A uniform lower bound for the Newton polygon of U_{p}.

Kevin Buzzard†

Let N be a positive integer, let k ≥ 2 be an integer, and let p be a prime not dividing N. We shall assume throughout that Np ≥ 4, but this assumption is only made for simplicity. In this note we give an elementary construction of an explicit quadratic lower bound for the Newton polygon of the Hecke operator U_{p} acting on the space of cusp forms of weight k and level Np. Moreover, this lower bound is independent of k. In down-to-earth terms, we obtain as corollaries, for all α ∈ ℚ, explicit bounds for the number of eigenforms of slope α, weight k, and level Np. Moreover, these bounds are independent of k. As far as the author knows, results of this kind, generalising those of Hida in the slope 0 case, were first proved in the unpublished [T]. Results of this kind can now be obtained using the rigid analytic techniques of [C] and [W]. However, the fact that they can be obtained in a purely elementary manner seems not to be widely known.

Some notation.

Let R = ℤ_{p}[ξ] denote the polynomial ring in one variable over ℤ_{p}. Let L be an Rmodule which as a ℤ_{p}-module is finitely generated and free, of rank t say. Define the Newton Polygon of the map ξ on L as follows. For simplicity, assume that det(ξ) ≠ 0. Write the characteristic polynomial of ξ on L ⊗ ℚ_{p} as ∑_{i=0}^{t} c_{n−i}X^{i}. Then let v_{p} denote the usual valuation on ℤ_{p}, and plot the points (i, v_{p}(c_{i})) in ℝ^{2}, for 0 ≤ i ≤ t, ignoring the i for which c_{i} = 0. Let Z denote the convex hull of these points. The Newton Polygon of ξ on L is the lower faces of Z, that is the union of the sides forming the lower of the two routes from (0, 0) to (t, v_{p}(c_{t})) on the boundary of Z. It is well-known that this graph encodes the p-adic valuations of the eigenvalues of ξ. In fact, if the Newton Polygon has a side of slope α and whose projection onto the x axis has side n, then there are precisely n eigenvectors of ξ with p-adic valuation equal to α.

It is our intention to say something about this polygon when ξ is the operator U_{p} acting on a certain lattice of modular forms. As a result we shall recover results about the p-adic valuations of the eigenvalues of U_{p}. We shall first establish a general result, after setting up some more notation.

Let L be an R-module which is free of finite rank over ℤ_{p} as before, and let K be a sub-R-module of L such that the index [L : K] is finite. Write Q for the quotient L/K. Then Q is a finite abelian p-group. Say

Q ≅ ∏_{i=1}^{r}(ℤ/p^{a_{i}}ℤ),

with a_{1} ≥ a_{2} ≥ a_{3} ≥ … ≥ a_{r} > 0. Write a_{i} = 0 for r < i ≤ t.

† The author would like to thank the Miller Institute and UC Berkeley for the financial support and hospitality they offered him whilst this note was being written.

Say there exists a positive integer n with the following property: n ≥ a_{1} and ξ(K) ⊆ p^{n}L. Then write b_{i} = n − a_{i} for 1 ≤ i ≤ t and let B(j) = ∑_{i=1}^{j} b_{i}. I claim

Lemma 1. The Newton Polygon of ξ on L lies on or above the points (j, B(j)) for 0 ≤ j ≤ t.

Proof. The proof is very easy, and shall merely be sketched. First choose a ℤ_{p}-basis (e_{i})_{1≤i≤t} for L such that (p^{a_{i}}e_{i})_{1≤i≤t} is a ℤ_{p}-basis for K. With respect to this basis, let ξ : L → L be represented by the matrix (u_{i,j}). One sees that the assumption ξ(K) ⊆ p^{n}L implies that p^{b_{j}} divides u_{i,j}. Then it is an immediate consequence, from the definition of the characteristic polynomial as det(ξ − X.1), that p^{B(j)} divides a_{j} for 0 ≤ j ≤ t, because the b_{i} are increasing. Moreover, again because the b_{i} are increasing, this suffices to prove the lemma.

We now show how to apply this in the case of classical modular forms. We follow [H, Chapter 6] for definitions and notation. Write Γ for Γ_{1}(Np), and let S_{k}(Γ) denote the space of classical cusp forms of weight k and level Np. Note that Γ is torsion-free and hence free. In fact, Γ is free on 2g(X(Γ)) + c − 1 generators, where g(X(Γ)) is the genus of the compactified modular curve associated to Γ, and c is the number of cusps added to make the compactification.

For simplicity of notation, let g = k − 2. For a commutative ring A, let V_{g}(A) = A^{g+1}, considered as column matrices of length g + 1. Let M denote the monoid consisting of two by two matrices with integer entries and positive determinant. Give V_{g}(ℤ) the unique left action of M with the following property: for all y, z ∈ ℤ we require

\begin{matrix}( \begin{matrix}a & b \\ c & d \end{matrix} ) & (y^{g}, y^{g−1}z, y^{g−2}z^{2}, …, yz^{g−1}, y^{g})^{T} = \\ ((ay + bz)^{g}, (ay + bz)^{g−1}(cy + dz), …, (ay + bz)(cy + dz)^{g−1}, (cy + dz)^{g})^{T} , \end{matrix}

where the superscript T denotes transpose. It is easily checked that indeed there is a unique action of M with this property check this!. Define an action of M on V_{g}(A) for any commutative ring A by extending the action on V_{g}(ℤ) A-linearly. Note that Γ ⊆ M and hence V_{g}(A) has the structure of a Γ-module.

There is a well-known isomorphism, due to Eichler and Shimura:

ES : S_{k}(Γ) → H_{P}^{1} (Γ, V_{g}(ℝ))

where H_{P}^{1} denotes parabolic cohomology. Composing ES with the natural inclusion H_{P}^{1} (Γ, V_{g}(ℝ)) → H^{1}(Γ, V_{g}(ℝ)) gives us an inclusion ES : S_{k}(Γ) → H^{1}(Γ, V_{g}(ℝ)).

The Hecke operator U_{p} acts (ℂ-linearly) on S_{k}(Γ), and we shall now recall the definition of an operator on H^{1}(Γ, V_{g}(ℝ)) that extends U_{p}.

Let A be any ring, and define a linear map ξ : H^{1}(Γ, V_{g}(A)) → H^{1}(Γ, V_{g}(A)) thus. Firstly note that

Γ ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) Γ = ∐_{i=0}^{p−1} Γ ( \begin{matrix}1 & i \\ 0 & p \end{matrix} ) ,

and write α_{i} for ( \begin{matrix}1 & i \\ 0 & p \end{matrix} ) . For γ in Γ and 0 ≤ i ≤ p − 1 there exists a unique 0 ≤ j ≤ p − 1

such that α_{i}γ = γ_{i}α_{j} for some γ_{i} ∈ Γ. First we shall define the map ξ on 1-cocycles. If u : Γ → V_{g} is a 1-cocycle, define v(γ) ∈ V_{g} by

v(γ) = ∑_{i=0}^{p−1} ( \begin{matrix}p & −i \\ 0 & 1 \end{matrix} ) u(γ_{i}).

One can check that v is a 1-cocycle and moreover that the association u ↦ v induces a map from H^{1}(Γ, V_{g}) to itself, which we define to be ξ.

Lemma 2. The diagram

S_{k}(Γ) ⟶^{ES} H^{1}(Γ, V_{g}(ℝ))
↓^{U_{p}}  ↓^{ξ}
S_{k}(Γ) ⟶^{ES} H^{1}(Γ, V_{g}(ℝ))

commutes.

This is proved in [S, Chapter 8]

Let α(X) denote the characteristic polynomial of U_{p} acting on the complex vector space S_{k}(Γ). It is the roots of this polynomial that we ultimately wish to study. Note first that the coefficients of α are rational. Hence α^{2} is the characteristic polynomial of U_{p} on S_{k}(Γ), considered as a real vector space, and hence if β(X) is the characteristic polynomial of ξ acting on the real vector space H^{1}(Γ, V_{g}(ℝ)) then we see that α^{2} divides β, as ES is an injection. Note that β is also the characteristic polynomial of ξ acting on the K-vector space space H^{1}(Γ, V_{g}(K)) for any field K.

We now pass to the case K = ℚ_{p}. Firstly note that the image of H^{1}(Γ, V_{g}(ℤ_{p})) in H^{1}(Γ, V_{g}(ℚ_{p})) is a lattice (although H^{1}(Γ, V_{g}(ℤ_{p})) itself may have some torsion). Let ϵ_{i} ∈ V_{g}(ℤ_{p}) denote the column vector with 1 in its ith row and zeroes elsewhere, and let W_{g}(ℤ_{p}) denote the ℤ_{p}-subspace of V_{g}(ℤ_{p}) generated by the vectors

{ϵ_{1}, ϵ_{2}, …, ϵ_{g−n−1}, ϵ_{g−n}, ϵ_{g+1−n}, pϵ_{g+2−n}, p^{2}ϵ_{g+3−n}, …, p^{n−1}ϵ_{g}, p^{n}ϵ_{g+1}}.

It is easily checked that W_{g}(ℤ_{p}) is preserved by Γ. Let I denote the quotient V_{g}(ℤ_{p})/W_{g}(ℤ_{p}) with its induced Γ-action. Taking the long exact sequence in group cohomology associated to the short exact sequence

0 → W_{g}(ℤ_{p}) → V_{g}(ℤ_{p}) → I ↦ 0

gives us an exact sequence of finitely-generated ℤ_{p}-modules

H^{0}(Γ, I) → H^{1}(Γ, W_{g}(ℤ_{p})) → H^{1}(Γ, V_{g}(ℤ_{p})) → H^{1}(Γ, I),

where the first and last terms in this sequence are finite. It is not hard to deduce from this that there is an exact sequence

0 → H^{1}(Γ, W_{g}(ℤ_{p}))^{T F} → H^{1}(Γ, V_{g}(ℤ_{p}))^{T F} → Q → 0

where Q is some subquotient of H^{1}(Γ, I) and the superscript T F denotes the maximal torsion-free quotient. In particular Q is finite (it would be good to have a more explicit description of Q). Now write L for H^{1}(Γ, V_{g}(ℤ_{p}))^{T F} and K for H^{1}(Γ, W_{g}(ℤ_{p}))^{T F} . The action of ξ on H^{1}(Γ, V_{g}(ℤ_{p})) induces an action of ξ on L and hence on K. Now we have

Lemma 3. ξ(K) ⊆ p^{n}L.

Proof. Let c : Γ → W_{g}(ℤ_{p}) be a 1-cocycle. It suffices to prove that the cocycle ξc is divisible by p^{n}. From the definition of ξc, this will follow if we can show that

( \begin{matrix}p & −i \\ 0 & 1 \end{matrix} ) W_{g}(ℤ_{p}) ⊆ p^{n}V_{g}(ℤ_{p})

for 0 ≤ i ≤ p − 1. But in fact one can check that

( \begin{matrix}a & b \\ c & d \end{matrix} ) W_{g}(ℤ_{p}) ⊆ p^{n}V_{g}(ℤ_{p})

if ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M and p divides both a and c, and we are home.

Using Lemma 1, we can deduce an explicit lower bound for the Newton polygon of ξ on L, which encodes the slopes of the eigenvalues of the polynomial β. But we remarked above that α^{2} divides β and so we are also getting explicit lower bounds for the Newton Polygon U_{p} on S_{k}(Γ). We do not know Q explicitly, we know that it is a subquotient of H^{1}(Γ, I), which is itself a quotient of the group of 1-cocycles from Γ to I. This group of cocycles is isomorphic to I^{m} where m is the rank of the free (because Np ≥ 4) group Γ. In fact, as mentioned earlier, m = dim_{ℂ} H^{1}(Γ, V_{0}(ℂ)) = 2g(X(Γ)) + c − 1. Moreover, I itself is isomorphic to a subgroup of ∏_{i=1}^{n}(ℤ/p^{i}ℤ), and indeed if n ≤ g then this is an equality. Note in particular that Q is bounded independent of k, and as a result our explicit lower bounds for the Newton Polygon of U_{p} will also be independent of k.

If x ∈ ℝ, write ⌊x⌋ for the largest integer which is at most x, and ⌈x⌉ for the smallest integer which is at least x. We see that if Q = ∏_{i=1}^{r}(ℤ/p^{a_{i}}ℤ), with a_{i} decreasing, then a_{i} ≤ ⌊n + 1 − i/m⌋ for 1 ≤ i ≤ mg and hence b_{i} = n − a_{i} ≥ ⌊(i − 1)/m⌋ (with notation as before Lemma 1.) In particular, choosing n sufficiently large, we see that for 1 ≤ i ≤ t we have b_{i} ≥ ⌊(i − 1)/m⌋. Thus we deduce from Lemma 1 the following result.

Proposition. The Newton polygon of ξ acting on L is bounded below by the function which goes through (0, 0), has slope 0 for 0 ≤ x ≤ m, slope 1 for m ≤ x ≤ 2m, slope 2 for 2m ≤ x ≤ 3m, and so on.

Note that this bound depends only on m and in particular only on N and p. Because of the relationship between S_{k}(Γ) and H^{1}(Γ, V_{g}(ℝ)) we deduce

Corollary. The Newton polygon of U_{p} acting on S_{k}(Γ) is bounded below by the function which goes through (0, 0), has slope 0 for 0 ≤ x ≤ m/2, slope 1 for m/2 ≤ x ≤ m, and in general has slope r for mr/2 ≤ x ≤ (mr + m)/2. Here m = 2g + c − 1, where g is the genus of X(Γ) and c is the number of cusps of X(Γ).

As a corollary, we can deduce for example that for α a rational, the space of forms of slope α, weight k and level Np has dimension bounded above by ⌈α + 1/2⌉m, independent of k. Uniform boundedness results like this were first obtained in [T], and they are also deducible from [W].

References.

[C] R. Coleman, p-adic Banach Spaces and Families of Modular Forms, preprint.

[H] Hida’s book

[S] Shimura’s book

[T] Richard’s thesis

[W] D. Wan, Dimension variation of Classical and p-adic Modular Forms, preprint.