The 2-adic eigencurve at the boundary of weight space.

Kevin Buzzard and L. J. P. Kilford

Abstract

We prove that near the boundary of weight space, the 2-adic eigencurve of tame level 1 can be written as an infinite disjoint union of “evenly-spaced” annuli, and on each annulus the slopes of the corresponding overconvergent eigenforms tend to zero.

1. Introduction

Coleman and Mazur constructed in [CM98] fundamental objects called eigencurves, parametrising finite slope overconvergent normalised p-adic eigenforms. In [CM98] many open questions are raised, one of which is whether the slopes on every component of the eigencurve tend to zero as one approaches the boundary of weight space. The object of this paper is to prove this assertion for E, the 2-adic eigencurve of tame level 1 (this notation will not be ambiguous because we shall not consider any other eigencurves in any detail in this paper). In fact we prove much more, explicitly computing the rigid space underlying E near the boundary of weight space. Note that strictly speaking this eigencurve was not constructed in [CM98], because of the running assumption p > 2, but see [Buz] for a construction of the eigencurve in this case (using Hecke algebras; the construction using deformation theory has not been written down for p = 2, as far as we are aware).

The first author has in recent years been making an explicit analysis of various aspects of E, on the basis that it should serve as a guide as to what to expect for eigencurves in general. The explicit questions about slopes in [Buz04] (which were not specific to the p = 2 case) were initially motivated by many computations of classical level 1 modular forms and the action of the operator T_{2} on these forms; these computations gave explicit information about E. Answers to the questions in [Buz04] would give information about E and other eigencurves near the centre of weight space, where the combinatorics of the slopes involved seems very complicated (although very little is known about the geometry of the abstract rigid space underlying an eigencurve, which might turn out to be very simple). Graham Herrick in [Her] seems to have made more conceptual and slightly more general conjectures about slopes in the central region, but the only theorem of which the authors are aware that computes all slopes at a point of weight space in this central region is the main result of [BC], where the slopes in weight 0 are explicitly computed for E.

It was the second author in his thesis [Kil02] who proved the first results indicating that nearer the boundary of weight space the story seemed much simpler. More precisely, he proved (see Theorem 3.2 of [Kil02]) that the sequence of slopes of the 2-adic overconvergent level 4 eigenforms of odd integer weight k ∈ Z was just the arithmetic progression 0, 2, 4, …, independent of k. He also obtained results about level 8 eigenforms, and later on dealt with some higher levels. These results led to the general belief that E may even be isomorphic to a disjoint union of copies of weight space, near the boundary of weight space. It is this result which we prove here.

We now set up some notation and state our main theorem precisely, and note a corollary. Let K denote any complete extension of Q_{2} (that is, K is a field complete with respect to a nonarchimedean

2000 Mathematics Subject Classification 11F33, 14G22 Keywords: eigencurve, slopes of modular forms, overconvergent modular forms

Kevin Buzzard and L. J. P. Kilford

valuation, containing a copy of Q_{2} and such that the norm on K induces the standard 2-adic norm on Q_{2}). Let W denote the open disc over K with centre 0 and radius 1. We make a slightly nonstandard definition: we define a weight to be a continuous group homomorphism κ : Z_{2}^{×} → K^{×} such that κ(−1) = 1. The condition κ(−1) = 1 is there because throughout this paper we are working in tame level 1, and hence there are no non-zero overconvergent modular forms of weight κ if κ(−1) = −1. If we identify a K-point w ∈ W with the unique weight κ : Z_{2}^{×} → K^{×} such that κ(5) = w + 1 then this gives a bijection between the K-points of W and the set of all weights. Note that the space we are calling W is the one denoted W^{+} in [CM98], because of our condition on κ(−1).

Note that if k = 2ℓ is an even integer then the function Z_{2}^{×} → K^{×} defined by x ↦ x^{k} is a weight, and this weight is denoted k throughout this paper. More generally, if χ : (Z/NZ)^{×} → K^{×} is a Dirichlet character of 2-power conductor N, and k is an integer satisfying (−1)^{k} = χ(−1), then classical modular forms of level N, weight k and character χ are, in this optic, modular forms of weight κ where κ(x) = x^{k}χ(x).

Let E denote the 2-adic eigencurve of tame level 1 as before. Then E is equipped with two canonical maps, a “weight” map π : E → W and a map U_{2} : E → G_{m}, sending a point e ∈ E to the U_{2}-eigenvalue of the corresponding finite slope eigenform. Let s : E(K) → R (s for slope) denote the composite of U_{2} with the valuation map v : K^{×} → R (normalised so that v(2) = 1).

Let W_{1} denote the open annulus {1/8 < |w| < 1} of W and let E_{1} denote π^{−1}(W_{1}).

Theorem A. The rigid space E_{1} can be written as a countably infinite disjoint union E_{1} = ∪_{i=0}^{∞}X_{i}, and the X_{i} have the following properties:

(i) π : X_{i} → W_{1} is an isomorphism for all i, and

(ii) if one identifies X_{i} with W_{1} via (i) above, then the induced map s : W_{1}(K) ≅ X_{i}(K) → R is the map sending w to iv(w).

We remark that the naive generalisation of theorem A to the annulus {1/8 ⩽ |w| < 1} is false; for example v(5^{10} − 1) = 3 but there is a level 2 weight 10 classical cuspidal eigenform q + 16q^{2} − 156q^{3} + 256q^{4} + … of slope 4, and hence the slopes are not all integer multiples of 3.

An elementary consequence of Theorem A is the following result:

Theorem B. If κ is a weight such that |κ(5) − 1| > 1/8, then the slopes of the overconvergent modular forms of weight κ are {0, t, 2t, 3t, …}, where t = v(κ(5) − 1), and each slope occurs with multiplicity 1.

Note that Theorem A trivially implies Theorem B. In fact the majority of this paper is a proof of Theorem B, and in the last section we show how to deduce Theorem A from Theorem B.

Theorem B has consequences in the classical theory of modular forms. Let k ⩾ 2 be an integer (even or odd) and let χ be a Dirichlet character of conductor N, where N ⩾ 4 is a power of 2. Assume furthermore that (−1)^{k} = χ(−1). Let M_{k}(N, χ) denote the space of weight k level N character χ classical modular forms, let S_{k}(N, χ) denote the subspace of cusp forms, and let U_{2} denote the classical Hecke operator on these spaces. One easily checks using Theorem 1 of [CO77] that dim M_{k}(N, χ) = 2 + dim S_{k}(N, χ) = \frac{(k−1)N}{8} + 1. Furthermore, the classical construction of Eisenstein series (see for example Theorems 7.1.3 and 7.2.12 of [Miy89], and a consideration of how the Diamond operators act on the Eisenstein series constructed there) shows that the space of Eisenstein series in M_{k}(N, χ) is 2-dimensional and has a basis consisting of the eigenforms

\frac{L(1 − k, χ)}{2} + ∑_{n⩾1} (∑_{d|n} χ(d)d^{k−1}) q^{n}

and

∑_{n⩾1} (∑_{d|n} d^{k−1}χ(n/d)) q^{n}.

By the slope of an eigenform we mean the 2-adic valuation of its U_{2}-eigenvalue. The two Eisenstein series above have slopes 0 and k − 1 respectively.

Corollary of Theorem B. (i) Fix an isomorphism C ≅ C_{2}. Then the 2-adic valuations of the eigenvalues of U_{2} on M_{k}(N, χ) are given by the set {0, t, 2t, 3t, …, (k − 1)} where t = 8/N. Each valuation occurs with multiplicity one.

(ii) There are \frac{(k−1)N}{8} + 1 normalised eigenvectors in M_{k}(N, χ) and, when regarded as modular forms with 2-adic coefficients, each of these eigenvectors has q-expansion in Q_{2}(χ)[[q]].

Remark. The case N = 4 of this result is Theorem 1.1 of [Kil02].

Proof of corollary. (i) Set K = C_{2}. Because χ is primitive, we know that χ(5) is a primitive (N/4)th root of unity. It follows that the corresponding point w = 5^{k}χ(5) − 1 of weight space satisfies |w| = 2^{−8/N} and hence 1 > |w| ⩾ 1/4 > 1/8, so we may apply Theorem B to deduce that the slopes of the overconvergent forms of weight k and character χ are {0, t, 2t, 3t, …}, each with multiplicity one. The values 0 and k − 1 correspond to the two Eisenstein series above, the slopes which are greater than k − 1 cannot be classical (see for example the proof of Theorem 4.6.17(1) of [Miy89], especially equation (4.6.30)), and there are only (k −1)/t−1 slopes left, namely {t, 2t, …, k −1−t}; each of these must hence be the slope of a classical cuspidal eigenform (note that this argument gives an alternative proof of the main theorem of [Col97] for p = 2 and κ as above).

(ii) There is a basis of M_{k}(Γ_{1}(N)) consisting of modular forms whose q-expansions have rational coefficients. By elementary group theory one deduces that M_{k}(N, χ) has a basis consisting of forms whose q-expansions have coefficients in Q(χ). Hence the characteristic polynomial of U_{2} on M_{k}(N, χ) has coefficients in Q(χ). Regarded as a polynomial in Q_{2}(χ) one sees from (i) that the valuations of the roots are all distinct. Hence all the roots must have degree 1 over Q_{2}(χ), because conjugates all have the same valuation. So the characteristic polynomial of U_{2} factors into distinct linear factors and the corresponding 1-dimensional subspaces of M_{k}(N, χ) must hence be Hecke equivariant and spanned by eigenforms.

We also note that our result implies that for χ a Dirichlet character of 2-power conductor N ⩾ 4, there are no weight 1 cusp forms of level N and character χ, because any such form would have slope at most 1 − 1 = 0, so the eigenvalue of U_{2} would be a 2-adic unit and our result implies that the only unit eigenvalues come from the Eisenstein family. On the other hand, this argument could already be extracted from Emerton’s thesis.

We finish this introduction with a brief sketch of the strategy of the proof of Theorem B. As Smithline observed, it is possible to “explicitly” write down the matrix coefficients (m_{i,j})_{i,j⩾0} of the U operator on certain (low tame level) spaces of weight κ overconvergent modular forms, with respect to a cleverly-chosen basis; the trick is that the power series ∑_{i,j} m_{i,j}X^{i}Y^{j} is the product of a relatively simple rational function of X and Y and a power series in X only (see Lemma 11 for an explicit form of this result in our case). This latter power series depends on κ and in our situation its coefficients are essentially those of the expansion of the overconvergent modular function E_{κ}/V (E_{κ}) (with E_{κ} an appropriate Eisenstein series) as a power series with respect to a certain parameter y on X_{0}(4). The reason for the appearance of the function E_{κ}/V (E_{κ}) is that it appears in “Coleman’s trick” for moving from forms of weight zero to forms of weight κ. Sections 2–6 of this paper produce bounds for these power series coefficients (see in particular Lemma 10(i)), for κ near the boundary of weight space. We remark that, almost paradoxically, these bounds are proved by a careful analysis of E_{κ}/V (E_{κ}) for κ near the centre of weight space (see Theorem 7 and its proof, which perhaps forms the technical heart of the paper). One one has these bounds one deduces certain upper and lower bounds for the valuations of the m_{i,j}. The miracle is that for κ near the boundary of weight space, these bounds are enough to tell us precise valuations for the coefficients of the characteristic power series of U (Proposition 15). This happens because our bounds are enough to ensure that in the infinite sums giving rise to each coefficient, one term has valuation strictly smaller than all of the others.

Acknowledgements

The genesis of this project came from two places: firstly Emerton’s thesis, where he computes some facts about the smallest slope cuspidal family, that is the region of the eigencurve denoted X_{1} above (indeed many of the computations in section 2–5 are variants on results in his thesis); and secondly the thesis of the second author (and also Smithline’s thesis), the techniques of which are used and extended a little in sections 7–8. Although none of the results in this paper rely on computer calculations,^{1} computers were used extensively throughout the work to spot conjectures about the growth of certain power series coefficients. Our computations were done using the MAGMA computer algebra system [BCP97], and many were done on a dual processor machine running Redhat Linux purchased with a grant awarded to the first author by the University of London Central Research Fund. Most of the write-up of the first version of this paper was done during a visit to the Banff International Research Station in December 2003. These organizations are all duly thanked. We remark that the first version of this paper used a completely different set of uniformisers and only treated the region 1/4 < |w| < 1, and it was Frank Calegari who encouraged us to extend the results to |w| = 1/4 and beyond. Finally, the first author would like to thank his mother for looking after his kids for a weekend in January 2004 whilst he changed all the variables in the paper to deal with the region 1/8 < |w| ⩽ 1/4 of weight space, and as a consequence had to recalculate and rewrite almost every lemma of the paper.

2. The coefficients of the power series defining certain modular functions

We start with some notation. All our rings are commutative and have a 1. If R is a ring, then we define two R-module homomorphisms U and V : R[[q]] → R[[q]] thus:

U (∑_{n⩾0}r_{n}q^{n}) = ∑_{n⩾0}r_{2n}q^{n},

and

V (∑_{n⩾0}r_{n}q^{n}) = ∑_{n⩾0}r_{n}q^{2n}.

It is an easy exercise to check that V is a ring homomorphism and furthermore that U(gV (h)) = hU(g) for g, h ∈ R[[q]]. The most common application of the latter results in this paper is the fact that if h ∈ R[[q]]^{×} then V (h) is too, and U(g/V (h)) = U(g)/h.

We now assign notation to some standard modular forms and functions that will play a key role in what follows.

1It would be tedious to prove Lemma 2 by hand, but it can be done…

If k ⩾ 2 is an even integer, then the function

E_{k} := 1 + \frac{2}{(1 − 2^{k−1})ζ(1 − k)} ∑_{n⩾1} ( ∑_{0<d|n,d odd} d^{k−1}) q^{n}

(where ζ(s) is the Riemann zeta function) is a modular form of level 2 and weight k. Note that U(E_{k}) = E_{k} and that if k ⩾ 4 then E_{k} is just the 2-stabilised ordinary oldform associated to the standard level 1 Eisenstein series of weight k (note however that we have dropped an Euler factor—E_{k} is not the standard level 1 Eisenstein series).

The function

Δ(q) = q ∏_{n⩾1}(1 − q^{n})^{24} = q − 24q^{2} + 252q^{3} − 1472q^{4} + …

is a level 1 weight 12 modular form; set

f = Δ(q^{2})/Δ(q) = q + 24q^{2} + 300q^{3} + 2624q^{4} + 18126q^{5} + …,

a level 2 modular function giving an isomorphism X_{0}(2) → P^{1} (this can be verified by observing that f : X_{0}(2) → P^{1} has a simple zero at the cusp ∞ and no other zeroes). We will make heavy use of f as a parameter in sections 2–5 to measure how far various modular functions, and families of modular functions, overconverge.

Define

y = \frac{(E_{2}/V (E_{2})) − 1}{24} = q − 20q^{3} + 462q^{5} − 10696q^{7} + …,

a level 4 modular function giving an isomorphism X_{0}(4) → P^{1} (this can be seen for example by noting (Lemma 2(iii)) that f = (y + 8y^{2})/(1 − 8y)^{2} and hence y : X_{0}(4) → P^{1} must have degree 1). We will make heavy use of y as a parameter in sections 6–8 when discussing explicit formulae for entries of matrices representing the U operator on spaces of modular forms.

Finally, define

G_{4} = 1 + 240 ∑_{n⩾1}σ_{3}(n)q^{n},

the level 1 weight 4 Eisenstein series.

A general convention in this paper is that if F = aq + … ∈ R[[q]] then R[[F]] will be considered as a subring of R[[q]] in the obvious way. Of course if a is a unit in R then R[[F]] = R[[q]]; for example because y = q + … ∈ Z[[q]] we see that R[[q]] = R[[y]] for any ring R. Another fact that we use over and over again is the following power series identity.

Lemma 1. The formal binomial expansion of (1 + 2s)^{ℓ} is in 1 + 2ℓsZ_{2}[[ℓ, s]].

Proof. This follows easily from the binomial theorem and the fact that v(n!) ⩽ n − 1 for n ∈ Z_{⩾1}, where v denotes the 2-adic valuation.

When we say that a result follows “from the binomial theorem”, we typically mean that it is a consequence of this lemma.

We now collect a few facts about the modular forms above.

Lemma 2. (i) U(y) = 0 and U(y^{2}) = (y + 8y^{2})/(1 + 24y)^{2}.

(ii) For m ∈ Z_{⩾0} we have U(y^{2m+1}) = 0 and U(y^{2m}) = (\frac{y+8y^{2}}{(1+24y)^{2}} )^{m} .

(iii) f = (y + 8y^{2})/(1 − 8y)^{2}, U(f) = 24f + 2048f^{2}, V (f) = y^{2}/(1 − 64y^{2}).

(iv) G_{4}/E_{2}^{2} = (1 + 256f)/(1 + 64f) and V (G_{4})/E_{2}^{2} = (1 + 16f)/(1 + 64f).

Proof. (i) We have 24y = (E_{2}/V (E_{2})) − 1, and hence 24U(y) = (E_{2}/E_{2}) − 1 = 0. The fact that U(y^{2}) = (y + 8y^{2})/(1 + 24y)^{2} could be proved theoretically by an analysis of the zeros and poles of U(y^{2}), but nowadays a much simpler proof is just to observe that U(y^{2}) and (y + 8y^{2})/(1 + 24y)^{2} are both meromorphic functions on X_{0}(4) with at worst 4 poles (counting multiplicities) and hence they will be equal if their q-expansions agree for the first few terms; on the other hand, checking that the first 100 terms of the q-expansions agree takes well under a second on a computer.

(ii) The fact that U(y) = 0 implies that y = qF (q^{2}) for some F ∈ Z[[q]]. In particular, y is an odd function of q and hence so is y^{2m+1}, so U(y^{2m+1}) = 0. On the other hand, U(y^{2}) = U(q^{2}F (q^{2})^{2}) = qF (q)^{2}, and so U(y^{2m}) = U(q^{2m}F (q^{2})^{2m}) = q^{m}F (q)^{2m} = U(y^{2})^{m} and the result follows.

(iii), (iv) As in (i) these follow either by a brute force calculation of q-expansions, easily checked by a machine computation, or by a careful analysis of zeroes and poles analogous to Lemma 2.2 of [Eme98].

We deduce some explicit results about how U and V affect overconvergence of modular functions (c.f. Lemma 3.1 of [Eme98]).

Corollary 3. Let R be a commutative ring, and let R[[2y]] and so on denote the obvious subrings of R[[q]]. Let r be 1, 2, 4, or 8. Then

(i) R[[rf]] = R[[ry]] and rfR[[rf]] = ryR[[ry]].

(ii) V (R[[r^{2}f]]) = R[[r^{2}y^{2}]] ⊆ R[[rf]] and V (r^{2}fR[[r^{2}f]]) ⊆ rfR[[rf]].

(iii) U(R[[rf]]) ⊆ R[[r^{2}f]] and U(rfR[[rf]]) ⊆ r^{2}fR[[r^{2}f]].

Remark. It is well-known (see Proposition 1 of the appendix of [BC] and the remarks that follow) that f can be used as a “measure of supersingularity” in a neighbourhood of the ordinary locus of X_{0}(1), and part (i) of this corollary shows that y can too. Part (ii) (resp. (iii)) translates into well-known properties of V (resp. U), namely that it decreases (resp. increases) overconvergence by a factor of 2 near the ordinary locus.

Proof. (i) follows easily from Lemma 2(iii), which shows that 8f = 8y + … ∈ 8yZ_{2}[[8y]] and hence that rf = ry + … ∈ ryZ_{2}[[ry]].

(ii) V is an R-algebra homomorphism, continuous with respect to the q-adic topology, so by Lemma 2(iii) we have V (R[[r^{2}f]]) = R[[V (r^{2}f)]] = R[[r^{2}y^{2}]]. Now R[[r^{2}y^{2}]] ⊂ R[[ry]] = R[[rf]] and the last part comes from observing that if a power series G has no constant term, then neither does V (G).

(iii) Note that R[[rf]] = R[[ry]] = R[[r^{2}y^{2}]] ⊕ ryR[[r^{2}y^{2}]]. If g ∈ R[[rf]] then write g = g_{e} + g_{o} with g_{e} ∈ R[[r^{2}y^{2}]] and g_{o} ∈ ryR[[r^{2}y^{2}]]. By Lemma 2(ii) we see that U(g_{o}) = 0. By part (ii) we can write g_{e} = V (h) with h ∈ R[[r^{2}f]]. Hence U(g) = U(g_{e}) = U(V (h)) = h and everything now follows easily.

3. Lemmas on powers of E_{2}

Here we prove a few results about the formal power series (E_{2})^{ℓ} ∈ Z_{2}[[ℓ, q]]. Before we introduce this power series, let us say a little more about weight space.

We have already introduced weight space W, the open unit disc; let w be the usual parameter on this disc. We regard elements of Z_{2}[[w]] as functions on W. We are also interested in some slightly bigger rings, corresponding to smaller discs in W: by the binomial theorem we have 5^{k} = (1 + 4)^{k} = 1+4k+8k(k−1)+… ∈ Z_{2}[[k]] and furthermore that 5^{k} −1 = 2(2k+4k(k−1)+…) is in 4kZ_{2}[[2k]], and in particular in the maximal ideal of Z_{2}[[k]]. This gives us a natural inclusion Z_{2}[[w]] → Z_{2}[[k]], sending w to 5^{k} − 1, and we shall henceforth regard Z_{2}[[w]] as a subring of Z_{2}[[k]] in this way. In fact one checks easily that the map Z_{2}[[w]] → Z_{2}[[k]] factors as

Z_{2}[[w]] ⊂ Z_{2}[[w/2]] → Z_{2}[[2k]] ⊂ Z_{2}[[k]],

where the outer inclusions are the natural ones and the inner map is an isomorphism, corresponding geometrically to the isomorphism of discs {k : k ∈ K : |k| < 2} → {w : w ∈ K : |w| < 1/2} sending k to w = 5^{k} − 1. The inverse of this isomorphism is the map Z_{2}[[2k]] → Z_{2}[[w/2]] sending 2k to 2 log(1 + w)/ log(5).

We will be working with powers of E_{2} so will be mostly concerned in the next couple of sections with even weight only. We introduce a new variable ℓ defined by 2ℓ = k, and we regard Z_{2}[[k]] = Z_{2}[[2ℓ]] as a subring of Z_{2}[[ℓ]]. Note that the induced ring homomorphism Z_{2}[[w]] → Z_{2}[[ℓ]] factors as

Z_{2}[[w]] ⊂ Z_{2}[[w/8]] → Z_{2}[[ℓ]],

where here the inclusion is the natural one and the map on the right is the isomorphism sending w/8 to (5^{2ℓ} − 1)/8 = 3ℓ + … ∈ ℓZ_{2}[[ℓ]].

Our general convention will be to use italics to denote modular forms of fixed weight, and bold face to denote families of modular forms. We introduce our first family here: recall that E_{2} = 1 + 24q + … ∈ 1 + 24qZ[[q]] is the weight 2 level 2 Eisenstein series defined above. Define

T = (E_{2})^{ℓ},

that is, T is the element (E_{2})^{ℓ} of 1+8ℓqZ_{2}[[ℓ, q]] ⊆ Z_{2}[[ℓ, q]]^{×}. One constructs T explicitly using the binomial theorem. If one evaluates T at ℓ = ℓ_{0} ∈ Z_{⩾1} one gets the classical modular form (E_{2})^{ℓ_{0}} of weight 2ℓ_{0}.

Lemma 4. (i) E_{2}/V (E_{2}) ∈ 1 + 8fZ_{2}[[8f]].

(ii) T/V (T) ∈ 1 + 8ℓfZ_{2}[[ℓ, 4f]].

(iii) U(T)/T ∈ 1 + 32ℓfZ_{2}[[ℓ, 16f]].

(iv) Let σ denote the Z_{2}[[ℓ]]-algebra automorphism of Z_{2}[[ℓ, q]] sending q to −q. Then we have σ(T)/T ∈ 1 + 16ℓfZ_{2}[[ℓ, 8f]].

(v) V U(T)/T ∈ 1 + 8ℓfZ_{2}[[ℓ, 8f]].

(vi) U(T)/V U(T) ∈ 1 + 8ℓfZ_{2}[[ℓ, 8f]].

Proof. (i) We have E_{2}/V (E_{2}) = 1 + 24y, and 24y ∈ 8fZ_{2}[[8f]] by Corollary 3(i).

(ii) This follows immediately from the binomial theorem and (i).

(iii) follows by applying U to (ii), and using Corollary 3(iii), by applying U.

(iv) Note first that σ fixes Im(V), and hence σ(T)/T = [σ(T/V (T))]/[T/V (T)]. Next recall that T/V (T) = (1 + 24y)^{ℓ}. Furthermore, by Lemma 2(i) we have U(y) = 0 and hence σ(y) = −y, and so σ(T/V (T)) = (1 − 24y)^{ℓ}. It suffices therefore, to prove that ((1 − 24y)/(1 + 24y))^{ℓ} ∈ 1 + 16ℓfZ_{2}[[ℓ, 8f]] = 1 + 16ℓyZ_{2}[[ℓ, 8y]], and this follows from the binomial theorem as the left hand side is (1 − 2(24y/(1 + 24y)))^{ℓ}.

(v) A formal q-expansion calculation shows that 2V U(T) = T + σ(T), so 2V U(T)/T ∈ 2 + 16ℓfZ_{2}[[ℓ, 8f]] by (iv) and the result follows.

(vi) follows from (iii) and (v).

4. Eisenstein series of fixed weight.

In this section we analyse the growth of the weight 4t Eisenstein series E_{4t}, for t ∈ Z_{⩾1}. This analysis will be used in the next section to deduce corresponding bounds on the growth of the full Eisenstein family.

Lemma 5. (i) There exists a positive integer constant c = c(t) such that cE_{4t}/(E_{2})^{2t} ∈ c + 16fZ_{2}[[16f]].

(ii) For any such c, we also have cE_{4t}/U((E_{2})^{2t}) ∈ c + 16fZ_{2}[[16f]].

Proof. (i) Firstly recall that there exists an invertible sheaf ω^{4} of degree 1 on the modular curve X_{0}(2), such that the sections of (ω^{4})^{⊗t} are weight 4t modular forms. Recall that G_{4} = 1 + 240 ∑_{n} σ_{3}(n)q^{n} ∈ Z[[q]] is a level 1 weight 4 Eisenstein series; because the space of weight 4 level 2 modular forms is 2-dimensional, a basis for this space must be G_{4} and V (G_{4}). Because X_{0}(2) has genus 0 we deduce that every weight 4t modular form of level 2 must be expressible as a homogeneous polynomial in G_{4} and V (G_{4}) of degree t. In particular E_{k_{0}} can be written as a degree t polynomial in G_{4} and V (G_{4}). All of these forms are defined over Q, and the argument above makes sense over Q, and hence the polynomial has coefficients in Q. Choose a positive integer c such that cE_{k_{0}} is a polynomial in G_{4} and V (G_{4}) with integral coefficients. By Lemma 2(iv), both G_{4}/(E_{2})^{2} and V (G_{4})/(E_{2})^{2} are in Z_{2}[[16f]], and hence cE_{k_{0}}/(E_{2})^{2t} ∈ Z_{2}[[16f]]. Now comparing constant terms gives the result.

(ii) By specialising Lemma 4(iii) to ℓ = 2t, we see that U((E_{2})^{2t})/(E_{2})^{2t} ∈ 1 + 16fZ_{2}[[16f]] and the result follows from this and (i).

We can do better than this however; we can eliminate the constant c. Keeping the above notation, set h = U((E_{2})^{2t})/V U((E_{2})^{2t}) and define an endomorphism Ũ of Z_{2}[[q]] by Ũ(g) = U(gh).

Lemma 6. If t ∈ Z_{⩾1} then E_{4t}/U((E_{2})^{2t}) ∈ 1 + 128fZ_{2}[[64f]], and furthermore E_{4t}/U((E_{2})^{2t}) is the unique element of 1 + 8fZ_{2}[[8f]] fixed by the operator Ũ.

Proof. We consider Ũ as an endomorphism of Z_{2}[[q]] = Z_{2}[[f]]. Set N_{+} = 8fZ_{2}[[8f]] and note that (N_{+})^{2} ⊆ N_{+} and that h ∈ 1 + 2N_{+} by Lemma 4(vi) specialised to ℓ = 2t. In particular multiplication by h induces an isomorphism N_{+} → N_{+}. We see from Corollary 3(iii) that Ũ(N_{+}) = U(N_{+}) ⊆ 64fZ_{2}[[64f]] ⊂ 8N_{+}. Furthermore, Ũ(1) = U(h) ∈ 1 + 2U(N_{+}) ⊆ 1 + 128fZ_{2}[[64f]] ⊆ 1 + 16N_{+}. Now if one defines g_{1} = 1 and g_{n+1} = Ũ(g_{n}) for n ⩾ 1, then g_{2} − g_{1} ∈ 16N_{+} ⊂ 8N_{+}, and applying Ũ^{n−1} we deduce that g_{n+1} − g_{n} ∈ 8^{n}N_{+}. We deduce that g = lim_{n} g_{n} exists in 1 + 16N_{+} and satisfies Ũ(g) = g. Now g′ := cE_{4t}/U((E_{2})^{2t}) ∈ c + N_{+} (notation as in the previous lemma) is easily checked to satisfy Ũ(g′) = g′, and hence if one sets g′′ := g′ − cg ∈ N_{+} then g′′ = Ũ(g′′), and by repeatedly applying Ũ we deduce g′′ ∈ 8^{n}N_{+} for all n ⩾ 1 and hence g′′ = 0. So E_{4t}/U((E_{2})^{2t}) = g′/c = g ∈ 1 + 16N_{+}. One final application of Ũ gets us a little more: we see E_{4t}/U((E_{2})^{2t}) = g = Ũ(g) = U(gh) ∈ U(1 + 2N_{+}) = 1 + 2U(N_{+}) ⊆ 1 + 128fZ_{2}[[64f]].

Finally, if e is any element of 1 + N_{+} such that Ũ(e) = e then g − e ∈ 8^{n}N_{+} for all n ⩾ 1 and hence e = g = E_{4t}/U((E_{2})^{2t}).

5. Families of Eisenstein series

In this section we use Lemma 6 to deduce an analogous result about the 2-adic family of Eisenstein series. We firstly recall the definition of the Eisenstein family.

By [Was97] (Theorem 7.10 and the remarks following Theorem 5.11), there is a unique meromorphic function ζ_{2} on W with the property that if k ⩾ 2 is an even integer, then ζ_{2}(5^{(1−k)} − 1) = (1 − 2^{k−1})ζ(1 − k). By [Was97] (Lemma 7.12 and the remarks following Proposition 7.9), we deduce that 2/ζ_{2} ∈ Z_{2}[[w]] (our w is Washington’s T). We define a meromorphic function ζ^{∗} on W by ζ^{∗}(w) = ζ_{2}(\frac{5}{1+w} − 1), and observe that ζ^{∗}(5^{k} − 1) = (1 − 2^{k−1})ζ(1 − k) for k ⩾ 2 an even integer, and 2/ζ^{∗} ∈ Z_{2}[[w]].

If κ : Z_{2}^{×} → K^{×} is a weight, then the formal power series

E_{κ} := 1 + \frac{2}{ζ^{∗}(κ)} ∑_{n⩾1} ( ∑_{0<d|n,d odd} κ(d)/d) q^{n}

is an overconvergent 2-adic Eisenstein series of weight κ (essentially by definition: see §2.4 of [CM98]). If we fix d ∈ Z_{2}^{×} then the function on W defined by κ ↦ κ(d) is of the form w ↦ (1 + w)^{u} for some u ∈ Z_{2}, and hence is an element of Z_{2}[[w]], and one deduces from this that there is a power series E ∈ 1 + wqZ_{2}[[w, q]] such that for any κ ∈ W the specialisation of E to κ is E_{κ}.

The fact that E ∈ 1 + wqZ_{2}[[w, q]] implies that E ∈ Z_{2}[[w, q]]^{×}, and hence V (E) ∈ Z_{2}[[w, q]]^{×}. Furthermore, because y = q + … ∈ Z[[q]] we can regard E/V (E) as an element of Z_{2}[[w, y]]. Write

E/V (E) = ∑_{i,j} a_{i,j}w^{i}y^{j}

with a_{i,j} ∈ Z_{2}. The result that the previous three sections have been building up to is

Theorem 7. If j ⩾ i ⩾ 0 then 8^{j−i}|a_{i,j}.

The proof is at the end of this section.

Remark. Results of a similar nature to the proposition were obtained by Emerton in [Eme98], and anyone who is familiar with this work will clearly see the debt which we owe it. Perhaps the idea that enables us to get a little further than Emerton’s thesis is that we use E/U(T) rather than E/T for our intermediary calculations; the point is that E/U(T) overconverges further than E/T.

We remark also that Theorem 7 gives information on how far the function E_{κ}/V (E_{κ}) overconverges for a given value of κ. This information will be crucial to us later.

The key intermediate step that we need to prove the theorem is the following proposition (c.f. Proposition 3.12 of [Eme98]). Regard E as an element of Z_{2}[[ℓ, q]] = Z_{2}[[ℓ, f]], the ring where T lives.

Proposition 8. E/U(T) ∈ 1 + 64ℓfZ_{2}[[ℓ, 64f]].

Proof. Let N_{+} denote the subset 8ℓfZ_{2}[[ℓ, 8f]] of Z_{2}[[ℓ, f]], and note that (N_{+})^{2} ⊂ N_{+}. Set H = U(T)/V U(T). Then H ∈ 1 + N_{+} by Lemma 4(vi), and hence multiplication by H induces an isomorphism N_{+} → N_{+}. Define an endomorphism \overline{U} of Z_{2}[[ℓ, f]] by \overline{U}(G) := U(GH).Then

\overline{U}(N_{+}) = U(N_{+}) = ℓU(8fZ_{2}[[ℓ, 8f]]) ⊆ 64ℓfZ_{2}[[ℓ, 64f]]

by Corollary 3(iii). In particular \overline{U}(N_{+}) = U(N_{+}) ⊂ 8N_{+}. Also \overline{U}(1) ∈ U(1 + N_{+}) ⊆ 1 + 8N_{+}. This is enough to prove, via a Hensel-like argument as in Lemma 6, that there is an element G ∈ 1+8N_{+} such that \overline{U}(G) = G. Indeed, if we set G_{1} = 1 and G_{n+1} = \overline{U}(G_{n}) for n ⩾ 1 then G_{2} − G_{1} ∈ 8N_{+} and hence G_{n+1} − G_{n} ∈ 8^{n}N_{+}. So G = lim_{n} G_{n} exists in 1 + 8N_{+} and satisfies \overline{U}(G) = G. Moreover we have G = \overline{U}(G) = U(GH) ∈ U(1 + N_{+}) = 1 + U(N_{+}) ⊆ 1 + 64ℓfZ_{2}[[ℓ, 64f]]. Hence it suffices to prove that E/U(T) = G, which we do by checking that their weight k specialisations agree for infinitely many classical weights.

So now let k_{0} = 2ℓ_{0} = 4t ∈ Z_{⩾1} be a multiple of 4. Specialising to ℓ = ℓ_{0} gives an element G(ℓ_{0}) ∈ 1 + 8fZ_{2}[[8f]] fixed by the endomorphism Ũ of Lemma 6. Hence by Lemma 6 we have G(ℓ_{0}) = E_{2ℓ_{0}}/U((E_{2})^{ℓ_{0}}), the specialisation of E/U(T) to weight k_{0}, and because there are infinitely many choices for t ⩾ 1 we deduce G = E/U(T). In particular, we deduce that E/U(T) ∈ 1 + 64ℓfZ_{2}[[ℓ, 64f]].

Corollary 9. (i) E/V U(T) ∈ 1 + 8ℓfZ_{2}[[ℓ, 8f]].

(ii) V (E)/V U(T) ∈ 1 + 8ℓfZ_{2}[[ℓ, 8f]].

(iii) E/V (E) ∈ 1 + 8ℓfZ_{2}[[ℓ, 8f]].

Proof. (i) follows from the Proposition and Lemma 4(vi).

(ii) follows from the Proposition and Corollary 3(ii).

(iii) follows from (i) and (ii).

Proof of Theorem 7. We know by the previous corollary and Corollary 3(i) that

E/V (E) ∈ Z_{2}[[ℓ, 8f]] = Z_{2}[[w/8, 8f]] = Z_{2}[[w/8, 8y]].

We also know that E/V (E) ∈ Z_{2}[[w, q]] = Z_{2}[[w, y]]. Hence if we write E/V (E) = ∑ a_{i,j}w^{i}y^{j} = ∑ b_{i,j}(w/8)^{i}(8y)^{j} then a_{i,j}, b_{i,j} ∈ Z_{2}. But b_{i,j} = 8^{i−j}a_{i,j} and the result follows.

6. The Eisenstein family near the boundary of weight space.

Having established the bounds on the coefficients of the Eisenstein family we require, we no longer need the family T or the uniformiser f, and now we concentrate more on the family E and the uniformiser y. We also return to our complete extension K of Q_{2}. Let O denote the integers of K and let F denote the residue field. As before, write E/V (E) = ∑_{i,j} a_{i,j}w^{i}y^{j}. Now specialise to some weight w_{0} ∈ O satisfying 1/8 < |w_{0}| < 1, and let κ denote the corresponding character. We deduce that E_{κ}/V (E_{κ}) ∈ O[[w_{0}y]]. Write E_{κ}/V (E_{κ}) = g_{κ}(w_{0}y) with g_{κ} ∈ O[[X]]. Let \overline{g}_{κ} ∈ F[[X]] denote the reduction of g_{κ} modulo the maximal ideal of O.

Lemma 10. (i) We have \overline{g}_{κ} = ∑_{i⩾0} X^{2^{i}−1}. In particular, \overline{g}_{κ} is independent of κ (for κ corresponding to w_{0} ∈ W with 1/8 < |w_{0}| < 1).

(ii) If n ∈ Z_{⩾0} then a_{n,n} ∈ Z_{2}^{×} iff n + 1 is a power of 2.

Proof. Fix κ and say g_{κ} = ∑ c_{n}X^{n}, with c_{n} = c_{n}(κ) ∈ O. Specialising E/V (E) = ∑_{i,j} a_{i,j}w^{i}y^{j} to weight w_{0} we see that we have c_{j}w_{0}^{j} = ∑_{i} a_{i,j}w_{0}^{i} and hence

c_{j} = ∑_{i} a_{i,j}w_{0}^{i−j}.

Now a_{i,j}w_{0}^{i−j} is in the maximal ideal of O if j > i, by Theorem 7 and the fact that |w_{0}| > 1/8. Also a_{i,j}w^{i−j} is in the maximal ideal of O if j < i, because a_{i,j} ∈ Z_{2}. Hence

\overline{c}_{n} = \overline{a}_{n,n} ∈ F

where here of course the bars denote reduction modulo the maximal ideal of O. In particular, \overline{c}_{n} is independent of the choice of κ, so \overline{g}_{κ} is independent of κ. Moreover, this shows that parts (i) and (ii) of the lemma are equivalent. To finish the proof of the lemma, we only have to verify that \overline{g}_{κ} = ∑_{i} X^{2^{i}−1} for one particular choice of κ. We make the choice κ = κ_{0} corresponding to w_{0} = 4, that is to modular forms of level 4, weight 1 and non-trivial character. The corresponding Eisenstein series is

E_{κ_{0}} = ∑_{a,b∈Z} q^{a^{2}+b^{2}},

and the corresponding ratio

f_{0} := E_{κ_{0}}/V (E_{κ_{0}}) = 1 + 4q − 16q^{3} + 56q^{5} − …

is a function on X_{0}(8) which can be checked to satisfy 4yf_{0}^{2}−(1+8y)f_{0}+(1+8y) = 0. We consider f_{0} as an element of Z_{2}[[y]]; then this last equation is an identity in Z_{2}[[y]], and setting X = 4y = w_{0}y, we deduce that the equation

Xg_{κ_{0}}(X)^{2} − (1 + 2X)g_{κ_{0}}(X) + 1 + 2X = 0

must be an identity in O[[X]]. Reducing modulo the maximal ideal of O we deduce

X\overline{g}_{κ_{0}}(X)^{2} + \overline{g}_{κ_{0}}(X) + 1 = 0

in F[[X]]. This quadratic equation is easily seen to have only one root in F[[X]], namely ∑_{i⩾0} X^{2^{i}−1}, which is what we needed to prove.

7. The U-operator near the boundary of weight space.

In this section we prove some results about the characteristic power series of U acting on overconvergent forms of weight κ, where κ corresponds to a point w_{0} in weight space with 1/8 < |w_{0}| < 1.

We briefly recall the definition of the spaces of overconvergent modular forms that interest us. It is well-known (see Proposition 2.2.7 of [CM98]) that V (E_{κ}) is an overconvergent modular form of weight κ. One checks easily that if c ∈ K with 1 > |c| > 1/8 then the region of the K-rigid space X_{0}(4) defined by |cy| ⩽ 1 is isomorphic to the region of X_{0}(2) defined by |cf| ⩽ 1 and hence powers of cy can be thought of as a Banach basis of a 2-adic Banach space M_{0} of weight 0 overconvergent modular forms (this space depends on the choice of c, but we shall suppress the choice of c from the notation). For c sufficiently close to 1, the space V (E_{κ})M_{0} of overconvergent weight κ modular forms will be closed under the action of the standard Hecke operators, and the operator U will be compact. This space has a Banach basis {V (E_{κ})(cy)^{n} : n = 0, 1, 2, …} and we shall analyse the characteristic power series of U on this space by proving facts about the matrix representing U with respect to this basis. So define m_{i,j} ∈ K, i, j ⩾ 0 by

U(V (E_{κ})(cy)^{j}) = V (E_{κ}) ∑_{i} m_{i,j}(cy)^{i}.

This identity can be thought of as taking place in K[[q]] or in K[[y]]. We analyse this matrix using the techniques of [Smi00] and [Kil02]. Recall that g_{κ} ∈ O[[X]] is defined by g_{κ}(w_{0}y) = E_{κ}/V (E_{κ}).

Lemma 11. The formal sum

∑_{i,j⩾0} m_{i,j}X^{i}Y^{j}

is equal to

\frac{g_{κ}(\frac{w_{0}}{c} X)(1 + \frac{24}{c}X)^{2}}{(1 + \frac{24}{c}X)^{2} − Y^{2}(cX + 8X^{2})}.

Proof. We have

∑_{i} m_{i,j}(cy)^{i} = (E_{κ}/V (E_{κ}))U((cy)^{j}).

Recall from Lemma 3 that U(y^{j}) = 0 if j is odd, and hence m_{i,j} = 0 if j is odd. For j = 2t even we have U(y^{j}) = ((y + 8y^{2})/(1 + 24y)^{2})^{t} and hence

∑_{i} m_{i,j}(cy)^{i} = g_{κ}(w_{0}y)(c^{2}(y + 8y^{2})/(1 + 24y)^{2})^{t}.

This is an identity in K[[y]] so substituting X for cy gives, for j = 2t,

∑_{i} m_{i,j}X^{i} = g_{κ}(w_{0}X/c)((cX + 8X^{2})/(1 + 24X/c)^{2})^{t}.

Multiplying by Y^{j} and summing over j gives

∑_{i,j} m_{i,j}X^{i}Y^{j} = g_{κ}(w_{0}X/c) ∑_{t⩾0}((cX + 8X^{2})Y^{2})/(1 + 24X/c)^{2})^{t}

and summing the geometric progression on the right hand side gives the result.

Recall that the m_{i,j} are the matrix coefficients of U acting on weight κ_{0} overconvergent modular forms, with κ_{0} corresponding to w_{0} ∈ W. It is well-known that the matrix (m_{i,j}) is compact for |c| < 1 sufficiently close to 1, and in fact if |c| > |w_{0}| then this can now be read off from the above lemma, using the fact that the coefficients of g_{κ} are integral, and w_{0}/c, 8/c and c all have norm less than 1.

8. The characteristic power series of U near the boundary of weight space.

Fix w_{0} with 1/8 < |w_{0}| < 1 as in the previous section, and let κ be the corresponding weight. We wish to analyse the characteristic power series of the matrix (m_{i,j})_{i,j⩾0} defined in the previous section, and we do this via Lemma 11 and the following three elementary but messy lemmas.

Fix s a positive integer, and 0 ≠ d ∈ O. let N = (n_{i,j})_{0⩽i,j⩽2s−1} be a 2s by 2s matrix with the property that n_{i,j} ∈ d^{j}O for all 0 ⩽ i, j ⩽ 2s − 1 (note that we index our matrices starting from (0, 0)). Assume that n_{i,j} = 0 if j is odd. Let P (T) = det(1 − T N) = 1 + … = ∑_{α⩾0} a_{α}T^{α} ∈ O[T] denote the “characteristic power series” (although of course it is a polynomial) of N. For 0 ⩽ β ⩽ s let T_{β} denote the β by β matrix whose i, jth entry, 0 ⩽ i, j < β, is n_{2i,2j}/d^{2j} ∈ O.

Lemma 12. We have a_{α}/d^{α(α−1)} ∈ O, and furthermore for α ⩽ s we have a_{α}/d^{α(α−1)} ∈ O^{×} iff det(T_{α}) ∈ O^{×}.

Proof. By definition of det(1 − T N), we see that (−1)^{α}a_{α} is the sum over S of d_{S}, where S ranges over the subsets of {0, 1, 2, …, 2s − 1} of size α, and

d_{S} = ∑_{σ:S→S} sgn(σ) ∏_{s∈S} n_{s,σ(s)}.

Note first that d_{S} = 0 if S contains any odd element of {0, 1, 2, …, 2s−1} and in particular a_{α} = 0 if α > s. If S contains only even elements then we see that d^{∑_{s∈S} s} divides d_{S} and ∑_{s∈S} s ⩾ α(α − 1), with equality iff S = S_{0} := {0, 2, 4, …, 2α−2}. Hence a_{α} is a sum of multiples of d^{α(α−1)}, all but one of which are multiples of d^{α(α−1)+1}. We deduce a_{α}/d^{α(α−1)} ∈ O and furthermore a_{α}/d^{α(α−1)} ∈ O^{×} iff d_{S_{0}}/d^{α(α−1)} ∈ O^{×}. But it is an easy consequence of the definitions that d_{S_{0}}/d^{α(α−1)} = det(T_{α}).

Our second lemma shows how to get a handle on the matrix T_{α} above.

Lemma 13. Define s_{i,j} ∈ F_{2}, 0 ⩽ i, j < ∞, by

∑_{0⩽i,j} s_{i,j}X^{i}Y^{j} = \frac{∑_{m⩾0} X^{2^{m}−1}}{1 + XY^{2}} ,

the equality taking place in F_{2}[[X, Y]]. Define t_{i,j} = s_{2i,2j} for 0 ⩽ i, j. Then

∑_{0⩽i,j} t_{i,j}X^{i}Y^{j} = \frac{1 + XY ∑_{m⩾0} X^{2^{m}−1}}{1 + XY^{2}} .

Proof. Define power series A(X, Y), B(X, Y) in F_{2}[[X, Y]] by

A(X, Y) = (∑_{m⩾0} X^{2^{m}−1})/(1 + XY^{2})

and

B(X, Y) = (1 + XY ∑_{m⩾0} X^{2^{m}−1})/(1 + XY^{2}).

Note that A is a function of X and Y^{2} and so the lemma will follow if one can show that X(A(X, Y)− B(X^{2}, Y^{2})) is in F_{2}[[X^{2}, Y^{2}]], and this follows from an explicit calculation; we see that X(A(X, Y)− B(X^{2}, Y^{2})) = (X^{2}Y^{2} + ∑_{m⩾0}(X^{2})^{2^{m}})/(1 + X^{2}Y^{4}).

Our final lemma will be used later on to verify an instance of the condition det(T_{α}) ∈ O^{×} appearing in lemma 12. The proof uses an idea explained to us by Robin Chapman.

Lemma 14. Fix an integer α ⩾ 0, and let \overline{T}_{α} be the α by α matrix (\overline{t}_{i,j})_{0⩽i,j<α} with entries in F_{2} defined via the following identity:

∑_{i,j} \overline{t}_{i,j}X^{i}Y^{j} = \frac{1 + XY (∑_{m⩾0} X^{2^{m}−1})}{1 + XY^{2}},

the equality taking place in F_{2}[X, Y]/(X^{α}, Y^{α}). Then det(\overline{T}_{α}) = 1.

Proof. Write ∑_{i,j} \overline{t}_{i,j}X^{i}Y^{j} = ∑_{j} f_{j}(X)Y^{j}, with f_{j}(X) ∈ V := F_{2}[X]/(X^{α}). It suffices to prove that the f_{j}(X), 0 ⩽ j < α, span V as an F_{2}-vector space. Write r = ∑_{m⩾0} X^{2^{m}} ∈ V. Then

∑_{j} f_{j}(X)Y^{j} = (1 + Y r)(1 + XY^{2} + X^{2}Y^{4} + X^{3}Y^{6} + …)

and one deduces, by comparing coefficients of powers of Y, that f_{2t}(X) = X^{t} and f_{2t+1}(X) = X^{t}r. Because r^{2} − r = X we see that f_{2t}(X) = (r^{2} − r)^{t} and that f_{2t+1}(X) = r(r^{2} − r)^{t}, and hence that as polynomials in r we have deg(f_{n}) = n. Hence the span of the f_{j} contains the image of F_{2}[r] in V. This is enough as r = X + … so this image is F_{2}[X].

We are now ready to prove the main result of this section. As usual let κ be a weight such that the corresponding w_{0} satisfies 1/8 < |w_{0}| < 1, and let (m_{i,j}) be the matrix representing U in weight κ as before.

Proposition 15. If P_{κ}(T) = ∑_{α⩾0} b_{α}T^{α} denotes the characteristic power series of U in weight κ, then |b_{α}| = |w_{0}|^{α(α−1)/2}.

Proof. If β ⩾ 0 and M_{β} denotes the truncated matrix (m_{i,j})_{0⩽i,j<β}, and if P_{β}(T) = det(1 − T M_{β}) is the characteristic power series of M_{β}, then the P_{β}(T) tend to P_{κ}(T), in the sense that if P_{β}(T) = ∑_{α} b_{α,β}T^{α} then lim_{β→∞} b_{α,β} = b_{α}. Hence it suffices to prove that |b_{α,β}| = |w_{0}|^{α(α−1)/2} for β > 2α and furthermore we may assume β = 2s − 1 is odd. Let N_{β} be the matrix with elements (n_{i,j})_{0⩽i,j<β} where n_{i,j} = m_{i,j}(c/w_{0})^{i−j}. Then N_{β} is easily checked to be a conjugate of M_{β}, so P_{β}(T) = det(1 − T N_{β}). Furthermore one easily checks that Lemma 11 implies (substituting X for w_{0}/cX and Y for c/w_{0}Y)

F (X, Y) := ∑_{0⩽i,j<β} n_{i,j}X^{i}Y^{j} = \frac{g_{κ}(X)(1 + 24X/w_{0})^{2}}{(1 + 24X/w_{0})^{2} − Y^{2}(w_{0}X + 8X^{2})}.

Note that F (X, Y) ∈ O[X, Y]/(X^{β}, Y^{β}). Extending K if necessary, we may assume that there exists d ∈ K with d^{2} = w_{0}. The fact that G(X, Y) := F (X, Y/d) satisfies

G(X, Y) = \frac{g_{κ}(X)(1 + 24X/w_{0})^{2}}{(1 + 24X/w_{0})^{2} − Y^{2}(X + 8X^{2}/w_{0})} ∈ O[X, Y]/(X^{β}, Y^{β})

shows us that n_{i,j}/d^{j} ∈ O for all i, j, and the fact that F (X, Y) is a function of X and Y^{2} implies that n_{i,j} = 0 if j is odd. We are hence in a position to apply Lemma 12 to deduce that |b_{α,β}| ⩽ |d|^{α(α−1)} = |w_{0}|^{α(α−1)/2}, with equality iff the matrix (n_{2i,2j}/d^{2j})_{0⩽i,j<α} has determinant a unit. Let T_{α} denote this matrix, and let \overline{T}_{α} denote its reduction modulo the maximal ideal of O.

Reducing G(X, Y) modulo the maximal ideal of O, it becomes

\overline{G}(X, Y) = \frac{\overline{g}_{κ}(X)}{1 + XY^{2}} ∈ F[X, Y]/(X^{β}, Y^{β})

and by Lemma 10 and Lemma 13 we deduce that \overline{T}_{α} = (\overline{t}_{i,j})_{0⩽i,j<α} with

∑_{0⩽i,j<α} \overline{t}_{i,j} = \frac{1 + XY ∑_{m} X^{2^{m}−1}}{1 + XY^{2}} ,

the equality taking place in F [[X, Y]]/(X^{α}, Y^{α}). Now Lemma 14 implies that det(\overline{T}_{α}) = 1 and hence det(T_{α}) ∈ O^{×}. This is what we wanted.

We now have enough to prove Theorem B.

Theorem B. If κ is a weight corresponding to w_{0} ∈ W with 1/8 < |w_{0}| < 1, and if v = v(w_{0}), then the slopes of U acting on overconvergent modular forms of weight κ are the arithmetic progression 0, v, 2v, 3v, 4v, …, each appearing with multiplicity 1.

Proof. This follows immediately from the previous Proposition and the theory of the Newton polygon.

9. The eigencurve near the boundary of weight space.

Theorem B is already enough to deduce the results about classical forms in the Corollary of section 1, so the reader that is only interested in these does not need to read this part of the paper. In this final section we deduce Theorem A from Theorem B.

Let A be a reduced affinoid algebra over K, and equip A with the supremum norm. Let P (T) = ∑_{n⩾0} a_{n}T^{n} be a power series with the property that |a_{n}|ρ^{n} tends to zero for all ρ ∈ R_{>0}. In particular P (a) converges for all a ∈ A. Assume furthermore that a_{0} = 1, and that for all x ∈ Sp(A) we have 1 = |a_{1}(x)| > |a_{2}(x)| > |a_{3}(x)| > . . ..

Lemma 16. One can factorise P (X) as

P (T) = (1 − vT)Q(T)

with v ∈ A^{×}, |v| = |v^{−1}| = 1, and Q(T) = ∑_{n⩾0} b_{n}T^{n}, b_{0} = 1, and |b_{n}(x)| = |a_{n+1}(x)| for all n ⩾ 0 and all x ∈ Sp(A).

Proof. After replacing K with a finite extension if necessary, we may choose ρ ∈ K with |a_{2}| ⩽ |ρ| < 1. Set u_{0} = −1/a_{1} and for n ⩾ 0 define u_{n+1} = u_{n} − P (u_{n})/P′(u_{n}). One proves by induction on n, using the fact that if u, h ∈ A and |u|, |h| ⩽ 1 then P (u + h) = P (u) + hP′(u) + h^{2}a for some a ∈ A with |a| ⩽ 1, that |P (u_{n})| ⩽ |ρ|^{2^{n}}, |P′(u_{n})| = 1, and |u_{n+1} − u_{n}| ⩽ |ρ|^{2^{n}}. Let u be the limit of the u_{n}; then P (u) = 0, and the fact that |u + 1/a_{1}| < 1 shows that |u(x)| = 1 for all x ∈ Sp(A). Set v = 1/u and define P (T)(1 + vT + v^{2}T^{2} + …) =: Q(T) =: ∑ b_{n}T^{n}. The fact that P (u) = 0 implies that |1 + a_{1}u(x) + a_{2}u^{2}(x) + … + a_{n}u^{n}(x)| = |a_{n+1}(x)| for all x ∈ Sp(A) and one easily deduces from this that |b_{n}(x)| = |a_{n+1}(x)| for all n ⩾ 0 and for all x ∈ Sp(A).

Corollary 17. If X = Sp(A) is an affinoid subdomain of W_{1}, with β : X → W_{1} the inclusion, and P_{X} (T) ∈ A[[T]] denotes the characteristic power series of U on weight X overconvergent modular forms, then P (T) = ∏_{n⩾0}(1 − λ_{n}T) with λ_{n} ∈ A and v(λ_{n}(x)) = nv(β(x)).

Proof. We repeatedly apply the lemma and then substitute P (T) for Q(T/W), where W ∈ A is the pullback of the parameter w on W_{1}. Proposition 15, and the fact that taking characteristic power series commutes with specialisation (see for example Corollary 2.9 of [Buz]) ensures that the conditions of the lemma are always satisfied.

Proof of Theorem A. The zero locus of P_{X} (T) is, by the previous corollary, a disjoint union of copies of X and letting X increase shows that the spectral curve corresponding to U is isomorphic to a disjoint union of annuli over weight space and that their valuations are as predicted in Theorem A. It remains to show that the eigencurve is isomorphic to the spectral curve over W_{1}, but this follows from the fact that the fibres of the spectral curve are reduced and hence that any overconvergent eigenform of weight in W_{1} is determined by its weight and its U_{2}-eigenvalue (see Proposition 6.3.2 of [CM98]).

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Kevin Buzzard Department of Mathematics , Imperial College , South Kensington , London , SW7 2AZ , United Kingdom

L. J. P. Kilford Department of Mathematics 253-37, California Institute of Technology, Pasadena, CA 91125 , United States of America