on X(Γ) ([16], Proposition 10.13.14). We define a classical modular form of level Γ and weight k, defined over Q_{p}, to be a global section of ω^{⊗k} on the generic fibre of X(Γ). If R is a Z_{p}-algebra then we denote by X(Γ)_{R} the base change of X(Γ) to R. The special fibre X(Γ)_{F_{p}} of X(Γ) is a smooth proper geometrically connected curve, and has finitely many supersingular points. Let P denote a supersingular point and say P is defined over the finite field F. Let W denote the Witt vectors of F, and let K denote the field of fractions of W. If X(Γ)^{an} denotes the rigid space over K associated to X(Γ) then there is a reduction map from X(Γ)^{an} to X(Γ)_{F}(\overline{F}), and the pre-image U of P is isomorphic to an open disc. The completed local ring of X(Γ)_{W} at P is a W -algebra isomorphic non-canonically to a power series ring W [[t]] in one variable; let us fix one such isomorphism. Then t can be thought of as giving an isomorphism from U to the rigid analytic open unit disc. Hence if r ∈ Q_{≥0} then we can talk about the open subdisc {x : |t(x)| < p^{−r}} of U. These subdiscs in general depend on the fixed isomorphism between the completed local ring and the power series ring, but if r < 1 then an easy calculation shows that they are independent of such choices—the point that we have chosen to be the centre of U is a K-point and the other K-points in U all have distance at most 1/p from our chosen centre, because K is an unramified extension of Q_{p}. See §3 of [2] for more details of this construction, or §2 of [4]). The universal formal deformation of the elliptic curve E_{0}/F corresponding to P is an elliptic curve over the completed local ring of X(Γ)_{W} at P, and hence can be regarded via our fixed isomorphism as an elliptic curve E/W [[t]]. Fix a basis η of H^{0}(E, Ω_{E/W [[t]]}^{1}), so

H^{0}(E, Ω_{E/W [[t]]}^{1}) = W [[t]] · η.

The Hasse invariant can be thought of as a mod p section of the (p − 1)st tensor power of this module, and hence an element A(t)η^{⊗(p−1)} of F[[t]] · η^{⊗(p−1)}. It is well-known that the Hasse invariant has a simple zero at every supersingular elliptic curve, which translates into the fact that A(t) is a uniformiser of F[[t]]. This provides the bridge between our point of view and Katz’. For example, Katz’ analysis of the p-divisible group associated to an elliptic curve and its relation to a lifting of the Hasse invariant in §3.7 of [15] shows that for a \overline{K}-point u ∈ U corresponding to an elliptic curve E_{u}, if |t(u)| > 1/p then |t(u)| is independent of all choices we have made, and depends only on the isomorphism class of E_{u} (and not on the level structure). Assume from now on that 0 ≤ r < 1, and that F is sufficiently large so that all the supersingular points in X(Γ)_{F_{p}} are defined over F. Choose parameters t as above for every supersingular point, and define X(Γ)_{K,≥p^{−r}} to be the rigid space over K which is the complement of the open discs {x : |t(x)| < p^{−r}} as above, as P ranges over all supersingular points of X(Γ)_{F}. If u is a \overline{K}-valued point of Y (Γ)_{K} then let E_{u} denote the fibre of the universal elliptic curve above x. So E_{x} is an elliptic curve defined over K(x). Now Katz’ arguments show that if E_{x} has good supersingular reduction and if |t(x)| > p^{−1}, then |t(x)| = |t(σx)| for any Q_{p}-automorphism of the field \overline{K(x)}. Hence for r ∈ Q with 0 ≤ r < 1 the rigid space X(Γ)_{K,≥p^{−r}} is the base extension to K of a rigid subspace X(Γ)_{≥p^{−r}} of X(Γ) defined over Q_{p}. Say r ∈ Q with 0 ≤ r < 1. We define a p^{−r}-overconvergent modular form of level Γ and weight k, defined over Q_{p}, is a section of ω^{⊗k} on X(Γ)_{≥p^{−r}}. Now let Γ be an arbitrary (not necessarily sufficiently small) compact open subgroup of GL_{2}(Ẑ), and let p be a prime not dividing the level of Γ. Choose a prime M > 2 dividing neither p nor the level of Γ, and let Γ′ denote Γ ∩ U_{M}. Then Γ′ is sufficiently small in the sense above, and so all of the definitions above apply to Γ′. Furthermore, Γ′ is normal in Γ; let G denote the quotient group. Then G is finite (in fact G is isomorphic to GL_{2}(Z/MZ)) and G acts on X(Γ) and ω. Moreover, because |t(x)| (notation as above) only depends on the elliptic curve E_{x} and not any level structure, X(Γ′)_{≥p^{−r}} is G-invariant if 0 ≤ r < 1. This motivates the following definitions. We define X(Γ) to be quotient of X(Γ′) by the finite group G; note that X(Γ′) is a projective curve so taking quotients is not a problem. In practice, one can form the quotient in the following manner: Y (Γ′) is affine, and is hence of the form Spec(R), R a ring with an action of G. One defines Y (Γ) = Spec(R^{G}) and then compactifies. Note that the sheaf ω will probably not in general descend to X(Γ). However, we can still define an r-overconvergent modular form of level Γ as being a Ginvariant element of H^{0}(X(Γ′)_{≥p^{−r}}, ω^{⊗k}), if 0 ≤ r < 1. Sometimes these spaces are zero for trivial reasons—for example if Γ contains −1 and if k is odd. If they are not zero, then they are always infinite-dimensional. One needs to check that these definitions are independent of the auxiliary choice of Γ′. This is not too difficult—if Γ′_{1} and Γ′_{2} are two choices for Γ′ with Galois groups G_{1} and G_{2} then one sets Γ′ := Γ′_{1} ∩ Γ′_{2} and checks that both definitions for X(Γ) coincide with the quotient of X(Γ′) by Γ/Γ′, and so on. One may check without too much difficulty that the weight 0 r-overconvergent forms of level Γ are just the functions on X(Γ)_{≥p^{−r}}; this comes from the fact that one can form quotients of affinoids by finite groups by looking at invariants, and compatibility of this with the analytification functor. One useful result is that if Γ ⊆ Δ both have level prime to p then the pre-image of X(Δ)_{≥p^{−r}} under the canonical (forgetful functor) map from X(Γ) to X(Δ) is X(Γ)_{≥p^{−r}}. Only a little harder is the fact that if γΓγ^{−1} ⊆ Δ and Γ, Δ have level prime to p, and γ_{p} = 1, then the same is true for the map X(Γ) → X(Δ) induced by γ. This is because |t(x)| depends only on the underlying p-divisible group of the elliptic curve. The above approach is good for theoretical purposes, but is too abstract in general to be of much computational use. We now show how to use these ideas to make the claims of this paper rigorous, thus turning the argument in this paper from a formal one into a rigorous one. We set Γ = GL_{2}(Ẑ) and p = 2, and write X_{0}(1) for X(Γ). We now explicitly evaluate X_{0}(1)_{≥p^{−r}} for 0 ≤ r < 3/4.

It is well-known that the j-invariant gives an isomorphism X_{0}(1) → P^{1} defined over Z_{2}. Let us set M = 3 (notation as above), so Γ′ = U_{3} and Y (3) := Y (Γ′) is the modular curve over Z_{2} parameterising elliptic curves equipped with two points of order 3 generating the 3-torsion of the curve. The generic fibre of this curve is not geometrically connected but this does not matter. The classical theta series θ := ∑_{a,b∈Z} q^{a^{2}+ab+b^{2}} = 1 + 6q + 6q^{3} + … is a modular form of level Γ′ (in fact it has level Γ_{1}(3)) and by the q-expansion principle it is a lift of the mod 2 Hasse invariant. The special fibre of X(3) := X(Γ′) has two geometric fibres, both of genus 0, both defined over F := F_{4}, and both with one F-valued supersingular point. Let P denote one of these supersingular points, and define W, K, U as above. Choose an isomorphism ι of the complete local ring of X(3)_{W} at P with W [[t]], and let E/W [[t]] denote the universal formal deformation of the elliptic curve corresponding to the point P. Fix a basis η of H^{0}(E, Ω_{E/W [[t]]}^{1}), so

H^{0}(E, Ω_{E/W [[t]]}^{1}) = W [[t]] · η.

Now if f is any modular form over W (\overline{F}) of level Γ′ and any weight k, then f(E) = hη^{⊗k} for some h ∈ W [[t]]. We now think of η as being fixed, and identify a modular form f with the corresponding function h as above. If u ∈ U then we define |f(u)| = |h(u)| and note that this is independent of the choice of η. Moreover, if |f(u)| > 1/2 then this value is also independent of the choice of ι. Via this fixed isomorphism, θ can be regarded as an element of W [[t]] and because θ lifts the Hasse invariant, we know that θ mod 2 in F[[t]] will be of the form ut + O(t^{2}) with u ≠ 0. Now consider the classical level one Eisenstein series E_{4} = 1 + 240(∑_{n≥1} σ_{3}(n)q^{n}). Note that the q-expansion of θ is congruent to 1 mod 2, and hence θ^{4} has q-expansion congruent to 1 mod 8. In particular θ^{4} ≡ E_{4} mod 8. This congruence can be thought be thought of as a congruence of elements of W [[t]]. Now for u ∈ U with |E_{4}(u)| > 1/8 we see that |θ(u)|^{4} = |E_{4}(u)| and hence |θ(u)| > 2^{−3/4} > 2^{−1}. So |t(u)| = |θ(u)| = |E_{4}(u)|^{1/4}. Conversely, if |t(u)| > 2^{−3/4} then |E_{4}(u)| = |t(u)|^{4} > 1/8. We conclude that if 0 ≤ r < 3/4 then X(3)_{≥2^{−r}} is the subregion of X(3) where |E_{4}| > 2^{−4r}. Moreover, because j = (E_{4})^{3}/Δ, and |Δ(u)| = 1 for all u ∈ U (as u corresponds to an elliptic curve with good reduction), we see that for 0 ≤ r < 3/4, we have that X(3)_{≥2^{−r}} is the region defined by |j| ≥ 2^{−12r}. We have proved

Proposition 1 If p = 2 and 0 ≤ r < 3/4 then X_{0}(1)_{≥p^{−r}} is the subdisc of the j-line defined by |j| ≥ 2^{−12r}.

We now let g denote the modular function 2^{6}Δ(2z)/Δ(z). Recall that this is an isomorphism X_{0}(2)_{Q_{2}} → P^{1}. Now one checks that 64/j = g/(4g + 1)^{3} and hence the map X_{0}(2)_{Q_{2}} → X_{0}(1)_{Q_{2}} induced by the forgetful functor sends the region {|g| ≤ 1} to the region {|j| ≥ |64|}. Moreover, this map preserves q-expansions, and induces an isomorphism between these two regions (one can write down an inverse, for example, to see this). We deduce that the disc {|g| ≤ 1} in X_{0}(2) is isomorphic to X_{0}(1)_{≥2^{−1/2}}, and that the functions {1, g, g^{2}, g^{3}, …} are a Banach basis for 2^{−1/2}-overconvergent level 1 weight 0 modular forms.

Furthermore, because g vanishes at infinity, the functions {g, g^{2}, g^{3}, …} form a basis for the space of 2^{−1/2}-overconvergent tame level 1 cusp forms of weight 0. Finally, we say a word about other weights k ≡ 0 mod 12. Write k = −12m and define h_{k} := Δ(q^{2})^{m}/Δ(q)^{2m}. Then h_{k} is a meromorphic section of ω^{⊗k} on X_{0}(2) and a computation of q-expansions shows that it is non-vanishing at infinity. It hence defines a section of ω^{⊗k} on X(3)_{≥2^{−1/2}} which is G := GL_{2}(F_{3})-invariant and non-vanishing, and hence a trivialisation of ω^{⊗k} on this rigid space. It is now easy to show that the G-invariant sections of ω^{⊗k} are exactly the sections of the form h_{k}.s for s a function on X_{0}(1)_{≥2^{−1/2}} and this establishes all the claims about explicit bases of overconvergent modular forms in this paper.

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