Bounding slopes of p-adic modular forms

Lawren Smithline

August 28, 2000

Abstract

Let p be a prime number and let N be an integer relatively prime to p. We investigate the Newton polygons of the Atkin U_{p} operator acting on p-adic tame level N modular forms of different weights for evidence in support of a conjecture of Gouvêa and Mazur. We refine results of Wan which support the conjecture. We establish more coincidence between Newton polygons of U_{p} for weights which are p-adically close and congruent modulo p − 1.

1 Introduction

1.1 Overview

Fix a prime p. Let K be a finite extension of the p-adic rationals, Q_{p}. Let C_{p} be the completion of the algebraic closure of Q_{p}. Let v_{p} denote the p-adic valuation, normalized so that v_{p}(p) = 1. Let O_{p} be the ring of integers of C_{p}, which is the set of elements of nonnegative valuation. Let B = K ∩ O_{p}. A tame level is an integer relatively prime to p.

For p ≥ 5, let E_{p−1} be the weight p − 1 Eisenstein series, which is a lift of the Hasse invariant on X_{0}(1)/F_{p} to characteristic zero. For p = 2 or 3, we use E_{4} or E_{6} as a lift of a power of the Hasse invariant.

Katz defines the space of overconvergent modular forms of weight k, tame level N over ring B, with growth condition r, M_{k}(N, B, r). The domain of f ∈ M_{k}(N, B, r) is a part of X_{1}(N) where v_{p}(E_{p−1}) < v_{p}(r).

For any r ∈ O_{p} of positive valuation, U_{p} is a compact operator on the Banach space M_{k}(N, B, r) preserving the subspace of cusp forms, S_{k}(N, B, r). The characteristic series of U_{p} restricted to S_{k}(N, B, r) is

det (1 − tU_{p}|_{S_{k}(N,B,r)}) = 1 + ∑_{n=1}^{∞} t^{n} tr (⋀^{n} U_{p}|_{S_{k}(N,B,r)}) .

Let M_{k}(N, B) = ⋃_{v_{p}(r)>0} M_{k}(N, B, r). The Atkin U_{p} operator acts on M_{k}(N, B) and preserves the subspace of cusp forms, S_{k}(N, B). Coleman and Mazur[CM] show the eigenforms of U_{p} are parametrized by a curve they call the Eigencurve.

Let P_{k}(t) be the characteristic series of U_{p} restricted to S_{k}(N, B). The Newton polygon of a power series P (t) = ∑_{n=1}^{∞} a_{n}t^{n} is the lower convex hull of the points {(n, v_{p}(a_{n}))}.

The slopes of the Newton polygon of P_{k}(t) are the slopes of the line segments composing the polygon. These slopes are also called slopes of U_{p} or slopes of overconvergent eigenforms. The multiplicity of a slope is the length of the segment projected to the first coordinate axis. Coleman[C] shows P_{k}(t) is not a polynomial, so this polygon continues indefinitely.

Wan shows for p ≥ 5, there is a parabola lying below the Newton polygon for U_{p} in all integral weights and levels. For tame level 1, and 12 | p − 1. Wan shows that there is a parabola with quadratic coefficient 6/(p+1) which lies below the Newton polygon.

Buzzard[B] extends Wan’s result to all primes p and congruence subgroups of SL_{2}(Z) of level prime to p. Let N be prime to p. Let S_{k}(Γ_{1}(N)) be the space of classical weight k cusp forms for the group Γ_{1}(N). Let

s_{N} = lim_{k→∞} k^{−1} dim S_{k}(Γ_{1}(N)),

which may be expressed in terms of the genus of X_{1}(N) and its ramification over the jline. The Newton polygon of the U_{p} operator on level N p-adic overconvergent modular forms lies above a parabola with quadratic coefficient

\frac{1}{2(p + 1)s_{N}}

in every integer weight.

We show this quadratic term is sharp for p = 3, N = 1 in the following sense.

Theorem 1.1.1 Let P (t) = 1 + ∑_{i=1}^{∞} a_{i}t^{i} be the characteristic series for weight zero tame level 1 U_{3}. The Newton polygon of P lies on or above the parabola

3\binom{i}{2} + 2i

and meets it infinitely often.

The methods of proof are explicit. The computational setup provides a basis for many numerical experiments, particularly for primes p in {2, 3, 5, 7, 13}, for which X_{0}(p) has genus zero.

In the rest of section 1, we define p-adic overconvergent modular forms and introduce the Gouvêa-Mazur conjecture[GM]. We state a result of [C] useful for relating U_{p} operators across different weights. We review the progress of Wan[W].

In section 2, we present the main technical theorem for extending the results discussed in section 1. We apply [Se] to [E] to extend Wan’s result to the primes 2 and 3, and tame level 1.

In section 3, we refine the main result of [W]. The improvement applies to all primes and levels in light of [B].

In section 4, we exhibit specific behavior of the Newton polygon for the U_{3} operator acting on overconvergent 3-adic modular forms of integral weight. In weight zero, the Newton polygon lies between two parabolas and coincides with the one below infinitely often. The methods apply to p-adic modular forms provided X_{0}(p) has genus 0.

1.2 Overconvergent modular forms

We outline Katz’s construction of p-adic overconvergent modular forms, as described in [W]. The construction shows that overconvergent modular forms include classical modular forms.

Let prime p ≥ 5, so that E_{p−1} exists.

Let B be the ring of integers of an extension of Q_{p}. Let N be a positive integer prime to p. Let k be an even integer. Let M_{k}(N, B) be the space of classical weight k Γ_{1}(N) modular forms whose q-expansions at the cusp \underline{∞} have coefficients in B. (Throughout this paper, all q-expansions are at \underline{∞}.) There is a free B submodule W_{k+p−1}(N, B) such that

M_{k+p−1}(N, B) = E_{p−1} · M_{k}(N, B) ⊕ W_{k+p−1}(N, B).

We define an r-overconvergent modular form of classical weight k as the “Katz” expansion

f = b_{0} + ∑_{j=1}^{∞} \frac{r^{j}b_{j}}{E_{p−1}^{j}} ,

where b_{0} ∈ M_{k}(N, B), b_{j} ∈ W_{k+j(p−1)}(N, B), and the supremum of the q expansion coefficients of b_{j} converges to zero as j increases to ∞. Let M_{k}(N, B, r) be the space of these r-overconvergent modular forms.

For p = 2, 3, notice that E_{4} is a lift of a power of the Hasse invariant to characteristic zero. A Katz expansion in these cases is a series

f = b_{0} + ∑_{j=1}^{∞} \frac{r^{4j/(p−1)}b_{j}}{E_{4}^{j}} .

1.3 Gouvêa-Mazur conjectures on p-adic families

We define the U_{p} operator. The curve X(p; N) has a proper model X over Z_{p}. Its reduction X̄ to F_{p} has points corresponding to supersingular elliptic curves. Let ϕ̄ : X̄ → X̄ be the Frobenius map. The Deligne-Tate map is a particular lifting of ϕ̄ to ϕ : X → X. Let ϕ^{∗} be the pullback of this map to functions and differentials on X. The equation

pU_{p} = tr ϕ^{∗}

defines U_{p}.

Let R be a ring. On the q-expansions R[[q]],

V : ∑ a_{n}q^{n} ↦ ∑ a_{n}q^{pn},

which agrees with ϕ^{∗} in its domain of definition, and

U_{p} : ∑ a_{n}q^{n} ↦ ∑ a_{pn}q^{n}.

The U_{p} eigenvalue of an overconvergent eigenform is the coefficient of q^{p} of the q-expansion around the cusp ∞. U_{p} is a compact operator, so it has a spectral theory. The roots of its characteristic series are reciprocals of its nonzero eigenvalues.

Every member of the Eisenstein family has U_{p} eigenvalue 1, and hence slope zero. Let E_{k}^{∗} be the p-adic weight k Eisenstein series. Let G_{k}^{∗} = ζ^{∗}(k)E_{k}^{∗}/2, which is the multiple of E_{k}^{∗} with q coefficient 1.

Gouvêa and Mazur[GM] review that for even integers k, k′, k ≡ k′ mod (p − 1)p^{n}, and (p − 1) not dividing k, the congruence

G_{k}^{∗} ≡ G_{k′}^{∗} mod p^{n+1}

holds term by term for the q-expansions.

Hida[H] shows that any slope zero eigenform lies in a p-adic continuous family of eigenforms. Gouvêa and Mazur[GM] conjecturally extend Hida’s observation. For prime p, tame level N, weight k and slope α, let S(k)^{α} be the space spanned by slope α p-adic tame level N eigenforms of weight k.

Conjecture 1.3.1 (Gouvêa-Mazur) Let k_{1}, k_{2}, n be integers such that k_{1} ≡ k_{2} modulo p^{n}(p − 1) and n ≥ α; then

dim S(k_{1})^{α} = dim S(k_{2})^{α}.

Conjecture 1.3.2 If dim S(k)^{α} = 1, then there is a continuous family f(κ) such that f(κ) spans S(κ)^{α}, and for integer ν ≥ α, integer m, f(k + mp^{ν}(p − 1)) spans S(k + mp^{ν}(p − 1)).

1.4 Prior art

Coleman introduces the change-of-weight similarity transformation. Let U_{k_{1}} and U_{k_{2}} be the U_{p} operator realized on modular forms of weights k_{1} and k_{2}. Let G be a modular form of weight k_{1} − k_{2} whose q expansion evaluates to a unit at q = 0.

Theorem 1.4.1 (Coleman) U_{k_{1}} is conjugate to U_{k_{2}}G/V (G).

Proof. Let f be a weight k_{1} eigenform. Then f/G is a weight k_{2} eigenform for the operator (1/G)U_{k_{1}}G. U is Frobenius-linear, so (1/G)U_{k_{1}}G = U_{k_{2}}(G/V (G)). □

Wan[W] shows for primes p ≥ 5, the Newton polygon for U_{p} lies above a parabola. We reexpress Wan’s conclusion below in Theorem 1.4.2. Let f be a function of a real variable. Let GM(f) be the proposition

(κ_{1} ≡ κ_{2} mod p^{f(α)}(p − 1)) ⇒ (dim S(κ_{1})^{α} = dim S(κ_{2})^{α}) .  (1)

Conjecture 1.3.1 is GM(⌈α⌉).

Theorem 1.4.2 (Wan) For p ≥ 5, tame level N, and integral weight, there is a quadratic polynomial f independent of the weight such that GM(f) holds.

The highest term of Wan’s quadratic polynomial has coefficient which is the reciprocal of the quadratic term of the parabola bounding the Newton polygon of U_{p}. Wan proves these results by examining U_{p} as it acts on a basis of modular forms derived from the Katz expansion.

Buzzard[B] extends Wan’s result.

Theorem 1.4.3 (Buzzard) The Newton Polygon of U_{p} acting on the space of classical modular forms of weight k and level Np is bounded below by an explicit quadratic lower bound independent of k.

Theorem 1.4.1 shows that Theorem 1.4.3 applies to overconvergent modular forms. The coefficient of the quadratic term found by Buzzard agrees with that found by Wan. Buzzard’s group cohomology approach avoids exceptions for p = 2, 3.

2 The technical part

2.1 Compact operators on p-adic Banach spaces

Wan’s analysis of slopes of U operators in different weights bounds the Newton polygon for the difference of two characteristic series below. He uses a linear bound. We show how to find a finer lower bound.

We name some parts of matrices. Let M be a matrix, possibly infintite rank. A n×n diagonal major of M is formed by choosing n distinct natural numbers and forming the n × n submatrix using those n rows and columns. A minor is the determinant of a major. A selection of a square matrix is a sequence of entries chosen so that there is exactly one from each row and one from each column.

The upper n × n diagonal major is the one formed by choosing the first n rows and columns.

Theorem 2.1.1 Let X, Y be compact operators on a p-adic Banach space B. Let Z = X − Y. Let P_{X}, P_{Y} be the characteristic series of X, Y. Let P_{X}(t) = ∑_{i=0}^{∞} a_{i}t^{i} and P_{Y} (t) = ∑ b_{i}t^{i}. Let Q = P_{X} − P_{Y}.

Let {e_{i}}_{i=1}^{∞} be a basis for B such that X = DM_{X} where D is the diagonal matrix diag(p^{α_{i}}) and the sequence {α_{i}} is a nonnegative nondecreasing and M_{X} preserves the O_{p}-lattice L generated by {e_{i}}..

Let Z = D′M_{Z} where M_{Z} preserves L and D′ = diag(p^{α_{i}+β_{i}}), {β_{i}} is a sequence of nonnegative real numbers. Let β = inf{β_{i}}.

Let f(n) be the Newton polygon of D. Then:

1. the Newton polygon of P_{X} lies on or above f(n);

2. the Newton polygon of P_{Y} lies on or above f(n);

3. the Newton polygon of Q lies on or above f(n) + β;

4. if v_{p}(a_{n}) < f(n) + β, then v_{p}(a_{n}) = v_{p}(b_{n}).

Proof. The first conclusion follows because any n × n diagonal minor of X has valuation at least f(n).

The second follows because β ≥ 0. Thus,

D′M_{Z} = DM′_{Z},

where M′_{Z} also preserves L. The equalities

Y = DM_{X} + DM′_{Z} = D(M_{X} + M′_{Z}),

show we may apply part 1 to Y.

Part three works because

(⋀^{n} X) − (⋀^{n} Y ) = ∑_{i=0}^{n−1} ( ⋀^{n−i−1} X) ∧ Z ∧ (⋀^{i} Y ) .

The trace of this difference is the coefficient (a_{n} − b_{n}) of t^{n} in Q(t). Reading the right side of the above equation, we see the trace is formed as the signed sum of products of selections of a variant on n×n diagonal majors, where n−i−1 rows are taken from X, one row from Z and i rows from Y. The valuation of each product is at least f(n) + β.

Part four restates the ultrametric property v_{p}(a_{n} − b_{n}) > v_{p}(a_{n}) implies v_{p}(a_{n}) = v_{p}(b_{n}). □

Corollary 2.1.1.1 With the hypotheses above, the sides of the Newton polygon of P_{X} which lie below the Newton polygon of Q coincide with sides of the Newton polygon of P_{Y}. The vertices of the Newton polygon of P_{X} below the Newton polygon of Q coincide with vertices of the Newton polygon of P_{Y}.

2.2 Uniformizers on genus zero X_{0}(p)

When X_{0}(p) has genus 0, it has a uniformizer d_{p} with simple zero at the cusp \underline{∞}, pole at the cusp \underline{0}, and leading q expansion coefficient 1. Let π : X_{0}(p) → X_{0}(1) be the map which ignores level p structure. Let ĵ = π^{∗}(j). Let Δ be the level 1 weight 12 cusp form. The map π : X_{0}(p) → X_{0}(1) is ramified above j = 0, 1728, ∞ only.

Shimura[Sh] shows

d_{p} = ^{p−1} \sqrt{V (Δ)/Δ},

and has q expansion in Z[[q]].

Proposition 2.2.1 There is a degree p + 1 polynomial H_{p} over Z with constant term 1 such that

d_{p}ĵ = H_{p}(d_{p}).

Proof. The map π has degree p + 1. The product d_{p}ĵ has a pole only at the cusp \underline{0}. Hence, there is a polynomial H_{p} satisfying the proposition.

H_{p} has integer coefficients, because d_{p}ĵ ∈ Z[[q]] and has constant term 1. □

The ramification degrees of π over j = 0 is 1 or 3, yielding roots of multiplicity 1 or 3 of H_{p}(d_{p}). Points over j = 1728 are roots of multiplicity 1 or 2 of H_{p}(d_{p}) − 1728d_{p}. We may calculate H_{p} by equating q-expansions.

For p = 2, 3, 5, X_{0}(p^{2}) also has genus zero, and by similar reasoning, d_{p^{2}} = (V (d_{p})d_{p})^{\frac{1}{p+1}} is a uniformizer with simple zero at the cusp ∞. The ramification of X_{0}(9) over X_{0}(3) shows that d_{3} is a polynomial in d_{9}; we compute

d_{3} = 27d_{9}^{3} + 9d_{9}^{2} + d_{9}.

In each case p = 2, 3, 5, we find d_{p}/d_{p^{2}} is a polynomial of degree p − 1 in d_{p^{2}} with Newton polygon a single segment of slope 12/(p^{2} − 1).

2.3 Computing tame level 1 U_{p} for p ∈ {2, 3, 5, 7, 13}

We describe the computation of U_{p} by using powers of d_{p} to form a basis for overconvergent modular forms. In this subsection, we assume p ∈ {2, 3, 5, 7, 13}.

Let t_{2} = 4, t_{3} = 3. For p ≥ 5, let t_{p} = 1. Let c_{2} = 0, c_{3} = 1728, c_{5} = 0, c_{7} = 1728, and c_{13} = 432000/691.

Let e = 12/(p^{2} − 1). The Newton polygon of H_{p} − c_{p}d_{p} has a single side of slope ep, and (H_{p}/d_{p} − c_{p})Δ is the weight 12 power of E_{t_{p}(p−1)}.

Proposition 2.3.1 Let r < p/(p + 1). The disc {z : z ∈ X_{0}(1), v_{p}(E_{tp(p−1)}(z)) < t_{p}r} is isomorphic to {z : z ∈ X_{0}(p), v_{p}(d_{p}(z)) > −er(p + 1)}.

Proof. When z ∈ X_{0}(1) is a point of supersingular reduction, Δ(z) is a unit. At a point of ordinary reduction, E_{t_{p}(p−1)}(z) is a unit and v_{p}(Δ(z)) ≥ 0. So the disc of X_{0}(1) is {z : v_{p}(j(z) − c_{p}) < er(p + 1)}.

The slope of the Newton polygon of (j − c_{p})d_{p} = H_{p} − c_{p}d_{p} shows that relation is uniquely invertible for d_{p} with v_{p}(d_{p}(z)) > −er(p+1), establishing the isomorphism. □

Corollary 2.3.1.1 M_{0}(1, Z_{p}) ⊂ Z_{p}[[d_{p}]]. U_{p} acts as a matrix M on a basis of powers of d_{p}.

Let W be the rigid subspace of X_{0}(p) where v_{p}(E_{t_{p}(p−1)}) < t_{p}/(p + 1).

We show how to compute U_{p}(d_{p}^{n}).

Lemma 2.3.2 The following equality holds on W:

V (ĵ) = ĵ∘ w_{p}.

Proof. The Atkin-Lehner involution acts as

w_{p} : (E, C) → (E/C, E[p]/C).

An elliptic curve corrseponding to a point of W has a canonical subgroup of order p, and

V : E → E/ ker ϕ^{∗},

which coincides with π ∘ w_{p}. □

We deduce the modular equation for X_{0}(p):

H_{p}(w_{p}^{∗}(d_{p}))V (d_{p}) = H_{p}(V (d_{p}))w_{p}^{∗}(d_{p}),

so

H_{p}(p^{12/(1−p)}/d_{p})V (d_{p}) − p^{12/(1−p)}H_{p}(V (d_{p})/d_{p} = 0.  (2)

We clear denominators and factor V (d_{p}) − w_{p}^{∗}(d_{p}) from Equation (2) to determine an algebraic relation

I_{p}(d_{p}, V (d_{p}))

between d_{p} and V (d_{p}) of degree p in d_{p}. The inverse of V applied to coefficient of d_{p}^{p−1} is tr V (d_{p}) = pU(d_{p}).

By standard manipulation of symmetric functions, we hash coefficients of I_{p} to determine U(d_{p}^{n}). The values of U(d_{p}^{n}) for n = 0 to p − 1 and the coefficients of I_{p} determine a recurrence for U(d_{p}^{n}) for n ≥ p.

Let M_{ij} be the coefficient of d_{p}^{i} in U(d_{p}^{j}).

Proposition 2.3.3 The p-adic valuation of M_{ij} is at least e(pi − j) − 1. The Newton polygon of U_{p} in weight zero lies on or above a parabola with quadratic coefficient 6/(p + 1).

Proof. The M_{ij} satisfy a generating function equation

∑_{i=1}^{∞} ∑_{j=1}^{∞} M_{ij}x^{i}y^{j} = \frac{y}{p} \frac{d}{dy} log I_{p}(y, x).  (3)

We calculate I_{p} for p = 2, 3, 5, 7, 13:

I_{2} = 1 − (2^{12}x^{2} + 3 · 2^{4}x)y − xy^{2},
I_{3} = 1 − (3^{12}x^{3} + 4 · 3^{8}x^{2} + 10 · 3^{3}x)y − (3^{6}x^{2} + 4 · 3^{2}x)y^{2} − xy^{3},
I_{5} = 1 − (5^{12}x^{5} + 6 · 5^{10}x^{4} + 63 · 5^{7}x^{3} + 52 · 5^{5}x^{2}I + 3 + 63 · 5^{2}x)y
−(5^{9}x^{4} + 6 · 5^{7}x^{3} + 63 · 5^{4}x^{2} + 52 · 5^{2}x)y^{2}
−(5^{6}x^{3} + 6 · 5^{4}x^{2} + 63 · 5x)y^{3} − (5^{3}x^{2} + 6 · 5x)y^{4} − xy^{5},
I_{7} = 1 − (7^{12}x^{7} + 4 · 7^{11}x^{6} + 46 · 7^{9}x^{5} + 272 · 7^{7}x^{4} +
845 · 7^{5}x^{3} + 176 · 7^{2}x^{2} + 82 · 7x)y...,
I_{13} = 1 − (13^{12}x^{13} + 2 · 13^{12}x^{12} + 25 · 13^{11}x^{11} + 196 · 13^{10}x^{10} +
1064 · 13^{9}x^{9} + 4180 · 13^{8}x^{8} + 12086 · 13^{7}x^{7} +
25660 · 13^{6}x^{6} + 39182 · 13^{5}x^{5} + 41140 · 13^{4}x^{4} +
27272 · 13^{3}x^{3} + 9604 · 13^{2}x^{2} + 1165 · 13x)y....

Let N_{ij} = p^{e(j−i)}M_{ij}. The matrix (N_{ij}) is similar to (M_{ij}) and v_{p}(N_{ij}) ≥ 12i/(p+1). □

Here is an example, the case for U_{3}. In weight zero on the basis {d_{3}^{n}}, the matrix for 3U_{3} has generating function

F (x, y) = (270xy + 72xy^{2} + 3xy^{3} + 4 · 3^{8}x^{2}y + 2 · 3^{6}x^{2}y^{2} + 3^{12}x^{3}y)/I_{3}(y, x). (4)

For p = 3, 12/(p^{2} − 1) = 3/2. Substitute α for 3^{−3/2}x and β for 3^{3/2}y into Equation (3) to find the generating function

∑_{i,j} N_{ij}α^{i}β^{j} = \frac{27(10αβ + 8\sqrt{3}αβ^{2} + 3αβ^{3}) + 3^{6}(4\sqrt{3}α^{2}β + 2α^{2}β^{2}) + 3^{9}α^{3}β}{1 − 3^{3}(10αβ + 4\sqrt{3}αβ^{2} + αβ^{3}) − 3^{6}(4\sqrt{3}α^{2}β + α^{2}β^{2}) − 3^{9}α^{3}β} . (5)

The valuation of N_{ij} is at least 12i/(p + 1) = 3i. We see this by substituting δ = 27α into Equation (5). The result

G(δ, β) = \frac{10δβ + 8\sqrt{3}δβ^{2} + 3δβ^{3} + 4\sqrt{3}δ^{2}β + 2δ^{2}β^{2} + δ^{3}β}{1 − 10δβ − 4\sqrt{3}(δβ^{2} + δ^{2}β) − (δβ^{3} + δ^{2}β^{2} + δ^{3}β)}  (6)

has coefficients of nonnegative valuation. So the Newton polygon of 3U_{3} is bounded below by the parabola 3\binom{n}{2} + 3n, and that of U_{3} by 3\binom{n}{2} + 2n.

Proposition 2.3.4 Let p ∈ {2, 3, 5, 7, 13}. The operators U_{p} and V preserve Z[[d_{p}]] and

U_{p}(p^{e+1}d_{p}O_{p}[[p^{e}d_{p}]]) ⊂ p^{pe}d_{p}O_{p}[[p^{pe}d_{p}]],
V (p^{pe}d_{p}O_{p}[[p^{pe}d_{p}]]) ⊂ p^{e}d_{p}O_{p}[[p^{e}d_{p}]].

Proof. The operators U_{p} and V preserve Z[[d_{p}]] because they preserve Z[[q]] and Z[[d_{p}]] = Z[[q]].

The valuations of matrix entries in Proposition 2.3.3 for the operator U_{p} verifies the statement about U_{p}.

Equation (2) shows

V (p^{pe}d_{p}) ∈ p^{pe}d_{p}^{p} + (p^{e}d_{p})^{p+1}O_{p}[[p^{e}d_{p}]],

so the statement about V follows. □

An appeal to Theorem 1.4.1 permits p-adic approximation of U_{p} as a matrix on the basis E_{k}^{∗}d_{p}^{n}/V (E_{k}^{∗}) to accuracy depending on the approximation of E_{k}^{∗}.

2.4 Tame level 1 and p = 2 or 3

Emerton[E] calculates the lowest positive slope 2-adic modular forms of every weight. Concise expressions for the q-expansions of a few forms facilitate computation.

Serre[Se] observes that for a compact operator M expressed as a matrix on a basis of a Banach space, if c_{i} is the infimum of the valuations of column i of M, then tr (∧^{n}M) has valuation at least the sum of the n smallest c_{i}.

Let k be an even weight. The operator U_{2} acting on S_{k}(1, O_{2}) is compact.

Proposition 2.4.1 There is an O_{2} basis {e_{n}}_{n≥1} of S_{k}(1, O_{2}) such that

U_{2}(S_{k}(1, O_{2})) ⊂ ⨁ 8^{n}e_{n}O_{2}.

Proof. This is a rewriting of Proposition 3.21 of [E] in language amenable to the noted observation of Serre. The basis element e_{n} is F_{k}d_{2}^{n} for a certain weight k form F. □

Corollary 2.4.1.1 The Newton polygon of U_{2} in even weight lies above a parabola with quadratic term 3/2.

Proof. Notice ∑_{i=1}^{n} 3i = 3\binom{n+1}{2}. □

Proposition 2.4.2 The Newton polygon of U_{3} in weight k lies above (n, 3\binom{n}{2}) for 3|k and above (n, \binom{n}{2} for k ∈ Z.

Proof. Let S = ^{8} \sqrt{Δ^{3}/V (Δ)}. Apply Theorem 1.4.1 using powers of S/V (S) and Proposition 2.3.4. □

3 Parabolas and the Newton polygon of U_{p}

3.1 A parabola below the Newton polygon

Wan shows the Newton polygon of P_{k} − P_{k′} lies above a horizontal line depending on v_{p}(k − k′). The Newton polygons of P_{k} and P_{k′} coincide when both lie below this line. He inverts this relationship to find a quadratic function such that GM(f) holds. Theorem 2.1.1 improves the bound on the Newton polygon of P_{k} − P_{k′}, yielding a smaller, but still quadratic, f.

In the case N = 1, we have s_{N} = 1/12 and for all primes p, there is parabola below the Newton polygon of U_{p} with quadratic coefficient 6/(p + 1).

We describe an increasing function lying below the Newton polygon of P_{k} − P_{k′}. Proposition 4.1.2 serves as a worked example.

Theorem 3.1.1 Let p be prime and N be an integer prime to p. Let v, l be integers and k and k′ be integers such that k − k′ = l(p − 1)p^{v}. Let P_{k} and P_{k′} be the characteristic series of the level N U_{p} operator on the respective weights. Let g(n) be a parabola with quadratic term 1/2(p + 1)s_{N} lying below the Newton polygon for P_{k}. Then g(n) lies below the Newton polygon for P_{k′} and

h(n) = v + g(n) + inf_{j≥n} (\frac{j − n}{(p + 1)s_{N}} ) − log_{p} j

lies below the Newton polygon for P_{k} − P_{k′}.

Proof. Let r = p^{1/(p+1)}. The quotient ϵ = E_{p−1}/V (E_{p−1}) is in M_{0}(N, O_{p}, r). Let U_{(k)} be U_{p} realized as a matrix on a Katz expansion basis for M_{k}(N, C_{p}, r). Multiplication by ϵ is a linear operator on M_{k}(N, C_{p}, r), which denote simply by ϵ. The operator U_{(k)}ϵ^{lp^{v}} is similar to U_{(k′)}.

Write

ϵ^{p^{v}} = 1 + ∑_{j=1}^{∞} r^{j}b_{j}/E_{p−1}^{j},

where b_{j} ∈ W_{j(p−1)}(N, B). The valuation of b_{j} is positive and at least v − log_{p} j. This much shows the Newton polygon of P_{k′} lies above g(n).

The matrix U_{(k)} has a factorization D^{∗}M_{A}, where D^{∗} is diagonal and both D^{∗} and M_{A} are over O_{p}. Theorem 1.4.3 shows we may choose D_{ii}^{∗} with valuation linear in i with slope 1/(p + 1)s_{N}. For any n > 0, we may change basis so that U_{(k)} factors as DM_{X}, for diagonal matrix D such that D_{ii} = D_{ii}^{∗} for i ≤ n and D_{ii} = D_{nn}^{∗} for i > n.

The difference U_{(k)}ϵ^{lp^{v}} − U_{(k)} factors as D′M_{Z} For i ≤ n, D′_{ii} has valuation

v + v_{p}(D_{ii}) + inf_{j>0}(v_{p}(D_{jj}) − v_{p}(D_{jj}^{∗}) − log_{p} j).  (7)

The maximum valuation of D′_{ii} results when n is as small as possible, n = i. So equation (7) simplifies to

v_{p}(D′_{ii}) ≥ v + v_{p}(D_{ii}^{∗}) + inf_{j>0} (\frac{j − i}{(p + 1)s_{N}} ) − log_{p} j,

The result follows by application of Theorem 2.1.1 for each n. □

3.2 A parabola above the Newton polygon?

Experiments with p = 3 and tame level 1 suggest that the Newton polygon of U_{p} lies between two parabolas with the same quadratic coefficient. We show that, if this is generally the case, there is a function f which is approximately linear and verifies GM(f).

Proposition 3.2.1 Let p, N, k, k′, l, v be given as for Theorem 3.1.1 and let

A = \frac{1}{2(p + 1)s_{N}} .

Let z > 0. Suppose the Newton polygons of U_{p} in weights k and k′ both lie in the region bounded by the parabolas P 1 with equation y = Ax^{2} + Bx and P 2 with equation y = Ax^{2} + Bx + zx. Let Q be the curve with equation

y = Ax^{2} + Bx + v − inf_{j≥x}(2A(j − x) − log_{p} j).

Suppose v is sufficiently large that Q intersects P 2 at x-coordinate x_{0} satisfying

zx_{0} = v − log_{p} x_{0}.

Then the Newton polygons of U_{p} in weights k and k′ coincide when their slopes are at most

2Ax_{0} + B + z − 2\sqrt{Bx_{0} + zx_{0}}.

Proof. Let (x_{0}, y_{0}) be the point on P 1 below the intersubsection of P 2 and Q. The tangent to P 2 through (x_{0}, y_{0}) with lower slope has equation

y = Ax_{1}(2x − x_{1}) + Bx + zx,

where

x_{1} = x_{0} − A^{−1}\sqrt{Bx_{0} + z_{0}}.

The slope of this tangent is a lower bound for the slope of the Newton polygons above x_{0}, because they are constrained between P 1 and P 2. The curves P 1 and P 2 are below Q to the left of x_{0}, so the Newton polygons coincide there. The slope of the tangent is

2Ax_{0} + B + z − 2\sqrt{Bx_{0} + zx_{0}}.
□

4 Example: tame level 1 U_{3}

4.1 A parabola below the Newton polygon of U_{3}

We show for p = 3, tame level 1, and weight 0 the Newton polygon of U_{3} lies on or above the parabola

g(n) = 3\binom{n}{2} + 2n

and coincides with that parabola infinitely often. As a consequence, there is a parabola lying strictly above the Newton polygon.

We show for 3^{v+1} | k, and nonnegative integer i < v, there are exactly 3^{i} overconvergent 3-adic modular forms with slopes in [(3^{i+1} + 1)/2, (3^{i+2} − 5)/2], with average slope 3^{i+1} − 1. This behavior is related to and consistent with the slopes predicted by conjectures of Gouvêa and Mazur.

In this subsection, the subscript k on P_{k} indicates the classical weight k. The narrowed band for the Newton polygon of P_{0} verifies a smaller, albeit O(n^{2}), f(n) for which GM(f) holds than previously reported. For weight k = 2 · 3^{i}, we show level 3 oldforms have slopes outside [k/4, 3k/4], supporting Gouvêa’s conjecture about where the slopes of classical level p oldforms are.

Recall S is a weight 3 overconvergent modular form and

S/V (S) ∈ 1 + (9d, 27d^{2})Z_{3}[[9d, 27d^{2}]].

Define

U_{(3l)} = U_{3} ∘ S^{l}/V (S)^{l}.

Equation (4) gives the generating function for U_{(0)} on the basis {d^{i}},

F (x, y) = −\frac{y}{3} \frac{d}{dy} log I_{3}(x, y).

Proposition 4.1.1 Let l ∈ Z_{3}. The Newton polygon of U_{(3l)} acting on overconvergent modular functions lies above the parabola g(n).

Proof. Let i, j > 0. The coefficient of d^{j} in 3^{3i/2}U_{(0)}(d^{i}) has valuation at least 3j/2 + 3j − 1. So we may express U_{(0)} in the basis \{(3^{\frac{3}{2}} d)^{i}\} as the product DM, where D is the infinite diagonal matrix with i, i entry 3^{3i−1} and M preserves the O_{3} lattice generated by the basis elements.

The parabola g(n) is below the Newton polygon of D, and the proposition follows from the first part of Theorem 2.1.1.

We observe S^{l}/V (S^{l}) ∈ 1 + (9d, 27d^{2})Z[[9d, 27d^{2}]], so the proposition holds for all l ∈ Z_{3}. □

Here is a special case of the phenomenon of Theorem 3.1.1.

Proposition 4.1.2 Let k = 3l be an integer and suppose v_{3}(l) = v. Let s be an integer, 0 ≤ s < 3^{v}. If the Newton polygon of P_{0} meets the parabola g(n) at (s, g(s)), so does the Newton polygon of P_{k}.

Proof. Express S^{l}/V (S^{l}) = (1+9d_{9} +27d_{9}^{2})^{−l} via the binomial expansion of (1+a)^{−l}. The coefficient of a^{j} has valuation at least v − v_{3}(j), which is at least 1 for j < 3^{v}.

Apply Theorem 2.1.1 to the operators U_{(0)} and U_{(0)}S_{l} on the basis

{3^{3i/2}d^{i}}_{i=1}^{3^{v}−1} ∪ {3^{3^{v+1}/2−1}d^{i}}_{i=3^{v}}^{i=∞}.

The observation about the binomial expansion shows β in part 4 of Theorem 2.1.1 is at least 1. □

The coefficient of d in S/V (S) has valuation 2, which is greater than \frac{3}{2}, so a result stronger than the previous proposition holds. That is, the range of s might be increased depending on k.

Let

m_{i} = ∑_{j=0}^{i−1} 3^{j} = \frac{3^{i} − 1}{2}.

Proposition 4.1.3 Let E be the set of n such that the Newton polygon of U_{(0)} has vertices on the same parabola f(n). The set E is {m_{i} : i ≥ 0}.

Proof. The leading coefficient of P_{0} is 1, so 0 ∈ E. We claim for all nonzero n, n ∈ E if and only if (n − 1)/3 ∈ E

By Proposition 4.1.1, U_{(0)} = DM, with D = diag(3^{3n−2}) and M preserving a certain lattice L. So n ∈ E if and only if the upper n × n diagonal minor of M has valuation zero. We claim n ∈ E if and only if there is a unique selection from the upper n × n diagonal major of M consisting of only 3-units.

As noted earlier, we compute U_{(0)}(d_{i}) in terms of U applied to lower powers of d. As a matrix, the entries of M satisfy linear recurrences. Equation (6) with (x, y) substituted for (δ, β) is

G(x, y) = \frac{10xy + 4\sqrt{3}xy(x + 2y) + xy(x^{2} + 2xy + 3y^{2})}{1 − xy(10 + 4\sqrt{3}(x + y) + x^{2} + xy + y^{2})}.

The coefficient of x^{i}y^{j} is the entry of M in row i and column j.

We are interested in finding 3-unit entries, so we consider G modulo 3. Let

R(i) = (1 + (xy + x^{3}y + x^{2}y^{2} + xy^{3}) + (xy + x^{3}y + x^{2}y^{2} + xy^{3})^{2})^{3^{i}}.

Over F_{3}, the exponent 3^{i} distributes. The reduction of G is

Ḡ = xy(1 − xy + x^{2}) ∏_{i=0}^{∞} R(i).

Let

Ḡ_{j} = xy(1 − xy + x^{2}) ∏_{i=0}^{j−1} R(i)

and

C_{j} = ∏_{i=j}^{∞} R(i).

Note that over F_{3}, C_{j}^{3} = C_{j+1}; the terms of C_{j} have exponents divisible by 3^{j}.

Define

Ḡ_{0}(x, y) = xy(1 − xy + y^{2}).

The polynomial G_{i} defines by its coefficients a 3^{i+1} × 3^{i+i} matrix which is a 3^{i} × 3^{i} array of interleaved 3 × 3 matrices.

The terms of R(i) have all exponents divisible by 3^{i}, so R(i) and the terms x^{c}y^{d} of G_{i} where a ≡ c and b ≡ d modulo 3^{i} determine the term x^{a}y^{b} of G_{i+1}.

We calculate

Ḡ_{1} = \begin{aligned}xy \\ & & & +x^{2}y^{4} \\ +x^{3}y & & +x^{3}y^{3} & & +x^{3}y^{5} & & +x^{3}y^{7} \\ & −x^{4}y^{2} & & −x^{4}y^{4} & & & & −x^{4}y^{8} \\ & & & & & & −x^{5}y^{7} \\ & +x^{6}y^{2} & & −x^{6}y^{4} & & −x^{6}y^{6} \\ & & & & +x^{7}y^{5} \\ & & & +x^{8}y^{4} \\ & & +x^{9}y^{3}. \end{aligned}

The typography suggests matrix coefficients; however, Ḡ_{j} is a polynomial. More concisely,

Ḡ_{1} = x^{−1}yḠ_{0}(x, y)^{3} + Ḡ_{0}(x, y)^{3} − xy^{−1}Ḡ_{0}(y, x) + y^{−2}Ḡ_{0}(y, x) + xy + x^{2}y^{4} + x^{6}y^{2}. (8)

We aim for a selection with nonzero product from the upper n × n diagonal major. All terms of Ḡ_{1} with exponent on y divisible by 3 are captured by Ḡ_{0}(x, y)^{3}. Every element of such a selection which is in a row number divisible by 3 must be in a column number divisible by 3, and also the one third of elements in the selection which are in the third rows must be further selected based on where G_{0}(x, y)^{3}C_{1} has nonzero entries.

Similarly, we determine that entries selected from rows numbered one more than a multiple of 3 must be in columns numbered one less than a multiple of 3.

If n ≡ 1 mod 3, the odd element must be taken from a row and column each one more than a multiple of 3. The only unit in row 1 is in column 1. Any selection of units must select that element. So if a selection of units in the upper n × n diagonal major exists, n ≡ 1 mod 3.

Equation (8) shows there is a selection of units from the upper n×n diagonal major exactly when there is a selection of units from the upper \frac{n−1}{3} × \frac{n−1}{3} diagonal major, and if such selection of units is unique in one case, it is unique in the other.

There is a unique selection of units for n = 1 (and n = 0 too). A diagonal minor is the signed sum of products of selections of a diagonal major. We have found which upper diagonal majors have selections of units, and each upper diagonal major has zero or one such selection. We have determined which upper diagonal minors have 3-adic valuation 0, establishing the claim. □

Corollary 4.1.3.1 Let k ∈ Z and suppose 3^{v+1} | k. Let 0 ≤ i < v. There are exactly 3^{i} overconvergent 3-adic modular forms of weight k with slope in [(3^{i+1}+1)/2, (3^{i+2}−5)/2], and these have average slope 3^{i+1} − 1.

Proof. The Newton polygon of P_{k} has among its vertices (m_{i}, g(m_{i})) and (m_{i+1}, g(m_{i+1})). There are 3^{i} = m_{i+1} − m_{i} slopes with multiplicity accounted for by the edges joining these vertices. The difference g(m_{i+1}) − g(m_{i}) is

3(m_{i} + 3^{i})(m_{i} + 3^{i} − 1)/2 + 2(m_{i} + 3^{i}) − 3m_{i}(m_{i} − 1)/2 + 2m_{i} =
3^{i}(3m_{i} − \frac{3}{2}) + 3^{2i+1}/2 + 2(3^{i}) =
(3^{i})(3^{i+1}/2 − 3 + 3^{i} \frac{3}{2} + 2) =
= 3^{i}(3^{i+1} − 1).

So the average slope is 3^{i+1}−1. The minimum of these slopes is at least g(m_{i}+1)−g(m_{i}) and the maximum at most g(m_{i+1}) − g(m_{i+1} − 1). □

Corollary 4.1.3.2 Let k ∈ Z and suppose

v_{3}(k) ≥ [g(m_{i+1}) + g(m_{i})]/2 − g([m_{i+1} + m_{i}]/2) + i + 1.

The Newton polygons of P_{0} and P_{k} coincide to the left of (m_{i+1}, g(m_{i+1})).

Proof. This number verifies a lower bound on the Newton polygon of Q which is above the secant of the parabola joining (m_{i}, g(m_{i})) and (m_{i+1}, g(m_{i+1})). □ This verifies a quadratic f for GM(f) to hold.

Affirming a pattern noticed by Gouvêa[G],

Corollary 4.1.3.3 Let k = 2 · 3^{v+1}. There are exactly 3^{v} classical weight k, level 3 modular forms of slope 3^{v+1} − 1. The oldforms have slopes outside [k/4, 3k/4].

Proof. Let l = k/3. The interesting term of the binomial expansion of S^{l}/V (S)^{l} is

(9d_{9} + 27d_{9}^{2})^{3^{v}},

as mentioned in Proposition 4.1.2 The coefficient of d_{9}^{j} in this term has valuation strictly more than 3j/2 for j < 2·3^{v}. Since m_{v+1} < 2 · 3^{v}, the coefficient of d_{9}^{j} in S^{l}/V (S)^{l} has valuation strictly greater

than 3j/2 for j < m_{v+1} So we see (m_{v+1}, g(m_{v+1})) is on the Newton polygon of P_{k}. □

4.2 A curve above the Newton polygon of U_{3}

For this case p = 3, N = 1, k = 0, experiments suggest the following hypothesis.

Proposition 4.2.1 Suppose l and v are integers, v > 0, k′, k weights such that k′ = k + l(p − 1)p^{v}. Suppose the Newton polygons of U_{p} in weights k and k′ lie below the parabola g(n) + n/2. Then the Newton polygons coincide when their slopes are at most v.

Proof. Applying Proposition 3.2.1 with A = 3/2, B = 1/2, z = 1/2, we derive

x_{0}/2 = v − log_{3} x_{0}.

The Newton polygons coincide when their slopes are at most

3x_{0} + 1 − 2\sqrt{x_{0}}.

We estimate x_{0} > v, and x_{0} > \sqrt{x_{0}}, so the Newton polygons coincide when their slopes are at most v. □

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Department of Mathematics Malott Hall Cornell University Ithaca, NY 14853 USA

lawren@math.cornell.edu