On the coefficients of the characteristic series of the U-operator

Abstract

A conceptual proof is given of the fact that the coefficients of the characteristic series of the U-operator acting on families of overconvegent modular forms lie in the Iwasawa algebra.

Introduction

In this document, I attempt to “explain” why the formula for the characteristic power series for the U-operator acting on families of completely continuous p-adic modular forms (see section B4 of ref. 2) looks the way it does. In other words, I give a conceptual proof of the part of theorem B6.1, when p is odd, which is evident from the explicit formulas (see appendix I of ref. 1) and which asserts that the coefficients of this series lie in the Iwasawa algebra Λ = Z_p[[Z*_p]]. I also prove that this series analytically continues to a larger space. This was asserted by this theorem and is not evident from the formulas (I have not proven this assertion when p = 2). I use the operator called U in section B4 of ref. 1, which is the U_p-operator on weight 0 overconvergent forms twisted by a family of Eisenstein series E (see section 1 below). The key point is that the q-expansion coefficients of E lie in Λ ⊂ Λ. This is enough to prove that the function E_p whose q-expansion is E(q)/E(q^p) lies in Λ⊗̂A⁰(Z) where Z is the connected component of the ordinary locus containing the cusp ∞ in X₁(q) [a sort of affinoid q-expansion principle (see Theorem 2.1 below)]. The operator U acts on Λ⊗̂A⁰(Z_N) and if it were completely continuous that would basically do it, but it’s not. I am forced into some technicalities to get around this difficulty in sections 3 and 4. I complete the proof in section 5, and in section 6, I prove theorem B6.2 of ref. 1, when p is odd, which asserts that this characteristic series “controls” forms of higher level.

Some notation: Fix a prime p. Let q = 4 if p = 2 and p otherwise. Let Λ = Z_p[[1 + qZ_p]].

If X is a rigid analytic space and Y is a reduced affinoid with good reduction, let Y_X = Y × X and A^†(Y_X/X) denote the ring of overconvergent rigid analytic functions on Y_X over X (see section A5 of ref. 1). If ℬ is the rigid space of continuous characters on 1 + qZ_p with values in C*_p, it is conformal over Q_p to the open unit disk. I can and do think of Λ as rigid functions on ℬ defined over Q_p bounded by 1. If Y is the affinoid unit disk with parameter T, let A⁰(X)[T]^† denote A^†(Y_x) ∩ A⁰(Y_x). Identifying ℬ with the open unit disk, we may regard Λ as Z_p[[S]]. Then, for each 0 < t < 1 and ∑_n^∞ b_nSⁿ ∈ Λ, set

If t ∈ |C_p|, this is the norm obtained upon mapping an element of Λ into A⁰(B[t]) and then taking the supremum norm of its image. Then, if t ≤ s < 1, log_t(s) > 0 and one can easily check

Let I be the maximal ideal in Λ. Suppose t < 1, then f ∈ Iⁿ implies

and if n ≤ Min{log_t(|f|_t), − log_p(|f|_t)}, f ∈ Iⁿ.

We deduce:

Proposition 1.1. All the norms | |_t, for 0 < t < 1, are equivalent and induce the I-adic topology on Λ.

Corollary 1.1.1. The image of Λ in A⁰(B) is closed.

I define Λ[X]^† to be the subring of Λ[[X]] consisting of elements of the form

for which there exists an a > 0 in R such that λ_n ∈ I^[an] for large n. Then, f(X) ∈ Λ[X]^† if and only if the image of f(X) in A⁰(B[t])[[X]] lies in A⁰(B[t])[X]^† for some t < 1 if and only if the image of f(X) in A⁰(B[t])[[X]] lies in A⁰(B[t])[X]^† for all t < 1. Thus

Lemma 1.2. A^†(B[1]_ℬ/ℬ)⁰ ≅ Λ[X]^†.

2. A q-Expansion Principle

In this section, I will prove:

Theorem 2.1 (q-expansion principle). Suppose, t ∈ |C_p| and 0 < t < 1. Then, if G ∈ A^†(Z_B[t]/B[t]) and G(q) ∈ Λ[[q]], G uniquely analytically continues to an element of A^† (Z_ℬ/ℬ)⁰.

Lemma 2.2. There exists a finite morphism f from Z^† onto B[1]^† such that f⁻¹(0) = ∞ and, is separated.

Proof: Let be the reduction of Z and D be the divisor of degree zero on , s[] − ∑_i=1^s[e_i], where {e₁, … , e_s} is the set of points at ∞ (the supersingular points) on . Then mD is principal for some positive integer m. Suppose m is minimal. If is a function on the completion of with divisor D, : → [1] is a finite separated morphism such that ⁻¹(0) = . We may now apply theorem A-1 of ref. 2 with A = Z_p, B = A⁰(B[1]^†), C = A⁰(Z^†) and D = Z_p[X]/X^m, thought of as the ring of the closed subscheme m∞ of Z^†, to conclude there is a lifting of f to an overconvergent function f on Z which gives a finite morphism of degree s from Z^† onto B[1]^† with the property f⁻¹(0) = ∞.▪

Proof of the q-expansion principle:

Let G be as in the statement of the theorem. Let f be as in the lemma. Suppose f has degree d. Let Tr_f denote the trace map from A(Z^†) to A(B[1]^†). Let X be the standard parameter on A¹. Regarding q as a parameter at ∞, the fact that f is totally ramified above 0 implies that Tr_f extends naturally to a map from Z_p[[q]] to Z_p[[X]]. Hence, we may write

where a_n,m ∈ Z_p. In fact, a_n,m = 0 for m < n/d. For r ∈ A⁰(Z^†), we may write

where λ_n ∈ Λ. Now f extends to a finite morphism from (Z_B[t]/B[t])^† to (B[1]_B[t]/B[t])^† and we extend Tr_f accordingly. Then,

We know, by the above, that, for each m, the coefficient of X^m is a finite sum so lies in Λ. We also know Tr_f(rG) ∈ A^†(B[1]_B[t]/B[t]). Since this is true for all r ∈ A⁰(Z^†) we conclude DG ∈ A^†(Z_ℬ/ℬ) where D generates the discriminant ideal in Z_p[X]^† of A⁰(Z^†)/Z_p[X]^†). Since is separated, p ∤D. The principle will follow from:

Lemma 2.3. Let t ∈ |C_p| ∩ (0, 1). Suppose a(X) ∈ A⁰(B)[X]^† and there exists a D(X) ∈ Z_p[X]^† such that p ∤D(X) and D(X)a(X) ∈ Λ[X]^†, then a(X) ∈ Λ[X]^†.

Proof: Let A = A⁰(B[t]). Suppose

where λ_n ∈ Λ and |λ_n|_t ≤ δⁿ for some δ < 1 and large n. Let d be the degree of the reduction of D(X) modulo p which is defined because ≠ 0. Using the division algorithm, we may write Xⁿ = D(X)h_n(X) + r_n(X) where r_n(X) is either 0 or a polynomial over Z_p of degree strictly less than d and h_n(X) ∈ Z_p[X]^†. [We first know we can do this with h_n(X) ∈ Z_p〈X〉. Then the equation Xⁿ − r_n(X) = D(X)h_n(X) implies h_n(X) ∈ Z_p[X]^†.] It follows that,

Since |λ_n|_t ≤ δⁿ for large n, we conclude both sums converge in A[X]^†. The second sum must be 0 since it has degree strictly less than d. Since A[X]^† is an integral domain, we conclude

The lemma follows from the fact that Λ is closed in A by Corollary 1.1.1.▪

Now suppose b₁, … , b_d is a basis for A⁰(Z^†) over Z_p[X]^†. We may write, uniquely,

where a_i(X) ∈ A⁰(B[t])[X]^†. Then I apply the lemma to a(X) = a_i(X) and deduce the theorem.▪

Let E(q) denote the element of Λ[[q]]* such that κ(E[q]) = E_κ(q). Recall, for t < |π|, I proved in corollary B4.1.2 of ref. 1, there exists a rigid analytic function F₀ on Z_B[t], overconvergent relative to B[t], such that F₀(κ, q) = E_κ(q)/E_κ(q^p) for κ ∈ B[t]. I deduce,

Corollary 2.1.1. There is an element E_p ∈ A^†(Z_ℬ/ℬ) bounded by 1 on Z_ℬ whose q-expansion is E(q)/E(q^p).

3. Continuous Versus Completely Continuous Operators

Suppose L is a complete subring of A, P and N are Banach modules over A and L, respectively, and ι:P ↪ N⊗̂_LA is a continuous injective homomorphism.

Proposition 3.1. Suppose u is a continuous linear operator on N such that u ⊗ 1 preserves ι(P) and ι⁻¹uι = u_Pis a completely continuous operator on P. Then, if there exists an orthonormal basis B := {b_i}_i∈I for N over L and a map r:B → A* such that B* = {r(b)b ⊗ 1:b ∈ B} is contained in ι(P) and ι⁻¹(B*) is an orthonormal basis for P, det(1 − Tu_P) ∈ L[[T]].

Proof: For b ∈ B, let b* = r(b)b ⊗ 1 and for a subset S of I, let π_s:P → P be the projector onto the subspace P_s spanned by {b*_i:i ∈ S} as defined in lemma A1.6 of ref. 1. Then by theorem A2.1 and lemma A1.6 of ref. 1,

as S ranges over finite subsets of I. Now since det(1 − T(π_S ○ u_P)|P_S) is independent of the choice of basis of P_S over A and its matrix with respect to the basis {b*_i/r(b_i):i ∈ S} has entries in L, we see that det(1 − T(π_S ○ u_P)|P_S) ∈ L[T]. Since L is a complete subring of A, the proposition follows.▪

We will be able to apply this to the operator U because,

Lemma 3.2. Suppose X is a minimal underlying affinoid of a basic wide open W. Then there exists an orthonormal basis B of A(X) and an underlying affinoid Y of W such that Y strictly contains X and there exists a map r from B to K* such that {r(e)e:e ∈ B} is an orthonormal basis of A(Y).

(Compare proposition 1 of ref. 3.)

This will be an immediate consequence of Corollary 4.2.1, which is a more precise version.

4. Orthonormal Bases of Wide Open Neighborhoods

Let K be a finite extension of Q_p contained in C_p, R the ring of integers of K, and F the residue field of R. Below, the symbol r will always refer to an element of |C_p|. Note, however, that for any given r one might have to replace K by a finite extension so that r ∈ |K|. Suppose that G is a finite Abelian group of order prime to p such that the |G|-th roots of unity are contained in K.

Suppose W is a basic wide open defined over K with minimal underlying affinoid X such that W − X has s connected components U₁, … , U_s (see ref. 4). Suppose in addition that G acts faithfully on W and preserves X. For 1 ≤ i ≤ s and σ ∈ G let 1 ≤ σ(i) ≤ s be such that σ(U_i) = U_σ(i). Let z_i:U_i → B(0, 1)/{0} be a uniformizing parameter such that the subset of U_i where |z_i| ≥ r is nonempty and connected to X for any r < 1. Suppose in addition that there exist c(σ, i) ∈ R such that σ* z_i = c(σ, i)z_σ(i) (this we can arrange by using appropriate projectors like Eq. 1 below corresponding to the fixers in G of elements 1 ≤ i ∈ s). It follows that c(σ, i) ∈ R*. For r ≤ 1, let X_r = W − ∪{x ∈ U_i:|z_i(x)| < r}. Then for r close to 1, r < 1, X_r is an underlying affinoid of W which is a strict neighborhood of X and is preserved by G.

The affine has s points at ∞, P₁, … , P_s corresponding to the U_i and is acted on faithfully by G [since (|G|, p) = 1]. For f ∈ F(), f ≠ 0 let

Let m(f) = {i: − v_Pi(f) = M(f)}. Let T_i be a parameter at P_i, which lifts to z_i and for i ∈ m(f), let c_i(f) ∈ F be such that

Let A be the ring F[y₁, y₂, … , y_s]/{y_iy_j:i ≠ j} and let G act on A so that

Also let λ(f) be the element of A,

It follows that deg(λ(f)) = M(f) and λ(σ* f)) = σ*λ(f). Let B be the subring of A generated by λ(f) where f ranges over 𝒪_x(), f ≠ 0. Then by Riemann–Roch B ⊃ I := ⊕y_i^N A for some positive integer N. Moreover, B/I is finite dimensional over F and is acted on by G. Let H be a basis of B/I each element of which is an eigenvector for the action of G. Then the set

is a basis of B. Let t be a map from T to F(), such that λ(t(a)) = a, σ*t(h) = et(h) if h ∈ H and σ*h = eh and

Then {t(a):a ∈ T} is a basis for F(). For ɛ ∈ H om(G, R*), let

1

Note that if a ∈ T, π_ɛa = 0 or deg(π_ɛa) = deg(a). It follows that if π_ɛ(a) ≠ 0,

2

Now, let f_i = t(y_i^N), g_ij = t(y_i^N+j) and k_l, 1 ≤ l ≤ m, be elements in F[{a_h, b_i, c_{j k}}_{h∈H,1≤i,j≤s,0<k<N}] which generate the ideal consisting of f such that

Let K_l be a lifting of k_l to R[{a_h, b_i, c_ij}_{h∈H,1≤i,j≤s,0<k<N}]. Then the equations

determine an affine scheme χ which lifts and so there is an isomorphism from X^† to its weak completion such that the pullbacks of a_h, b_i, and c_ij are liftings of t(h), t(y_i^N) and t(y_i^N+j). For u ∈ T call the lifting of t(u) in X^† made by taking the appropriate product of these pullbacks t̃(u). Let V = {t̃(u):u ∈ H or u = f_i^N+j, 1 ≤ i ≤ s, 0 ≤ j < N}. Let U_ir = U_i ∩ X_r.

Lemma 4.1. If r is close enough to 1, r < 1, and J ∈ V, r^M()|J|U_ir equals 1 if i ∈ m() and is strictly less than 1 otherwise. Moreover, if i ∈ m(),

It follows that if r is close to 1 and J ∈ V that |J|_Xr = r^−M(J̄) and, in particular, if |a| = r,

Proposition 4.2. For r close enough to 1, r ≤ 1, the R-algebra A⁰(X_r) is the completion of the subalgebra generated over R by the elements {a^M(J)J:J ∈ V} and if r < 1 its reduction, (X_r) is G-isomorphic to B.

Proof: This proposition is immediate when r = 1 so suppose r < 1. Let C be the above complete subalgebra. We know, for r close to 1, C ⊗ Q_p = A(X_r) so by lemma 3.11 of ref. 5 we only have to prove: (i) for all f ∈ C, there exists a c ∈ R such that f/c ∈ C − mC, (ii) A⁰(X_r) is integral over C and (iii) C/mC is reduced. Now, (i) follows after making a finite extension if necessary, (ii) follows from proposition 6.3.4/1 of ref. 7, and the above description of A(X_r) and finally, (iii) (as well as the second part of the proposition) will follow, once we exhibit a G-isomorphism C/mC → B.

To see the latter, first note that elements in A⁰(U_ir) may be written in the form ∑_−∞^∞a_nz_iⁿ where a_n∈R and |a_n|rⁿ → 0 as |n| → ∞ and so (U_ir) is isomorphic to F((z_i)). If we map C in to ⊕_iA⁰(U_ir) and then reduce we get, after mapping the reduction of z_itoy_i, a homomorphism

Using the previous lemma, we see that for r close to 1, this factors through a surjection onto B which is a G-homomorphism by construction.

Now we produce the inverse to this homomorphism. For J ∈ V, let J_a = a^M()J. Consider the correspondence λ() ↦ J_a mod mC from to V mod mC. It suffices to show that for r sufficiently close to 1 this extends to an R-algebra homomorphism B → C/mC. Let Y_m be the subset of B consisting of elements of the form ∏_{f ∈} f^n(f)such that ∑_f∈ n(f)M(f) = m and let Y = ∪_m Y_m. If z ∈ Y_m we will say deg(z) = m. Then ℐ is generated by a finite set of relations of the form

(These relations may include single monomial relations.) For each relation of this form, there must be a relation of the form

on F(). If ã_y and b̃_y are liftings of the coefficients and ỹ and z̃ are the liftings of the monomials y and z obtained by lifting t(u) to t̃(u) for u ∈ T. Then, because χ lifts X, there must be a relation of the form

where h is a polynomial in {v ∈ V} with coefficients in R and α ∈ R, |α| < 1. It follows that

Since rⁿũ for u ∈ V_n is a product of elements of the form J_a for J ∈ V and r^mαh is in mC for r close to one, since h is a polynomial, we see that for r close to 1 we have a homomorphism from B onto C/mC which takes λ() to J_a as desired.

For a character ɛ ∈ Hom(G, R*) and an R module M on which G acts, set M(ɛ) = π_ɛM.

Corollary 4.2.1. Let r be as in the proposition and suppose |a| = r. Then the set

is an orthonormal basis for A(X_r). Moreover, if ɛ ∈ Hom(G, R*) and S ⊂ T is such that {π_ɛ(s):s ∈ S} is a basis for B(ɛ), then

is an orthonormal basis for A(X_r)(ɛ).

5. End of Proof

Fix a positive integer N prime to p. Let X be connected component of the ordinary locus of X₁(Nq) containing the cusp ∞ and U be the operator on A^†(X_ℬ/ℬ),

where U₍₀₎ is the weight zero U-operator, which is an operator on A(X)^†. This is the analytic continuation of the operator with the same name in remark B4.2 of ref. 1. We have a natural action of (Z/pZ)* on X₁(Nq) via diamond operators. (Note that this is intentionally trivial when p = 2, unfortunately.)

Let D be a disk around zero contained in ℬ and Y a strict affinoid neighborhood of X stable under the action of (Z/pZ)* such that E_p converges on Y_D. (This exists, by Corollary 2.1.1.) Let ɛ: (Z/pZ)* → Z*_p be a character. By Corollary 4.2.1 (and the properties of U₍₀₎), we may suppose we have an orthonormal basis B of A⁰(X)(ɛ) over K⁰ (we allow K to be as large as necessary and then eliminate this choice later) satisfies the hypotheses of Proposition 3.1, with A = K, L = K⁰, N = A⁰(X)(ɛ) and P = A(Y)(ɛ). It follows that {1 ⊗ b:b ∈ B} is an orthonormal basis for A⁰(X_D)(ɛ) over A⁰(D) which also satisfying these hypotheses with N = A⁰(X_D)(ɛ) and P = A(Y_D)(ɛ). We conclude from Proposition 3.1 that the characteristic series of U acting on A(Y_D)(ɛ), which is the series labeled P_ɛ(s, T) in section B3 of ref. 1 when p ≠ 2 and is the series labeled there P_N(s, T) when p = 2, lies in A_K⁰(D)[[T]]. Since this is true for all D, we see that it lies in A_Cp⁰ (ℬ)[[T]]. However, we know, a priori (using arguments as in the proof of theorem B3.2 of ref. 1, that it lies in Q_p[[S, T]]. Hence,

Theorem 5.1. (i) If p is odd the characteristic series of U acting on A(X_𝒲)^† over A(W), Q_N(T) (see lemma B3.7 of ref. 1), lies in Λ[[T]] and converges on W × C_p. (ii) If p = 2, the characteristic series of U acting on A(X_ℬ)^† over A(ℬ), P_N(s, T), lies in Λ[[T]] and converges on ℬ × C_p.

In fact, if we were only worrying about modular forms on Γ₀(N) (i.e., without character), we could have used a Katz basis (see section 2.6 of ref. 6). Indeed, suppose, for now, p ≥ 5. Let {b_a,1 … , b_a,ni} be a Z_p basis for B(N, 0, a). Then we have an orthonormal basis for A(X_r) = S(K, r, N, 0)

for all r ∈ R, r ≠ 0. This is good enough to apply the results of section 3 in this case.

6. Higher Level

In this section, I prove theorem B6.2 of ref. 1 when p is odd. That is, I prove,

Theorem 6.1. Suppose p is odd. If κ(x) = χ(x)〈〈x〉〉^k where k is an integer, χ:Z*_p → C*_p is a character of finite order and pⁿ = LCM(p, f_χ), then κ(Q_N)(T) is the characteristic series of the U-operator acting on overconvergent modular forms of level N pn, weight k, and character χ.

Proof: The proof of this is very simple, given what we now know. Let α be the character on Z*_p, x ↦ κ(〈〈x〉〉) and ψ = κ/α. Then ψ = τⁱ for some i. If M(N pⁿ, k, χ) denotes the Banach space of overconvergent modular forms of level Npⁿ, weight k and character χ (of some fixed yet to be determined radius), then, the map

is an isomorphism from M(N pⁿ, k, χ) onto the Banach space M(N p, 0, ψ) and thus the characteristic series of U on M(Npⁿ, k, χ) is the characteristic series of the operator G ↦ U₍₀₎ ((E_α(q)/E_α(q^p))G) acting on M(N p, 0, ψ). Since α(E_p(q)) = E_α(q)/E_α(q^p), the theorem follows.▪