Aug. 15, 1994

P-adic Banach Spaces and Families of Modular Forms Robert F. Coleman

This paper is dedicated to Bernard Dwork on the occasion of his seventieth birthday

0. Introduction Let p be a prime, C_{p} the completion of an algebraic closure of the p-adic numbers Q_{p} and K a finite extension of Q_{p} contained in C_{p}. Let v be the valuation on C_{p} such that v(p) = 1 and let | | be the absolute value on C_{p} such that |x| = p^{−v(x)} for x ∈ C_{p}.

Suppose N a positive integer prime to p. Let X_{1}(Np) denote the modular curve over K which represents elliptic curves with Γ_{1}(Np)-structure and let U_{p} be the Hecke operator on modular forms on X_{1}(Np) which takes a form with q-expansion ∑_{n} a_{n}q^{n} to the modular form with q-expansion ∑_{n} a_{np}q^{n}.

A modular form F is said to have slope α ∈ Q if there is a polynomial R(T) over C_{p} such that R(U_{p})F = 0 and such that the Newton polygon of R(T) has one side of slope −α.

Now (Z/NpZ)^{∗} acts on modular forms of level Γ_{1}(Np) via the diamond operators and we identify (Z/pZ)^{∗} with a subgroup of (Z/NpZ)^{∗} in the natural way. if χ is a C_{p}^{∗} valued character on (Z/pZ)^{∗}, we will say a modular form F is of character χ if F |⟨d⟩ = χ(d)F for d ∈ (Z/pZ)^{∗}. Let ω: (Z/pZ)^{∗} → μ_{p−1}(Q_{p}) denote the Teichmüller character.

For simplicity, now suppose p is odd. We prove in §8;

Theorem A. Suppose α ∈ Q and ϵ : (Z/pZ)^{∗} → C_{p}^{∗} is a character. Then there exists an M ∈ Z which depends only on p, N, ϵ and α with the following property: If k ∈ Z, k > α + 1 and there is a unique normalized cusp form F on X_{1}(Np) of weight k, character ϵω^{−k} and slope α and if k′ > α + 1 is an integer congruent to k modulo p^{M+n}, for any positive integer n, then there exists a unique normalized cusp form F′ of weight k′, character ϵω^{−k′} and slope α. Moreover, this form satisfies the congruence

F′(q) ≡ F (q) mod p^{n+1}.

Both F and F′ must be eigenforms for the full Hecke algebra of the respective weight because these algebras are commutative and therefore preserve the space of forms of a given slope. It is (a slight generalization of) a conjecture of Gouvêa-Mazur [GM-F] that M may be taken to be zero as long as and n ≥ α. When α = 0 this is a theorem of Hida [H-GR]. In this paper, we obtain no information about M.

The following example of this phenomenon was used by Serre in his work on p-adic L-functions [S]: Suppose k is a positive even integer. Then

\frac{L_{p}(1 − k, ω^{k})}{2} + ∑_{n≥1} σ_{k−1}^{∗}(n)q^{n},

where L_{p} is the p-adic L-function, and σ_{d}^{∗}(n) is the is the sum of the d-th powers of the divisors of n not divisible by p, is the q-expansion of a weight k normalized Eisenstein series G_{k}^{∗} on X_{1}(p) with trivial character and slope 0. For non-negative integers k, k′ and n such that k is even, k ≢ 0 mod p − 1, and k′ ≡ k mod (p − 1)p^{n}

G_{k}^{∗}(q) ≡ G_{k′}^{∗}(q) mod p^{n+1}.

(Note: In this case, ω^{k} = ω^{k′}.) The forms G_{k}^{∗} are not cuspidal and the failure of the members of the family of G_{k}^{∗} where k ≡ 0 mod p − 1 to satisfy this congruence gives some indication why we needed to make the cuspidal hypothesis in the statement of the theorem. In fact, we can remove the cuspidal hypothesis if we assume ϵ is non-trivial. (It goes without saying that if F is non-cuspidal F′ won’t be either.)

We may use the weight 12 level 1 modular form Δ and Theorem A to obtain other examples. Write

Δ(q) = ∑_{n≥1} τ(n)q^{n}.

Then τ(7) = −7 · 2392. The above theorem implies, for any positive integer k divisible by 6 and close enough 7-adically to 12, that there exists a unique normalized weight k level 1 cuspidal eigenform F_{k} over Q_{p} such that F_{k}|T_{7} = aF_{k} where a ∈ Z_{7} and v(a) = 1 (one first gets old eigenforms on X_{0}(7)). Moreover, for any positive integer n, if k is sufficiently close 7-adically to 12

F_{k}(q) ≡ Δ(q) mod 7^{n}.

The following is one important ingrediant in the proof of Theorem A:

Theorem B. For integers 0 ≤ i < p − 1 there exist series P_{N,i}(s, T) ∈ Z_{p}[[s, T]] which converge on the region |s| < |p^{(p−2)/(p−1)}| in C_{p}^{2} such that for integers k, P_{N,i}(k, T) is the characteristic series of Atkin’s U-operator acting on overconvergent forms of weight k and character ω^{i−k}.

This considerably strengthens the main result of [GM-CS].

Now let M_{k,cl} denote the space of classical modular forms and weight k on X_{1}(Np) defined over K. Then the eigenvalues of U_{p} on M_{k,cl} have valuation at most k − 1. For a character ϵ on (Z/pZ)^{∗} we also let M_{k,cl}(ϵ) denote the subspace of forms of weight k and character ϵ and set d(k, ϵ, α) equal to the dimension of the subspace of M_{k,cl}(ϵω^{−k}) consisting of forms of slope α.

As a corollary of Theorem 8.1 of [C-CO] we obtain:

Theorem C. With notation as above, the set of zeroes of P_{N,i}(k, T) of valuation > 1 − k is the same as the set of inverse eigenvalues with valuation > k − 1 of U_{p} acting on M_{k,cl}(ω^{i−k}) (counting multiplicity in both cases).

We are able to deduce, in §8, from this another result conjectured in more precise form by Gouvêa and Mazur:

Theorem D. If ϵ is a C_{p}^{∗}-valued character on (Z/pZ)^{∗}, and k and k′ are integers strictly bigger than α + 1 and sufficiently close p-adically

d(k, ϵ, α) = d(k′, ϵ, α).

Moreover, the closeness sufficient for this equality only depends on α.

Since Z_{p} is compact this implies what is called a “control theorem;” that is, for a fixed slope α, the dimension of the space of forms of a given weight and slope α is bounded independantly of the weight. We treat what are called R-families by Gouvêa and Mazur [GM-F] in §10. We also prove a consequence of the Gouvêa-Mazur conjectures; namely that the set of slopes of modular forms of level Γ_{1}(N) (and arbitrary weight) is a discrete subset of the real numbers in §8.

We are also able to deal with the prime 2. In the appendix, we show how to use the trace formulas of Eichler-Selberg and Reich-Monsky to prove that there exist overconvergent forms of any integral weight of arbitrarily large slope.

What foreshadows our proofs is the study of the k-th Hecke polynomial (see Eichler [E], Sato [Sa], Kuga [Ku] and Ihara [I])

H_{k}(u) = det((1 − T_{p}u + p^{k−1}u^{2})|S_{k})

where S_{k} is the space of weight k cusp forms of level one defined over C and T_{p} is the p-th Hecke operator. It was used to relate the Ramanujan-Petersson conjecture to the Weil conjectures (see the introduction to [I] for more history). Ihara applied the Eichler-Selberg trace formula to this effort. This line of research was continued by Morita, Hijikata and Koike ([M], [Hj], [Ko1] and [Ko2]). Dwork began another approach to the study of these polynomials using what is now known as the theory of overconvergent p-adic modular forms and also the Reich-Monsky trace formula ([D1] and [D2]). This work was continued by Katz [K] and Adolphson [A]. Both of these lines of research seem to have stopped in the mid-seventies. Hida [H1], [H2] developed to great utility the theory of “ordinary” modular forms which in this optic are modular forms of slope zero. In particular, he proved what may now be interpreted as the slope zero part of the aforementioned conjectures.

Our approach is a continuation of that introduced by Dwork. In fact, the inspiration for this paper arose in an attempt to to interpret Dwork’s paper “On Hecke polynomials” [D1] in terms of the point of view developed in [C-CO]. The key idea in Gouvêa-Mazur’s paper “On the characteristic series of the U operator” [GM-CS] provided the bridge between [D1] an [C-CO]. What we will do is show that the spaces of weight k overconvergent forms with their U operators naturally live in a family. We prove our main results by verifying in sections 1-5 that Serre’s p-adic Banach-Fredholm-Riesz theory [S] works in a family.

1. Families of Banach spaces Suppose A is a ring with unit, complete and seperated with respect to a non-

trivial ultrametric norm | | (see [BGR §1.2]). I.e., |1| = 1,

|a + b| ≤ Max{|a|, |b|}, |ab| ≤ |a||b|,

for a and b ∈ A, and moreover, |a| = 0 if and only if a = 0. We will call such a ring, a Banach algebra.

We will call an ultrametrically normed complete module E over A, such that that |ae| ≤ |a||e| if a ∈ A and e ∈ E a Banach module over A. Let A^{0} denote the subring of A consisting of elements a such that |a| ≤ 1 and E^{0} the A^{0} submodule in E consisting of all e such that |e| ≤ 1. An element a in A is called multiplicative if |ab| = |a||b| for all b ∈ A. We say | | is multiplicative if every element in A is (in [BGR] such a norm is called a valuation). Let A^{m} be the group of multiplicative units in A. Let I =: I(A) denote the set of finitely generated ideals I of A^{0} such that I^{n} is a basis of open neighborhood of the origin in A^{0}. We will suppose throughout this article that I ≠ ∅. Clearly, if there exists an a ∈ A^{m} such that |a| < 1, then aA^{0} ∈ I. We will also frequently make the following hypothesis:

Hypothesis H. |A^{m}| ≠ 1 and

|A^{m}| ∪ {0} = |A|.

If there is an isometry of a multiplicatively normed field K into A such that the image of K^{∗} is contained in A^{m}, we will call A a K-Banach algebra.

An orthonormal basis for a Banach module E over A is a set {e_{i} : i ∈ I} of elements of E for some index set I such that every element m in E can be written uniquely in the form ∑_{i∈I} a_{i}e_{i} with a_{i} ∈ A such that lim_{i→∞} |a_{i}| = 0 and

|m| = Sup{|a_{i}| : i ∈ I}.

We say E is orthonormizable if it has an orthonormal basis. Clearly if E is orthonormizable |E| = |A| and |ae| = |a||e| if a ∈ A is multiplicative.

Lemma 1.0. If E is orthonormizable, and I ∈ I, I^{n}E^{0} is a basis of open neighborhoods of the origin in E^{0}.

Proof. For ϵ ∈ R, ϵ > 0, let E(ϵ) = {e ∈ E: |e| < ϵ}. What we must prove is that the collections {E(ϵ)} and {I^{n}E^{0}} are cofinal. Since the collection {I^{n}}_{n≥0} is a basis of neighborhoods of 0 in A^{0}, it is clear that if ϵ > 0, there exist an n ∈ Z, n ≥ 0 such that I^{n}E^{0} ⊆ E(ϵ). Now suppose n ∈ Z, n ≥ 0. Then there exists an ϵ > 0 such that A(ϵ) ⊆ I^{n}. Claim: E(ϵ) ⊆ I^{n}E^{0}. Suppose a_{1}, …, a_{m} generates I^{n} over A^{0} and {e_{i}: i ∈ J} is an orthonormal basis for E. Let e ∈ E(ϵ). Write,

e = ∑_{J} b_{i}e_{i}.

Then b_{i} ∈ I^{n} for all i. In fact, since |b_{i}| → 0, b_{i} ∈ I^{n+m(i)} where m(i) ≥ 0 and m(i) → ∞ as i → ∞, so we may write

b_{i} = ∑_{j=1}^{m} c_{ij}a_{j}

where c_{ij} ∈ I^{m(i)}. It follows that |c_{ij}| → 0 as i → ∞. Hence

e = ∑_{j=1}^{m} a_{j} ∑_{i∈J} c_{ij}e_{i}

and the inner sums converge in E. Thus e ∈ I^{n}E^{0}.

Using this we see that if S = {e_{i}} is an orthonormal basis for E and I ∈ I then the reduction of S modulo I^{n}E^{0} is an algebraic basis for E^{0}/I^{n}E^{0} over A^{0}/I^{n}A^{0}.

One has using the same line of reasoning as in the proof of [S, Lemma 1].

Lemma 1.1. Suppose (A, | |) satisfies hypothesis H and |A| = |E|. Then a subset S of E is an orthonormal basis for E if and only if S ⊂ E^{0} and the image of S in E^{0}/aE^{0} is a basis of this module over A^{0}/aA^{0} for some a ∈ A^{m}, |a| < 1.

Suppose M and N are Banach modules over A. Then we put a semi-norm on N ⊗_{A} M by letting |c|, for c ∈ N ⊗_{A} M, equal the greatest lower bound over all representations c = ∑_{i} n_{i} ⊗ m_{i} of

Max_{i}{|n_{i}||m_{i}|}.

We then let N ˆ⊗_{A}M denote the completion of N ⊗_{A} M with respect to this seminorm. If B is a complete normed A algebra such that the structural morphism from A to B is contractive, then B is a Banach module over A and B ˆ⊗_{A}M is naturally a Banach module over B.

Proposition 1.2. If {e_{i}} is an orthonormal basis for M over A then {1 ⊗ e_{i}} is an orthonormal basis for B ˆ⊗_{A}M over B.

Proof. Let N = B ˆ⊗_{A}M. First, every element n of N can be written in the form ∑_{i} b_{i} ⊗_{A} e_{i} where b_{i} ∈ B and b_{i} → 0. We claim |n| = Sup|b_{i}| := P. We have,

|n| = lim_{S} |n_{S}|

where

n_{S} = ∑_{S} b_{i} ⊗ e_{i}

and S ranges over finite sets. It follows that |n| ≤ P. Now, fix j ∈ I and let h: M → A be the A homomorphism which takes ∑_{I} a_{i}e_{i} to a_{j}. Then h is continuous, in fact |h(m)| ≤ |m|. Let h_{B} denote the extension by scalars of h to a morphism from B ⊗_{A} M to B. Suppose j ∈ S. Then h(n_{S}) = b_{j}. If n_{S} = ∑_{i} c_{i} ⊗ m_{i} where c_{i} ∈ B and m_{i} ∈ M. Then b_{j} = h_{B}(n_{S}) = ∑_{i} c_{i}h_{B}(m_{i}). Hence, using the contractivity of A → B,

|b_{j}| ≤ Max_{i}{|c_{i}|_{B}|h_{B}(m_{i})|_{B}} ≤ Max_{i}{|c_{i}|_{B}|h(m_{i})|_{A}} ≤ Max_{i}{|c_{i}|_{B}|m_{i}|}.

Since this is true for all representations of n_{S}, all j ∈ S and all S, it follows that P ≤ |n|. This establishes the claim.

The remainder of the proposition follows easily.

If J is a closed ideal of A for a ∈ A/J we set

|a|_{J} = Inf{b : b ∈ A, b ≡ a mod J}.

Then | |_{J} is a norm on A/J which we call the residual norm. We set E_{J} = (A/J) ˆ⊗E. If J is finitely generated and E is orthonormalizable, E_{J} = E/JE. We note that, since A is complete, maximal ideals are automatically closed [BGR 1.2.4/5]. Since the map A → A/J is contractive, as an immediate corollary of the previous proposition, we obtain;

Corollary 1.2.1. If J is a closed ideal of A, {1 ⊗ e_{i}} is an orthonormal basis for E_{J} over A/J.

Example. We will later see many examples of Banach algebras satisfying hypothesis H, but one which does not and will be very important for us in the future is the Iwasawa algebra, Λ =: Z_{p}[[Z_{p}^{∗}]]. One complete norm on Λ may be described as follows: Suppose κ: Z_{p}^{∗} → C_{p}^{∗} is a continuous character. Then κ extends by linearity and continuity to a continuous ring homomorphism of Λ into C_{p}. For λ ∈ Λ, let

|λ| = sup_{κ} |κ(λ)|

where κ runs over the set of C_{p}^{∗} valued characters on Z_{p}^{∗} which are analytic on 1 + qZ_{p}, where q = 4 if p = 2 and p otherwise. Then | | is a complete multiplicative norm on Λ, Λ^{0} = Λ, |Λ^{m}| = 1 and (p, 1 − [1 + q]) ∈ I(Λ). Homomorphisms.

If M and N are Banach modules over A, and L: M → N is a continuous Ahomomorphism we set

|L| = sup_{m≠0} \frac{|L(m)|}{|m|}.

This determines a topology on the set of continuous A-homomorphisms. The homomorphism L is said to be completely continuous if

L = lim_{j→∞} L_{j}

where L_{j} is a continuous A-homomorphism from M to N whose image is contained in a finitely generated submodule of N. Let C(M, N) denote the Banach module of completely continuous A-homomorphisms from M to N.

It is easy to see;

Lemma 1.3. If A → B is a contractive map of Banach algebras and L: M → N is a completely continuous homomorphism between Banach modules over A, then 1 ˆ⊗L: B ˆ⊗M → B ˆ⊗N is a completely continuous homomorphism over B.

Remarks. (i) In general, (A^{0})^{∗} = {a ∈ A^{m}: |a| = 1}. (ii) When A is field, | | is multiplicative and |A^{∗}| ≠ {1} the above is the theory discussed by Serre [S]. However, Serre’s theory works without change even when |A^{∗}| = {1}. It’s only easier. Indeed, in this case, an orthonormal basis is a basis and a completely continous linear map is a linear map of finite rank. (iii) When A is a field, | | is multiplicative, |A^{∗}| ≠ 1, and V is a finite dimensional subspace of M, then Serre proves that there exists a continuous projector from M onto V with norm less than 1 whose kernel is orthonormalizable. We cannot prove this in our general context. (iv) If L: M → N is as above and f: M′ → M and g: N → N′ are continuous A-homomorphisms of A-Banach modules then g ∘ L ∘ f is completely continuous.

Suppose {e_{i}}_{I} is an orthonormal basis for M and {d_{j}}_{J} is an orthonormal basis for N. Suppose

L(e_{i}) = ∑_{j} n_{i,j}d_{j}.

Then as in Serre

Lemma 1.4. The linear map L is completely continuous if and only if

lim_{j→∞} Sup_{i∈I}|n_{i,j}| → 0.

Proof. First suppose the matrix for L is as above. Then for each finite set S of J let

L_{S}(e_{i}) = ∑_{j∈S} n_{i,j}d_{j}.

It is clear that the L_{S} converge to L.

Another way to formulate this lemma is as follows: For S ⊂ I, let π_{S}: E → E be the projector

∑_{i∈I} a_{i}e_{i} ↦ ∑_{i∈S} a_{i}e_{i}.

Then, the lemma asserts L is completely continuous if and only if the directed set {π_{S} ∘ L}, where S ranges over finite subsets of I, converges to L.

Now suppose L is completely continuous. Then for each ϵ > 0 there exists an Alinear map L′: M → N whose image is contained in a finitely generated submodule N′ and is such that |L − L′| < ϵ. Since N′ is finitely generated there exists a finite subset T of J such that if π_{T} is the projection from N onto the span of {d_{j}}_{j∈T} |π_{T} |_{N′} − id_{N′}| < ϵ. It follows that

|L − π_{T} ∘ L′| < ϵ.

This implies |n_{i,j}| < ϵ for j ∉ T which concludes the proof.

Alternatively: For an orthonormalizable Banach module E, let E′ denote the continuous dual of E with the norm | |′ defined by

|h|′ = sup{|h(x)|: x ∈ E^{0}}

for h ∈ E′. This is well defined and if S is an orthonormal basis for E, |h|′ = sup{|h(e)|: e ∈ S}.

Lemma 1.5. If M and N are orthonormalizable Banach modules over A, M′ ˆ⊗N is naturally isomorphic to C(M, N).

Proof. Suppose {e_{i}}_{I} is an orthonormal basis for M and {d_{j}}_{J} is an orthonormal basis for N.

We can write any y ∈ M′ ˆ⊗N, uniquely, as

∑_{J} h_{j} ⊗ d_{j}

where h_{j} ∈ M′, |h_{j}|′ → 0. Now if m ∈ M, we set

y(m) = ∑_{J} h_{j}(m)d_{j}.

This clearly well defines a linear map from M to N and, since |h_{j}|′ → 0, is completely continuous by the previous lemma.

Now let e′_{i} be the element of E′ such that e′_{i}(e_{j}) = δ_{i,j}. We can represent any h ∈ M′ as ∑_{I} a_{i}e′_{i} where a_{i} ∈ A and the set {|a_{i}|: : i ∈ I} is bounded. If, on the other hand, L ∈ C(M, N) has the matrix (n_{i,j})_{I,J} let y = ∑_{J}(∑_{I} n_{i,j}e′_{i})⊗d_{j} which, using Lemma 1.4, we see is an element of M′ ˆ⊗N. Clearly, y maps to L.

The map M′ ˆ⊗N to C(M, N) is independant of the choice of the bases because it is the natural map on M′ ⊗ N and is continuous.

We say two norms on a ring are equivalent, if they induce the same topology. We say a normed ring A satisfies hypthesis J if:

Hypothesis J. The intersection of the maximal ideals of A is 0 and if m is a maximal ideal, m is finitely generated and the residual norm on A/m is equivalent to a multiplicative norm.

Examples. If A is a reduced affinoid algebra over a complete multiplicatively normed field and the norm on A is the spectral norm, then A satisfies hypothesis J (see [BGR Prop. 6.1.1/3 and Cor. 6.1.2/3]).

We note that if a norm | | on a field is equivalent to a multiplicative norm | |′, then there exists a positive constant c such that | |′ ≤ c| | (see [B]). We do not know an example of a complete normed field whose norm is not equivalent to a multiplicative norm.

2. The Fredholm determinant Suppose A is a Banach algebra and E is a Banach module over A with an

orthonormal basis B. If L is a completely continuous operator on E, and either |L| ≤ 1 or |A^{m}| ≠ 1, one can translate the discussion in Serre to produce a characteristic series P_{L}(T) of L, with respect to B, which morally is det(1−T L). The key point is to first suppose |L| ≤ 1 and then observe, if I ∈ I, that Lemma 1.4 implies L(E^{0}) mod I^{n}E^{0} is contained in a free direct factor of E^{0}/I^{n}E^{0} of finite rank over A^{0}/I^{n}A^{0}. We can also prove: Theorem 2.1. If L has norm at most |a| where a ∈ A^{m} then P_{L}(T) is an element of A^{0}[[aT]] and is entire in T (i.e. if P_{L}(T) = ∑_{m≥0} c_{m}T^{m}, |c_{m}|M^{m} → 0 for any real number M). Also, P_{L}(T) is characterized by:

(i) If {L_{n}}_{n≥0} is a sequence of completely continuous operators on E, either |L_{n}| ≤ 1 or |A^{m}| ≠ 1 and L_{n} → L then P_{L_{n}} → P_{L} coefficientwise.

(ii) If the image of L in E is contained in an orthonormizable direct factor F of finite rank over A of E such that the projection from E onto F has norm at most 1 then

P_{L}(T) = det(1 − T L|F).

Proof. This follows by translating the arguments in [S]. E.g. suppose the hypotheses of (ii) Let π: E → F be the projection. After changing L by a homothety in A^{m} if necessary we may assume |L| ≤ 1. Let F^{0} = {x ∈ F: |x| ≤ 1}. Let I be an element of I. Then, since |π| ≤ 1, F_{I} =: F^{0}/IF^{0} injects onto a free direct factor of finite rank in E^{0}/IE^{0} over A^{0}/I. It follows that

P_{L}(T) ≡ det(1 − T L|F_{I}) mod I.

Assertion (ii) follows.

We will also use the notation det(1 − T L) for P_{L}(T).

Remark. It follows from (i) and (ii) of the theorem that P_{L}(T) does not depend on the choice of the orthonormal basis but it may depend on the norm on E and not just the topology (see below).

Just as in [S], one may deduce;

Corollary 2.1.1. If u and v are completely continuous operators on E,

det(1 − T u) det(1 − T v) = det((1 − T u)(1 − T v)).

Proposition 2.2. Suppose E_{1} and E_{2} are orthonormalizable Banach modules over A. Suppose u is a completely continuous homomorphism from E_{1} to E_{2} and v: E_{2} → E_{1} is a continuous homomorphism. Then

P_{u∘v}(T) = P_{v∘u}(T).

Proof. Recall, for a Banach A-module E, E′ denotes the continuous A-module dual of E, so that if S is an orthonormal basis of E, E′ may be identified with the set of bounded A-valued functions on S. The point is that the natural diagram

E′_{1} ˆ⊗E_{2} ⟶^{v′ ˆ⊗id} E′_{2} ˆ⊗E_{2}
↓_{id ˆ⊗v}  ↓
E′_{1} ˆ⊗E_{1} ⟶  A

commutes and u may be identified with an element of E′_{1} ˆ⊗E_{2} by Lemma 1.5 (see also [S, §4]). The result follows from the fact that the bottom and right arrows may be interpreted as traces.

Lemma 2.3. If M = N ⊕ F where N and F are closed orthonormalizable submodules of M stable under L. Then

P_{L}(T) = P_{L|_{N}} (T)P_{L|_{F}} (T).

It follows easily using Proposition 1.2 and Lemma 1.3 that Lemma 2.4. Suppose ϕ : A → B is a contractive map of Banach algebras, then

ϕ(det(1 − T L|E)) = det(1 − T (L ˆ⊗1)|B ˆ⊗_{A}E)

Proposition 2.5. Suppose A satisfies hypothesis J, E is an orthonormalizable Banach module over A and L is linear operator on E whose image in E is contained in a free direct factor F of finite rank such that the projector from E onto F is continuous. Then P_{L}(T) = det(1 − T L_{F} |F).

Proof. Let {e_{i}} be an orthonomal basis for E. Let m be a maximal ideal of A, k = A/m and | |_{m} a multiplicative norm on k equivalent to the residue norm. Then the natural map from A to k is contractive with respect to this norm so 1 ⊗ e_{i} is an orthonormal basis for k ˆ⊗E. Also, if π: E → F is a continuous procector id ⊗ π: k ⊗ E → k ⊗ F is a continuous projector and the elements 1 ⊗ π(e_{i}) are bounded in k ⊗ F. Since m is finitely generated, k ⊗ E = k ˆ⊗E.

The result now follows from Lemma 2.4 applied to B = A/m and Remark 1) of [S, §5] and [S, Prop. 7d)].

This proposition together with part (i) of the theorem implies that

Corollary 2.5.1. When A satisfies hypothesis J, P_{L}(T) only depends on the topology of E.

We do not know whether this is true more generally.

Example. Suppose A is a Banach algebra, M is an orthonormalizable Banach module over A and u and v are two completely continuous operators on M over A. Then if A⟨X, Y ⟩ is the ring of restricted power series over A, the operator Xu + Y v is a completely continuous on M̃ =: M ˆ⊗A⟨X, Y ⟩ over A⟨X, Y ⟩ (which is given the Gauss Norm). Hence we have a characteristic series P_{u,v}(X, Y, T) = det ((1 − T (Xu + Y v))|M̃) such that

P_{u,v}(x, y, T) = det ((1 − T (xu + yv))|M)

whenever x, y ∈ A and both |x| and |y| are at most 1. Clearly when A^{m} ≠ 1, P (X, Y, T) continues to a series entire in X and Y.

Now suppose A is an algebraically closed field with a multiplicative norm. Then if u and v commute

P_{u,v}(X, Y, T) = ∏_{i} (1 − (a_{i}X + b_{i}Y)T )

where a_{i} and b_{i} are elements of A which tend to zero. This is a consequence of the fact that the generalized eigenspaces of u are stablized by v and vice versa.

3. Resultants Suppose (A, | |) is a Banach algebra and |A^{m}| ≠ 1. We let A[[T]]^{ent} denote the set of power series ∑_{n=0}^{∞} a_{n}T^{n} over A such that

|a_{n}|M^{n} → 0 for all M ∈ R.

Lemma 3.1. If G(T) is a polynomial whose leading coefficient is multiplicative and H(T) ∈ A⟨T ⟩ such that G(T)H(T) ∈ A then G(T) ∈ A or H(T) = 0. Proof. Let a ∈ A^{m}, |a| > 1. Replacing G(T) by G(a^{M}T) for some positive integer M we may assume that the absolute value of the leading coefficient c of G is greater than all its other coefficients. Suppose n = deg(G) > 0 and H ≠ 0. Suppose H(T) = ∑_{k} b_{k}T^{k} and m ≥ 0 is such that |b_{m}| ≥ |b_{k}| for all k with strict inequality for k > m. It follows that the coefficient of T^{n+m} has absolute value equal to |cb_{m}| = |c||b_{m}| ≠ 0. Remark. This is also true if we suppose instead of the hypothesis G(T) ∈ A[T] that G(T) ∈ A[[T]]^{ent} and either all the coefficients of G are multiplicative or A satisfies hypothesis J.

Suppose

F (e_{1}, …, e_{n}) = ∑_{I} B_{I}e^{I}

where I runs over n-tuples of positive integers (i_{1}, …, i_{n}) and

e^{I} = e_{1}^{i_{i}} ⋯ e_{n}^{i_{n}}.

Then we want |B_{I}|M^{s(I)} to go to zero as I → ∞ for all M ∈ R. This defines A[[e_{1}, …, e_{n}]]^{ent}. Let

Q(T) = T^{n} − a_{1}T^{n−1} + ⋯ + (−1)^{n}a_{n}

be an element of A[T].

Lemma 3.2. If S(e_{1}, …, e_{n}) ∈ (Q(T_{1}), …, Q(T_{n})) ∩ A[[e_{1}, …, e_{n}]]^{ent} where the e_{i} are the elementary symmetric polynomials in the T_{i} then S(a_{1}, …, a_{n}) = 0.

Suppose P (T) ∈ A[[T]]^{ent} and P (T_{1}) ⋯ P (T_{n}) = H(e_{1}, … e_{n}). Then we defiine the resultant of Q and P to be

Res(Q, P) = H(a_{1}, …, a_{n}).

(See also [L, V §10].) Then

Res(Q, 1) = 1  (0)
Res(Q, aP) = a^{n}Res(Q, P)  (1)
Res(Q, P R) = Res(Q, P)Res(Q, R)  (2)
Res(Q, P + BQ) = Res(Q, P)  (3)

if a ∈ A and R, B ∈ A[T]^{ent}. If P (T) = ∑_{n≥0} b_{n}T^{n}. One can show Res(Q, P) is the limit as m goes to infinity of the determininants of the (n + m) × (n + m) matrices:

\begin{matrix}m \{ \\ n \{ \end{matrix} ( \begin{matrix}1 & −a_{1} & … & … & (−1)^{n}a_{n} \\ & 1 & −a_{1} & … & … & (−1)^{n}a_{n} \\ & & … & … & … & … & … \\ & & & 1 & −a_{1} & … & … & (−1)^{n}a_{n} \\ b_{m} & b_{m−1} & … & … & b_{0} \\ & b_{m} & b_{m−1} & … & … & b_{0} \\ & & … & … & … & … & … \\ & & & b_{m} & b_{m−1} & … & … & b_{0} \end{matrix} )

in which there are m rows of a’s and n rows of b’s. If S is a monic polynomial of degree m

Res(SQ, P) = Res(S, P)Res(Q, P)  (4)
Res(Q, S) = (−1)^{mn}Res(S, Q)  (5)
Res(Q, S^{∗}) = Res(S, Q^{∗})  (6)

where if F (T) is a polynomial of degree d, F^{∗}(T) = T^{d}F (T^{−1}). We can also interpret the resultant as a norm. Indeed, consider the extension B =: A[T]^{ent}/(Q(T)) of A. This extension is isomorphic to A[T]/(Q(T)) which is finite and free and the resultant of Q and P is the norm of the image of P in B to A. Lemma 3.3. The resultant of Q and P is a linear combination of Q and P. If Q and P have a non-constant polynomial common factor G whose leading term is multiplicative, then the resultant of Q and P is zero. Proof. When P is a polynomial, the first statement follows from [L Chapt. V, §10]. In general, we can write P as BQ + R where R is a polynomial and B ∈ A[[T]]^{ent} and then apply formula (3) above.

Now it follows that G(T) divides the resultant. However, the resultant lies in A, and this and Lemma 3.1 implies the resultant is zero.

Remark. By the previous remark, the conclusion of this lemma is still true if we only assume G(T) ∈ A[[T]]^{ent} as long as A satisfies hypothesis J. Lemma 3.4. Res(Q, P) is a unit if and only if Q and P are relatively prime. Proof. One direction follows imediately from the previous lemma. Therefore suppose fQ + gP = 1 where f, g ∈ A[[T]]^{ent}. Then using (0), (2) and (3)

1 = Res(Q, fQ + gP) = Res(Q, gP) = Res(Q, g)Res(Q, P).

Suppose B and P are polynomials over A and

P (T) = 1 − a_{1}T + ⋯ + (−1)^{n}a_{n}T^{n}

Then we set

D(B, P)(T) = ∏_{i=1}^{n}(1 − T B(T_{i})))_{e_{i}=a_{i}}.

If B and P are in A[[T]]^{ent}, B(0) = 0 and P (0) = 1 then we set

D(B, P)(T) = lim_{n→∞} D(B_{n}, P_{n})(T)

where, for an element F (T) = ∑_{k=0}^{∞} c_{k}T^{k} ∈ A[[T]], F_{n}(T) = ∑_{k=0}^{n} c_{k}T^{k}. It is easy to see that D(B, P)(T) ∈ A[[T]]^{ent}. Moreover, Lemma 3.5. If P (T) = R(T)S(T), R, S ∈ A[[T]]^{ent} and R(0) = S(0) = 1, then we have,

D(B, P) = D(B, R)D(B, S).  (7)

and if Q is a monic polynomial,

D(1 − Q^{∗}, P)(1) = Res(Q, P).  (8)

Proof. The first formula is obvious. For the second, observe that it follows from the definitions and (6) that

D(1 − Q^{∗}, P_{n})(1) = Res(T^{n}P_{n}(T^{−1}), Q^{∗}(T))
= Res(Q, P_{n}).

Hence the lemma follows by taking a limit.

Theorem 3.6. If u is a completely continuous operator on an orthonormalizable Banach module E over A and B ∈ T A[[T]]^{ent} then

P_{B(u)}(T) = D(B, P_{u})(T).  (9)

Proof. We point out that B(u) is completely continuous by remark (iv) of §1 so P_{B(u)} makes sense. We may apply a homothety and assume that the norms of u and B are at most one. Suppose I ∈ I. Consider the operator u_{I} induced by u on E_{I} =: E^{0}/IE^{0}. The corresponding formula is true for det(1 − u_{I}T |E_{I}) which is congruent to P_{u}(T) modulo I. Hence (9) follows by a limiting argument.

4. Riesz Theory Suppose (A, | |) satisfies hypothesis H. Let u be a completely continuous operator on an orthonormalizable Banach module E over A. As in Serre, we can define the Fredholm resolvant R(T, u) of u and use it and the theory of resultants to prove:

Lemma 4.0. Suppose Q(T) ∈ A[T] is a monic polynomial. Then Q and P_{u} are relatively prime if and only if Q^{∗}(u) is an invertible operator on E.

Proof. Let v = 1 − Q^{∗}(u). Then v is completely continuous and we have

(1 − vT)R(T, v) = P_{v}(T) = D(1 − Q^{∗}, P_{u})(T)

by Theorem 3.6 and so using Theorem 3.5

Q^{∗}(u)R(1, v) = (1 − v)R(1, v) = Res(Q, P_{u}).

Thus it follows from Lemma 3.4 that if Q and P_{u} are relatively prime, Q^{∗}(u) is invertible. If on the other hand the exists an operator w on E such that Q^{∗}(u)(1 − w) = 1, then we find that w is completely continuous and we deduce using Corollary 2.1.1

det(1 − v) det(1 − w) = 1

but by Theorem 3.6 and Lemma 3.5,

det(1 − v) = D(1 − Q^{∗}, P_{u})(1) = Res(Q, P_{u}).

Hence Q and P_{u} are relatively prime by Lemma 3.4.

Let Δ^{s} denote the operator on power series in T which takes ∑_{n} a_{n}T^{n} to ∑_{n} \binom{n}{s}a_{n}T^{n−s}. Suppose a ∈ A. Then we say a is a zero of H(T) ∈ A[[T]]^{ent} of order h if Δ^{s}H(a) = 0 for s < h and Δ^{h}H(a) is invertible.

Using the previous lemma and following the same line of reasoning as in [S, §7] one obtains: Proposition 4.1. Suppose a ∈ A is a zero of P_{u}(T) of order h. Then we have a unique decompositon

E = N(a) ⊕ F (a)

into closed submodules such that 1−au is invertible on F (a) and (1−au)^{h}N(a) = 0. Theorem 4.2. Suppose P_{u}(T) = Q(T)S(T) where S ∈ A[[T]]^{ent} and Q is a polynomial of whose leading coefficient is a unit which is relatively prime to S. Then there is a unique direct sum decomposition

E = N(Q) ⊕ F (Q)

of E into closed submodules such that Q^{∗}(u)N(Q) = 0 and Q^{∗}(u) is invertible on F (Q). Proof. We note that Q(0)S(0) = 1. Let B(T) = 1 − Q^{∗}(T)/Q^{∗}(0) and v = B(u). Then, by (3.7)

P_{v} = D(B, P_{u}) = D(B, Q/Q(0))D(B, S/S(0)).

We have D(B, Q/Q(0))(T) = (1 − T)^{n} and

D(B, S/S(0))(1) = Res(Q/Q^{∗}(0), S/S(0))

by (3.8) which is a unit using Lemma 3.4. Now apply Proposition 4.1 to the operator v and the zero 1 of P_{v}(T).

Remarks. (1) Let B_{Q} = A[X]/Q^{∗}(X) ≅ A[Y]/Q(Y). Then N(Q) is a B_{Q} module, via

Xn = un

for n ∈ N(Q). (2) Following Serre we have explicit formulas for the projectors from E onto the subspaces N(Q) and F (Q). For example, let v be as above, then

(\frac{(1 − v)Δ^{n}R(1, v)}{Δ^{n}P_{v}(1)})^{n}

is a formula for the projector onto F (Q) with kernel N(Q).

Theorem 4.3. Suppose A satisfies hypothesis J and Q has degree r. Then under the hypotheses of Theorem 4.2 the A module N(Q) is projective of rank r.

Proof. First suppose A is a field and | |_{m} is a multiplicative norm on A equivalent to | |. Let | |_{m} also denote the norm on E with respect to the contractive map (A, | |) → (A, | |_{m}). Then by Corollary 1.2.1, an orthonormal basis for E with respect to | | is an orthonormal basis with respect to | |_{m}. Since the topology on E with respect to | |_{m} is the same as the topology on E with respect to | | the result of Serre applies and establishes our result in this case.

Let N = N(Q) and F = F (Q). Let R be a maximal ideal of A. Then because E = N + F, E_{R} = N_{R} + F_{R} and Q^{∗}(u) is zero on N_{R} and invertible on F_{R} so that this decomposition is the one established by Theorem 4.2. It follows from the above and the hypotheses on A that that N_{R} is a vector space of dimension r over k_{R}, the residue field at R. Now, let

f_{i} = ∑_{j∈I} a_{i,j}e_{j}

1 ≤ i ≤ n be elements of N which form a basis of N_{R} modulo R. Then, there exist j_{1}, …, j_{r} in I such that

g = det ((a_{i,j_{k}})_{i,k})

is not zero at R. Let U be the affine open subscheme of Spec(A) where g is invertible. It follows that the f_{i} are a basis for N_{P} for every closed point P of U. We claim {f_{i}} is a basis for N_{U}.

Indeed let h ∈ N_{U}. Then because g is invertible on U, there exist a_{i} ∈ A_{U} such that the coefficient of e_{j_{k}} in the expansion of

a_{1}f_{1} + ⋯ + a_{r}f_{r} − h

is zero for 1 ≤ k ≤ r. If follows that this element vanishes at every closed point P of U. Thus by the hypotheses this element vanishes on U. If h = 0, it follows that the a_{i} vanish at every closed point P in U and hense a_{i} = 0 for all i. Thus N is locally free, so projective.

Corollary 4.3.1. Suppose A satisfies hypothesis J. If B_{Q} is étale over A (i.e., if (Q(T), ΔQ(T)) = 1) then N(Q) is a locally free B_{Q} module of rank 1.

Proof. This is true when A is a field. It follows more generally when A satisfies hypothesis J by the same kind of resoning which established the theorem.

Corollary 4.3.2. If A is a reduced affinoid algebra and the norm on A is the spectral norm, then the conclusion of the above theorem holds.

Proof. Indeed, as we have pointed out, such rings sastisfy hypothesis J.

In fact, in this case, we can strengthen Corollary 4.3.1. Suppose A is a reduced affinoid algebra and B_{Q} is étale over A. Then B_{Q} is also a reduced affinoid algebra and the spectral norm on B_{Q} extends the spectral norm on A. Let E_{Q} = B_{Q} ˆ⊗E and let u also denote the completely continuous operator u ⊗ 1 on E_{Q}. Then Y is a zero of P_{u}(T) of order 1 as

ΔP_{u}(Y) = ΔQ(Y)S(Y)

which is a unit since B_{Q} is étale over A and (Q, S) = 1 so the subspace N(T − Y) of E_{Q} is locally free of rank one over B_{Q}. In other words, locally on B_{Q} there exists a generator F over B_{Q} of N(T − Y) which satisfies

uF = XF.

This is the genesis of our work on R-families (see §7).

More generally, suppose Q = R^{m} where m deg R = deg Q, R^{∗}(u)N(Q) = 0 and B_{R} is étale over A. Let C = (B_{R})^{m}. Then C is a reduced affinoid algebra and the spectral norm on Y extends the specral norm on A.

5. Rigid Theory Let K be either C_{p} or a complete discretely valued subfield of C_{p} and | | be the absolute value on K such that |p| = p^{−1} (or more generally we may suppose that K is a complete stable valued field (see [BGR §3.6.1 Def. 1])). Let R = {a ∈ K: |a| ≤ 1} be the ring of integers in K and ℘ = {a ∈ R: |a| < 1} the maximal ideal of R.

If Y is a rigid space over K, A(Y) will denote the ring of rigid analytic functions on Y, we let | | also denote the spectral norm on A(Y) and A^{0}(Y) will denote the subring of power bounded functions {f ∈ A(Y): |f| ≤ 1} on Y. The spectral norm is a non-trivial ultrametric norm on A(Y) if A(Y) is reduced [BGR Prop. 6.2.1/4]. Also, we set t(A(Y)) = {f ∈ A(Y) : |f| < 1}, the topologically nilpotent elements of A(Y) and Ȳ = Spec(A^{0}(Y)/t(A(Y))). In general, if X → Y is a morphism of rigid spaces and Z is a subspace of Y, then X_{Z} will denote the pullback of X to Z (the “fiber” of X → Y over Z).

Suppose X → Y is a morphism of reduced affinoids over K. Then (A(Y), | |) is a Banach algebra and (A(X), | |) is a Banach module over (A(Y), | |).

If A^{0}(Y)/℘A^{0}(Y) is reduced then |A(Y)| = |K| so (A(Y), | |) satisfies hypothesis H. In this case, ℘A^{0}(Y) = t(A(Y)) so the reduction of Y, Ȳ, equals Spec(A^{0}(Y)/℘A^{0}(Y)) =: Ỹ. If Y is reduced, this occurs after a finite base extension. We will suppose for the rest of this section that Y is a reduced affinoid such that Ỹ is also reduced.

One can show, using Lemma 1.1; Lemma 5.1. Suppose K is discretely valued, X → Y is a morphism of reduced affinoids over K and A^{0}(X)/℘A^{0}(X) is free over A^{0}(Y)/℘A^{0}(Y). Then the Banach module A(X) over A(Y) is orthonormalizable.

The simplest case of this phenomenon occurs when X = Z×_{K}Y where Z is a reduced affinoid over K. This will, in fact, be the case of interest for us.

Definition. If f: Z → X is a morphism of affinoids over Y then we say, f is inner if the image of \overline{Z} in \overline{X} is finite over \overline{Y}. This a slight generalization of Kiehl’s notion of inner which is called relatively compact in [BGR §9.6.2].

Proposition 5.2. Suppose f: Z → X is an inner map of reduced affinoids over Y, X̃ is reduced and A(X) is orthonormalizable over A(Y). Then the map f^{∗} from A(X) to A(Z) of Banach modules over A(Y) is completely continuous.

Proof. Let B = A^{0}(Y), C = A^{0}(Z) and D = A^{0}(X). Let x_{1}, …, x_{n} be elements of D such that the map from B⟨T_{1}, …, T_{n}⟩, T_{i} ↦ x_{i} is surjective onto D (this exists by [BGR, Thm. 6.4.3/1] using the fact that under our hypothesis |D| = |K|). Let π be a uniformizing parameter of R. Then the hypotheses imply that there exist monic polynomials g_{i}(S) ∈ A[S] such that f^{∗}g_{i}(x_{i}) ∈ πC. We can write any element of D as

∑_{I,N} a_{I,N}x^{I}g(x)^{N},

where x = (x_{1}, …, x_{n}), g = (g_{1}, …, g_{n}), I and N are multi-indices ordered lexographically, I < deg(g) and a_{I,N} ∈ R. It follows that the image of D in C/π^{n}C is spanned by the images of f^{∗}(x^{I}g(x)^{N}) where I < deg(g) and ∑ N < n. Now let {e_{i}}_{i∈I} be an orthonormal basis for A(X) over A(Y). (We may work locally on Y.) Now e_{i} ∈ D. Let F_{i,n} be an element in the B-span of {f^{∗}(x^{I}g(x)^{N}): I < deg g and ∑ N < n} such that F_{i,n} ≡ f^{∗}e_{i} mod π^{n}C. There exists a unique continuous B-linear map L_{n}: A(X) → A(Z) such that L_{n}(e_{i}) = F_{i,n}. Then L_{n} converges to f^{∗} and the image of L_{n} is contained in a submodule of C finitely generated over A(Y).

We will also need in sections 9 and 10, the following definition of relative overcvonvergence: Definition. If X → Y is a morphism of rigid spaces over K, we say that X is affinoid over Y if for each open affinoid Z in Y, X_{Z} is an affinoid. Suppose W → Y is a map of rigid spaces and X ⊆ W is affinoid over Y, then we say that a rigid space V ⊆ W is a strict neighborhood of X over Y in W if there exists a neighborhood Z of X in V affinoid over Y such that X → Z is inner over Y. Finally, if X, W and Y are as above, we say that a rigid function f on X is overconvergent in W over Y if f extends to some strict neighborhood of X in W over Y. When Y is Spec(K), we just say f is overconvergent in W.

Suppose for the rest of this section that Y is a reduced irreducible affinoid. Suppose P (T) is an entire power series over Y (like the characteristic series of a completely continuous operator on a Banach space over Y). Suppose r ≤ s are real numbers in |K|. Then the subset of Y × A_{K}^{1} determined by the inequalities r ≤ |T | ≤ s is the affinoid Y ×A[r, s], the fiber product of Y and the annulus of radii r and s which is irreducible. The subspace of this affinoid determined by P (T) = 0 is an affinoid Z of dimension equal to that of Y. Moreover, the map Z → Y is finite to one if P (0) = 1. We will investigate this situation in the abstract. I.e., suppose Z → Y is a quasi-finite morphism of affinoids over K such that dim(Z) = dim(Y). We will prove, in particular, Proposition 5.3. Suppose Y = B_{K}^{1}. Then for each integer i ≥ 0 the set of closed points x of Y such that deg(f^{−1}(x)) ≥ i is the set of closed points of an affinoid subdomain Y_{i}. Moreover, Y_{i} = ∅ for large i. Lemma 5.4. Suppose g: W → B_{K}^{1} is a non-constant morphism of affinoids over K and W is irreducible. Then the image of g is an affinoid subdomain of B_{K}^{1}. Proof. We may suppose W is reduced and absolutely irreducible. We may also extend scalars to C_{p} so that W̃ = \overline{W}. After a translation and a homothety we may suppose ḡ is non-constant. If every point whose reduction is in the image of ḡ is in the image of g we have nothing to prove since the image of ḡ is an affine open and its inverse image under reduction is an affinoid subdomain. Therefore suppose 0 is not in the image of g but is in the image of ḡ. Then there exists a b ∈ C_{p}^{∗} such that |b| < 1 and |b/g| = 1. Let h = b/g. Then as h, g ∈ A^{0}(W), |g| = |h| = 1 and |gh| = |b| < 1, it follows that \overline{W} is not irreducible. Thus the lemma is true in the case when \overline{W} is irreducible and in this case g(W) = B^{1} − ⋃_{a∈T} B(a, 1) where T is some finite subset of B^{1}(C_{p}).

Now let Z be an irreducible component of \overline{W}. Let Z^{0} be the complement in Z of the other irreducible components of \overline{W} and Z′ = red^{−1}Z^{0}. Then the rigid space Z′ is an irreducible open in W and since Z^{0} is a finite union of irreducible affine opens in \overline{W}, Z′ is a finite union of affinoid subdomains with irreducible reduction It follows from the above discussion that g(Z) = B[a_{Z}, r_{Z}] − ∪B(b_{Z,j}) for some r_{Z} ∈ |C_{p}|, a_{Z} ∈ B^{1}(C_{p}) and some finite set of b_{Z,j} ∈ B^{1}(C_{p}).

Let S = {B(x, r) : x ∉ g(W), r = |(g −x)^{−1}|^{−1}}. Clearly, g(W) = B^{1} −⋃ S. We claim S ⊆ \{B(b_{Z,j}, r_{Z}) : Z is an irreducible component of \overline{W} \}. This will complete the proof of the lemma as the latter set is finite. Let B(x, r) ∈ S. In particular, r ≤ 1. After a translation we may suppose x = 0. Let |b| = r and h = b/g. Then h̄ is non-constant by the above reasoning if r < 1 and as an immediate consequence of the above in the case r = 1. Therefore, there exists an irreducible component Z of \overline{W} such that h̄|_{Z} is non-constant. It follows, that (g/b)|_{Z} is non constant. Thus B[a_{Z}, r_{Z}] = B[0, r] and so B(0, r) = B(b_{Z,i}, r_{Z}) for some i, which is what we wanted.

Now we define a descending tower Z_{i}, i ≥ 1 of affinoid subdomains of Z such that, if Y_{i} = f(Z_{i}), x ∈ Y_{i} if and only if deg f^{−1}(x) ≥ i. The Y_{i} are affinoid subdomains of B_{K}^{1} by the lemma as quasi-finiteness implies f is not constant on any irreducible component of Z_{i}. We take Y_{0} = Y.

Let X denote the affinoid subspace of Z^{k}, k ≥ 1, determined by the equations f ∘ π_{i}(x) = f ∘ π_{j}(x), 1 ≤ i ≤ j ≤ k, where the π_{i} are the k projections from Z^{k} to Z. Since f is quasi-finite, X is one dimensional. Let X_{k} denote the one dimensional affinoid consisting of the union of the irreducible components of of X not contained in any hyperdiagonal, π_{i}(x) = π_{j}(x) for some i ≠ j, of Z^{k} and set Z_{k} = π_{1}(X_{k}). It follows that Z_{k} satisfies the required conditions.

Finally, we sketch two proofs of the fact that Y_{i} = ∅ for large i. First, extend scalars to a maximally complete algebraically closed field Γ containing K. Maximal completeness implies there exists an x ∈ ⋂ Y_{i}(Γ) if Y_{i} ≠ ∅ for all i. But then deg f^{−1}(x) = ∞ which contradicts the quasi-finiteness of f.

The other proof uses the stable reduction theory of curves. There exists a semistable model of f over a finite extension of K. I.e. there exist semi-stable formal scheme models Y and Z of Y and Z over R and an extension of f to a morphism F from Z to Y such that F̄ is quasi-finite. It follows that for each irreducible component X of \overline{Z}, the map \overline{F}_{X} has finite generic degree d(X) for some nonnegative integer d(X). Suppose x ∈ X. Let x̄ denote its image in \overline{X}. Then one can show

deg f^{−1}(x) ≤ ∑_{X} d(X)

where X runs over the irreducible components of \overline{Z} which meet \overline{F}^{−1}(x̄).

Corollary 5.3.1. For each x ∈ B_{K}^{1}(K), there exists an affinoid ball B ⊆ B_{K}^{1} over K containing x such that g: W_{B} → B is finite.

Remarks. (i) Using the stable reduction theory of curves, one can check this proposition remains true whenever dim Y = 1. (ii) Is the proposition true when Y has dimension greater than one if the phrase “an affinoid subdomain” in this proposition is replaced with “a finite union of affinoid subdomains?” (iii) It is clear that the results of this section can be globalized to arbitrary rigid spaces over K. One only has to replace the notion of orthonormalizability with local orthonormalizability.

(iv) Suppose Z is an irreducible component of the zero locus of P (T). Then as Liu has pointed out the map from Z to Y is not necessarrily quasi-finite. In general Z corresponds to an an irreducible factor of P (T). Now suppose

P (x, T) = xT + ∏_{n}(1 − p^{n}T).

Then P (x, T) is irreducible over K⟨x⟩ and P (0, T) has infinitely many zeroes. Moreover, he has observed that the image in B^{1}(C_{p}) of an irreducible component is the compliment of a finite set of points. Corollary 5.3.2. Suppose K is discretely valued. Let T be an invertible rigid function on Z defined over K. Then the subset of R,

{v(T (z)): z ∈ Z(C_{p}), f(z) ∈ Y (K)},

is finite. Proof. Since the degree of f^{−1}(y) for y ∈ Y (C_{p}) is bounded and for y ∈ Y (K) the set of points of f^{−1}(y) is closed under Gal(K̄/K), the points in f^{−1}(y) for y ∈ Y (K) are all defined over a finite extension of K. The result follows from this and the fact that T is bounded above and below on Z.

We can also show: Proposition 5.6. Suppose X is an irreducible component of the rigid space P (T) = 0. Then if T is bounded on X(C_{p}), the map X → B^{1} is finite. Proof. We may suppose |T |_{X} < 1. Now restrict P (T) to the ball B^{2}, |T | ≤ 1. Then X is still an irreducible component of the P (T) = 0 in B^{2}. Since K⟨S, T ⟩ is factorial [BGR, 5.2.6/1], it follows that there is a G ∈ K⟨S, T ⟩ such that the affinoid cut out by G is X. We may suppose the Gauss norm of G is 1. Since |T |_{X} < 1 it follows that the zero locus of Ḡ is T = 0 and so Ḡ = cT^{n} for some positive integer n and some non-zero constant c. Thus G is distinguished [BGR, 5.2.1/1] and so by the Weiersträss preparation theorem [BGR, 5.2.2/1], X → B^{1} is finite.

6. Eisenstein series For the statements on Eisenstein series discussed in this section see [H-LE, Chapt. 5 §1 and Chapt. 9 §4] as well as [S-MZp, §3]. For the statements on p-adic Lfunctions see [L, Chapt 4] and [W, Chapts. 5 and 7]. For a character χ: Z_{p}^{∗} → C_{p}^{∗}, let f_{χ} denote the smallest positive integer such that χ is trivial on 1 + f_{χ}Z_{p} if one exists, if not, let f_{χ} = ∞. We call f_{χ} the conductor of χ.

Let ρ denote the number of roots of unity in Q_{p}, τ : Z_{p}^{∗} → μ(Q_{p}) be the character of smallest conductor which restricts to the identity on μ(Q_{p}) and q = f_{τ}. Then ρ = 2, τ is the character d ↦ (−1)^{(d−1)/2} and q = 4, if p = 2. Otherwise, ρ = p − 1, τ is the composition of reduction and the Teichmüller character and q = p. For d ∈ Z_{p}^{∗}, let ⟨d⟩ = d/τ(d) which is congruent to 1 modulo q. (Sometimes we will denote this by ⟨⟨d⟩⟩ to avoid confusion with the corresponding diamond Hecke operator (see §8).) Also fix a p − 1-st root π of −p in C_{p}.

We set C equal to the set of continuous C_{p}^{∗} valued characters on Z_{p}^{∗}. We note that Z injects naturally into C; indeed send k ∈ Z to the character which sends a ∈ Z_{p}^{∗} to a^{k}. Let 1 denote the trivial character a ↦ 1. For κ ∈ C, κ ≠ 1, and n ≥ 1 ∈ Z let

σ_{κ}^{∗}(n) = ∑_{d|n} ; (d,p)=1 κ(d)d^{−1} and ζ^{∗}(κ) = \frac{1}{κ(c) − 1} ∫_{Z_{p}^{∗}} κ(a)a^{−1}dE_{1,c}(a)

in the notation of [L-CF, Chapt. 4 §3] for any c ∈ Z_{p}^{∗} such that κ(c) is not 1. So that, when κ(a) = ⟨a⟩^{s}χ(a) where s ∈ C_{p}, |s| < |π/q|, and χ is a character of finite order

ζ^{∗}(κ) = L_{p}(1 − s, χ)

where L_{p} is the p-adic L-function. (This number is zero when χ(−1) = −1.) If κ ≠ 1 let

G_{κ}^{∗}(q) = \frac{ζ^{∗}(κ)}{2} + ∑_{n≥1} σ_{κ}^{∗}(n)q^{n}.

Then when κ(a) = ⟨a⟩^{k}χ(a), where k is an integer and χ is a character of finite order on Z_{p}^{∗} such that χ(−1) = 1, G_{κ}^{∗}(q) is the q-expansion of a weight k overconvergent modular form G_{κ}^{∗} of level Γ_{1}( LCM(q, f_{χ})) and character χτ^{−k}, where f_{χ} ∈ Z_{>0} is the conductor of χ. It is classical if k is at least 1.

Whenever ζ^{∗}(κ) ≠ 0 and κ ≠ 1, let E_{κ}^{∗}(q) = 2G_{κ}^{∗}(q)/ζ^{∗}(κ). We also set E_{1}^{∗}(q) = 1. Suppose κ ∈ C and κ is trivial on μ_{ρ}, then |ζ^{∗}(κ)/2| > 1 and

|E_{κ}(q)^{∗} − 1| < 1.

Remark. We may regard C rigid analytic covering space of C_{p} whose fibers are principal homogenous spaces for the group Hom(Z_{p}^{∗}, μ_{ρp^{∞}}). Indeed the covering map is given by

κ ∈ C ↦ log(κ(a))/ log(a)

for any a ∈ 1 + qZ_{p}, a ≠ 1 (|κ(a) − 1| < 1 since κ is a continuous homomorphism). The space C has ρ connected components (one for each element of C trivial on 1 + qZ_{p}) each conformal to the open unit disk. In view of this, ζ^{∗} may be thought of as a rigid analytic function on a covering space of C_{p}.

Let B = B(0, |π/q|) and W = B × Z/ρZ. We identify a point s = (t, i) in W with the character κ: a ↦ ⟨a⟩^{t}τ^{i}(a) (and will denote this latter expression a^{s}) and will also write, in this case, G_{s} = G_{κ}^{∗} and E_{s} = E_{κ}^{∗}. Let E denote the weight one modular form E_{1,0} which natually lives on X_{1}(q). We signal,

E(q) = 1 + \frac{2}{L_{p}(0, 1)} ∑_{n≥1} ( ∑_{d|n} ; (d,p)=1 τ^{−1}(d))q^{n}.  (1)

Note that E(q) ≡ 1 mod q because L_{p}(0, 1) ≡ 1/p mod Z_{p}.

For an integer n ≥ 0 and a positive integer N prime to p let Z_{1}(Np^{n}) denote the rigid connected component of the ordinary locus in X_{1}(Np^{n}) containing the cusp ∞. In particular, Z_{1}(Np^{n}) is an affinoid.

Lemma 6.1. Suppose κ(a) = ⟨a⟩^{k}χ(a) where k is an integer and χ is a character in C of finite order which is trivial on μ_{ρ}. Then E_{κ}^{∗} (which converges on) does not vanish on Z_{1}(p^{n}) where p^{n} = LCM(q, f_{χ}).

Proof. First E_{κ}^{∗} converges on Z_{1}(p^{n}) because it is overconvergent. Next, the lemma is true for E; i.e., E does not vanish on Z_{1}(q), because E^{p−1} reduces to the Hasse invariant on the component of the reduction of the Deligne-Rapoport/Katz-Mazur model of X_{1}(q) containing ∞. Now observe that F = E_{κ}^{∗}/E^{k} is a function on Z_{1}(p^{n}) whose q-expansion is congruent to 1. It follows that F is congruent to 1 on all of Z_{1}(p^{n}) and so doesn’t vanish there. Hence, E_{κ}^{∗} = F E^{k} does not vanish on this affinoid.

7. General Setup In this section we will lay the groundwork needed to be able to study overconvergent forms in all levels for all primes.

Suppose N > 4, n ≥ 1 are integers and (Nn, p) = 1 and there is a lifting A of the Hasse invariant to X_{1}(Nn). Such a lifting always exists if p > 3 (indeed, in such a case, one can take A = E_{p−1}) and exists for suitable n for p = 2 or 3. For v ≥ 0 ∈ Q let X_{Nn}(v) denote the affinoid subdomain of X_{1}(Nn), v(A(y)) ≤ v. (In particular, X_{Nn}(0) = Z_{1}(Nn).) Let E_{Nn}(v) denote the pullback of E_{1}(Nn) to X_{Nn}(v). Then by Katz, [K-pMF], if v > 1/(p + 1) there is a commutative diagram of rigid morphisms;

E_{Nn}(v) ⟶^{Φ} E_{Nn}(pv)
↓  ↓
X_{Nn}(v) ⟶^{ϕ} X_{Nn}(pv)

We will think of this diagram as a morphism Φ/ϕ from E_{Nn}(v)/X_{Nn}(v) to E_{Nn}(pv)/X_{Nn}(pv), which it is in an appropriate category. Let X_{N}(w) be the affinoid subdomain of X_{1}(N) which is the image of X_{Nn}(w) in X_{1}(N) and E_{N}(v) the pullback of E_{1}(N) to X_{N}(v).

Proposition 7.1. There is a unique morphism

Φ′/ϕ′ : E_{N}(v)/X_{N}(v) → E_{N}(pv)/X_{N}(pv)

such that

E_{Nn}(v)/X_{Nn}(v) ⟶^{Φ/ϕ} E_{Nn}(pv)/X_{Nn}(pv)
↓  ↓
E_{N}(v)/X_{N}(v) ⟶^{Φ′/ϕ′} E_{N}(pv)/X_{N}(pv)

commutes, where the verticle arrows are the natural forgetful projections.

Proof. It is enough to check this on C_{p}-valued points. Let x be a point of X_{N}(v) and (E, α: μ_{N} ↪ E) the corresponding elliptic curve with level structure. Let y be a point of X_{Nn}(v) above x. Then y corresponds to (E, β) where β: μ_{Nn} ↪ E is an injective homomorphism such that β|_{μ_{N}} = α. It follows that ϕ(y), corresponds to (E′, α′) where E′ = E/K_{y} and α′ is the composition of α and the natural map ρ: E → E/K_{y} where K_{y} is the canonical subroup of E. Moreover, ρ = Φ_{E}. The proposition follows from the fact that K_{y} is independant of the choice of y. Indeed, K_{y}(C_{p}) is the set of points of E[p] closest the origin. (If v(A) = w < p/(p + 1) and X is a local uniformizer on E at zero the points of K_{y} are the points P of E such that v(X(P)) ≥ (1 − w)/(p − 1) and at the other points P ∈ K_{y}, X(P) has valuation w/(p^{2} − p) [K-pMF, Thm. 3.10.7].)

Henceforth we will denote Φ′/ϕ′ by Φ/ϕ.

This proposition is already enough to allow us to establish the main results of [C-CO] for the primes 2 and 3. In particular, for any prime p we can define a U operator on overconvergent forms of level N and assert that any such form of weight k and slope stricly less than k − 1 is classical.

As in [C-CO, §8], we may and will regard X_{N}(v) as an affinoid subdomain of the modular curve X(N; p) = X(Γ_{1}(N) ∩ Γ_{0}(p)).

For n a positive integer, denote the set {v ∈ Q : 0 ≤ p^{n−1}v < p/(p + 1)} by the expression I_{n} and I_{n} − {0} by I_{n}^{∗}. Fix a finite extension K of Q_{p} contained in C_{p}. All our constructions will be over K. We will employ the notation and definitions of [C-CO]. Fix a positive integer N > 4 prime to p. For v ∈ I_{1}, let X_{pN}(v) be the affinoid subdomain of X_{1}(pN) which is the inverse image of X_{N}(v) under the natural forgetful map to X(N; p). For k ∈ Z let M_{k}(v) be the space of modular forms of weight k and level Γ_{1}(pN) which converge on X(v). In other words, M_{k}(v) = ω^{k}(X(v)). Now, M_{k}(v) has a natural structure as a Banach space and when v > 0 there is a completely continuous operator on this space denoted by U_{(k)} in [C-CO].

Now we will recall and modify some constructions carried out in [C-HCO]. Let π_{n,1}: X_{1}(Np^{n}) → X_{1}(Np) denote the map which sends the point correspnding to triples (E, α, β), where E is an elliptic curve and where α: μ_{p^{n}} ↪ E and β: μ_{N} ↪ E are embeddings, to the point correponding to (E, α|_{μ_{p}}, β). Suppose, v ∈ I_{n}. Let X_{Np^{n}}(v) denote the affinoid subdomain of X_{1}(Np^{n}) consisting of points x corresponding to triples (E, α, β) such that π_{n,1}(x) ∈ X_{Np}(v), Φ^{n−1}(α(μ_{p^{n−1}})) = 0 and (Φ^{n−1}E, Φ^{n−1} ∘ α, Φ^{n−1} ∘ β) corresponds to ϕ^{n−1}(f(x)). Let E_{Np^{n}}(v) denote the pullback of E_{1}(Np^{n}) to X_{Np^{n}}(v).

If p^{n−1}v < 1/(p + 1), we have a lifting of Φ/ϕ to a morphism from E_{Np^{n}}(v)/X_{Np^{n}}(v) to E_{Np^{n}}(pv)/X_{Np^{n}}(v), which takes (E, α, β) to (ΦE, α′, Φ ∘ β) where α′: μ_{p^{n}} ↪ E is determined by the requirements that (ΦE, α′, Φ ∘ β) corresponds to a point in X_{Np^{n}}(v) and α′(ζ) = Φ(Q), if Q is a point of E such that pQ = α(ζ) for ζ ∈ μ_{p^{n}}, ζ ≠ 1. We will denote these liftings by the same symbols. The context will make it clear which spaces we are dealing with.

Let ω =: ω_{Np^{n}} equal the direct image on X_{1}(Np^{n}) of the sheaf Ω_{E_{1}(Np^{n})/X_{1}(Np^{n})}^{1}. For k ∈ Z, v ∈ I_{n}, we set M_{k}^{n}(v) =: M_{Np^{n},k}(v) = ω^{k}(X_{Np^{n}}(v)). These spaces may be considered as Banach spaces over K and when v > 0, we have a completely continuous operator, which we will still denote by U_{(k)}, acting on M_{k}^{n}(v) defined as in [C-CO] (see also [C-HCO]).

We can deal with N ≤ 4 along the same lines as discussed in the remark at the end of §6 of [C-CO]. In partcular, if A, B ∈ Z, A, B > 4, (A, B) = N, we identify M_{N,k}(v) as the intersection of the images of M_{A,k}(v) and M_{B,k}(v) in M_{AB,k}(v).

8. Twists of U Fix a positive integer N, (N, p) = 1. For v ∈ I_{1}, let X(v) = X_{Nq}(v) and

M_{k}(v) = M_{Nq,k}(v).

Suppose v ∈ I_{1} and F ∈ M_{k−r}(v) be an overconvergent form of weight k − r which has an inverse in M_{r−k}(v). Then the map from M_{r}(v) to M_{k}(v), h ↦ hF, is an isomorphism of Banach spaces. Moreover, the pullback of U_{(k)} on M_{k}(v/p) to M_{r}(v/p) is the map

h ↦ F^{−1}U_{(k)}(hF) = U_{(r)}(hF/σ(F)).

by [C-CO, 3.3]. Thus, in this case, since the restriction map M_{k}(v) → M_{k}(v) is completelely continuous, this formula, together with Proposition 2.2, implies the Fredholm theory of the operator U_{(k)} on M_{k}(v) is equivalent to that of the completely continuous operator U_{(r)} ∘ m_{f} on M_{k}(v/p) where f = F/σ(F) and m_{f} is the operator “multiply by f.” (Recall, the q-expansion of σ(F) is F (q^{p}).)

Recall E is the weight one modular form E_{1,0} of level Γ_{1}(q) with character τ^{−1} described in §6. It follows that there is an overconvergent function e on X(0) with q-expansion E(q)/E(q^{p}). Since E(q) ≡ 1 mod q, we see that |e − 1|_{X(0)} ≤ |q|. It follows that for any ε ∈ R, |q| < ε, there exists a v ∈ I_{1}, 0 ≠ v such that e is defined on X(v) and |e − 1|_{X(v)} ≤ ε.

Recall, π is a (p − 1)-st root of −p. For s ∈ C_{p}, |s| < |π/q|, let u_{s} be the operator on M_{0}(v), for any v ∈ I_{2}^{∗} such that |e − 1|_{X(v)} < |π/s|, defined by

u_{s}(h) = U_{(0)}(he^{s}).

Then from the above discussion if k ∈ Z

det(1 − T u_{k}|M_{0}(v)) = det(1 − T U_{(k)}|M_{k}(v)).  (1)

Let B = B(0, |π/q|). Now we think of s as a parameter on B so that we may view e^{s} as a rigid analytic function on the rigid analytic subspace V of A^{1} × X_{∞} which is admissably covered by the affinoids Z_{t}(v) =: B[0, t] ×_{K} X(v) where v ∈ I_{2}^{∗} and t ∈ |C_{p}| ∩ [1, |π/q|) such that |e − 1|_{X(v)} < |π|/t. Let T be the set of ordered pairs (t, v) satisfying these conditions. (The set T is not empty, in fact, the first projection to |C_{p}| ∩ [1, |π/q|) is a surjection.) Since U_{(0)} extends uniquely to a continuous A(B[0, t]) linear map from A(Z_{t}(v)) to A(B[0, t]) for (t, v) ∈ T , we may now view u_{s} as a family of operators, i.e. there is a compatible collection of operators {U_{(t,v)}: (t, v) ∈ T }, where U_{(t,v)} is the operator on A(Z_{t}(v)), whose restriction to the fiber above s is u_{s}. By Proposition 5.2, if M(t, v) =: A(Z_{t}(v)), for (t, v) ∈ T U_{(t,v)} is a completely continuous operator on M(t, v) over A(B[0, t]). We will abuse notation and write U for U_{(t,v)} when the context makes it clear we are talking about an operator acting on M(t, v). Also as remarked after Lemma 5.1, M(t, v) is orthonormalizable over A(B[0, t]) = C_{p}⟨X/m⟩ where m ∈ C_{p} |m| = t. Thus we have characteristic series det(1 − T U|M(t, v)) for any (t, v) ∈ T . It follows from the Proposition 2.2 and Proposition 5.2 that det(1 − T U|M(t, v)) is independant of the choice of v and it is independant of the choice of t, by Lemma 2.4, in the sense that if t ≤ t′ < |π/q| the restriction of det(1 − T U|M(t′, v)), which is analytic on B[0, t′] × C_{p}, to B[0, t] × C_{p} is det(1 − T U|M(t, v)).

Theorem 8.1. There is a unique analytic function P (s, T) =: P_{N}(s, T) on B × C_{p} such that for k ∈ Z and v ∈ Q such that 0 < v < p/(p + 1),

det(1 − T U_{(k)}|M_{k}(v)) = P (k, T).

Proof. The existence of the function P (s, T) follows from the above discusion combined with formula (1) and Lemma 2.4. Its uniqueness follows from the fact that any rigid analytic function on an irreducible rigid space which vanishes on an infinite set of points in that space is zero.

Let

P (s, T) = ∑_{n≥0} f_{n}(s)T^{n}.

At this point we could prove that the coefficients of the series f_{n}(s) lie in Q_{p} and the numbers |f_{n}|_{B} are bounded independantly of n, in fact, by 1, if p ≥ 7, using known properties of U_{(0)} (e.g. Lemma 3.11.7 of [K-pMF]) but later we will prove much more. In the appendix we will give explict formulas for the f_{n}(s) which imply that they can be interpreted in terms of Iwasawa functions. Ultimately we will give a conceptual proof of this.

As Z_{t}(v) = B[0, t] × X(v) for (t, v) ∈ T , the diamond operators ⟨b⟩. b ∈ (Z/NqZ)^{∗} act on M(t, v), Recall, D = (Z/qZ)^{∗}, and we regard D as a subgroup of (Z/NqZ)^{∗}. We will also regard D as a quotient of Z_{p}^{∗}. Then D acts via the diamond operators ⟨d⟩ for d ∈ D on all the spaces M_{k}(v) and M(t, v). For a ∈ Z_{p}^{∗} we set ⟨a⟩ = ⟨a mod q⟩. For each integer k, character ϵ ∈ D̂ = Hom(D, μ_{ρ}) and v ∈ I_{2}^{∗}, let M_{k}(v, ϵ) denote the subspace of M_{k}(v) of forms with eigencharacter ϵ for this action. Similarly, let M(v, ϵ) denote the subspace of M(t, v) with eigencharacter ϵ. Then,

\begin{aligned}M_{k}(v) & = ⨁_{ϵ} M_{k}(v, ϵ) and \\ M(t, v) & = ⨁_{ϵ} M(t, v, ϵ) \end{aligned}

where the direct sums range over ϵ ∈ D̂. Moreover these direct sums are stablized by the respective operators U_{(k)} and U. We thus have, by Lemma 2.3, the formulas

det (1 − T U_{(k)}|M_{k}(v)) = ∏_{ϵ} det (1 − T U_{(k)}|M_{k}(v, ϵ)) and
det (1 − T U|M(t, v)) = ∏_{ϵ} det (1 − T u_{s}|M(t, v, ϵ)).

Let P_{ϵ}(s, T) be the function on B × C_{p} characterized by the identities:

P_{ϵ}(s, T)|_{B[0,t]×C_{p}} = det(1 − T U|M(t, v, ϵ))

for all (t, v) ∈ T . Then we have

P_{ϵ}(k, T) = det (1 − T U_{(k)}|M_{k}(v, ϵτ^{−k}))  (2)

and

P (s, T) = ∏_{ϵ} P_{ϵ}(s, T)  (3)

for k ∈ Z. This implies Theorem B of the introduction, as well as its extension to p = 2, taking P_{N,i} = P_{τ^{i}}. Theorem D follows from (2), Theorem C and Proposition 5.3. The fact that the set of slopes of classical modular forms of level Γ_{1}(N) is discrete in R follows from Corollary 5.3.2.

Let S(t, v, ϵ) denote the subspace of cusp forms in M(t, v, ϵ) (i.e. the subspace of functions vanishing at the cusps). Then S(t, v, ϵ) is stable under U and we can proceed as above and let P^{0}(s, T) be the function characterized by the identities:

P_{ϵ}^{0}(s, T) = det(1 − T U|S(t, v, ϵ))

for all (t, v) ∈ T . Moreover,

P_{ϵ}^{0}(k, T) = det(1 − T U|S_{k}(v, ϵ))  (4)

for k ∈ Z. Finally, let M(ϵ) =: M_{1}(Np, ϵ) = lim_{↼}_{T} M(t, v, ϵ), which is a subring of

A(V), and S(ϵ) =: S_{1}(Np, ϵ) = lim_{↼}_{T} S(t, v, ϵ), which is an ideal in M(ϵ).

We will now prove Theorem A of the introduction as well as a qualitative version of the Gouvêa-Mazur conjecture about “R-families” [GM-F, Conj. 3] in the case in which U_{p} “acts with multiplicity one” on a slope α subspace. To treat the general case we will use the ring of Hecke operators in §10.

Recall, for a character ϵ on D with values μ_{ρ}, d(k, ϵ, α) denotes the dimension of the slope α subspace of M_{k,cl}(ϵω^{−k}). Let d(k, ϵ, α) denote the dimension of the slope α subspace of M_{k}(ϵτ^{−k}). Also let bd^{0}(k, ϵ, α) and d^{0}(k, ϵ, α) denote the dimensions of the spaces of cusp forms in M_{k,cl}(ϵω^{−k}) and M_{k}(ϵτ^{−k}). Then, by [C-CO, Thm. 8.1 and Lemma 8.7], we know,

\begin{aligned}d(k, ϵ, α) & = d(k, ϵ, α) \\ d^{0}(k, ϵ, α) & = d^{0}(k, ϵ, α). \end{aligned} if k > α + 1  (5)

Fix ϵ a character on D with values μ_{ρ}. Let Z be the zero locus of P_{ϵ}^{0}(s, T) in B × A^{1} and for α ≥ 0 ∈ Q let Z(α) the affinoid subdomain of Z such that v(T (z)) = −α for z ∈ Z. This, in fact, is a defined over Q_{p}. Let r ≥ 0 ∈ Z and S_{r}(ϵ, α) be the subset of j ∈ Z_{p} such that d(j, ϵ, α) = r. It follows from Corollary 5.3.1 that S_{r}(ϵ, α) is compact and if k ∈ S, there exists an affinoid ball B =: B[k, p^{−m}] ⊂ B containing k such that the map Z(α)_{B} → B is finite of degree r. Thus there is a monic polynomial Q(T) of degree r with coefficients in K⟨(s − k)/p^{m}⟩ such that P_{ϵ}(s, T)_{B} = Q(T)S(T) where S(T) ∈ K⟨(s − k)/p^{m}⟩[[T]] is entire in T and prime to Q(T). By Theorem 4.2

M_{B} = N(Q) ⊕ F (Q).

Let

B_{Q} = Q_{p}⟨(s − k)/m⟩[X]/Q^{∗}(X).

We know N(Q) is a B_{Q} module.

Suppose that (ΔQ(k, T), Q(k, T)) = 1. This will automatically be true when r = 1 and more generally when the eigenvalues of U_{p} as an operator on M_{k}(ϵ, α) over C_{p} are distinct. Then after shrinking B if necessary we may suppose that (ΔQ(T), Q(T)) = 1 and using Corollary 4.3.1 that N(Q) is a locally principal B_{Q} module.

Suppose for the moment that r = 1 and suppose k is an integer and F is nonzero overconvergent cusp form of level Γ_{1}(Np) of weight k, character ϵτ^{−k} and slope α. Because all the Hecke operators preserve the space of slope α, character ϵτ^{−k} modular forms, F must be an eigenform. It is non-constant because it vanishes on the cusps, so we may suppose it is normalized. If F |U_{p} = aF, then x = (k, a) is a point of Z(α). Thus the map Z(α)_{B} → B has degree 1. It follows that there exists a function f on B such that f(k) = a, v(f(s)) = α and 1 − f(s)T divides the restriction of P_{ϵ}(s, T) to B × A^{1}. This latter is the characteristic series of the restriction of U to the space M ˆ⊗B by Lemma 2.4 and ΔP_{ϵ}(s, 1/f(s)) is invertible on B. Hence our Riesz theory implies that there exists an analytic function G on B ×W_{0}^{†} which vanishes at the cusps and spans the kernel of U −f(s) in S(ϵ) Thus if k′ ∈ Z, k′ ≡ k mod p^{m}, F_{k′} =: E^{k′}G(k′) is a non-zero overconvergent modular form of weight k′, slope α and character ϵτ^{−k′}. Moreover, if k′ > α + 1, F_{k′} is classical by [C-CO, Theorem 8.1]. Now let

G(s) = ∑_{n≥1} a_{n}(s)q^{n}

be the q-expansion of G(s). The a_{n}(s) are rigid analytic functions on B. We must have F_{k} = a_{1}(k)F and so a_{1}(k) ≠ 0. Hence after shrinking B, we may suppose a_{1}(s) is invertible and so equal to 1. In particular, now F_{k} = F. Since G is bounded on the affinoidB × W_{0}, being a rigid function, the a_{n}’s are uniformly bounded on B. Hence, there exists a constant M ≥ 0 such that for all r ≥ 0 and all n ≥ 0 and all a ∈ Z

|a_{n}(k + p^{r+M}a) − a_{n}(k)| ≤ |qp^{r}|.

As E^{p^{r}}(q) ≡ 1 mod qp^{r}, this implies

F_{k′}(q) ≡ F_{k}(q) mod qp^{r}

if k′ ≡ k mod p^{r+M}. Since S_{r}(ϵ, α) is compact we see that we may pick an M that only depends on α. This yields the statement of Theorem A as well as its extension to p = 2.

All of the above will go through with M(ϵ) and P_{ϵ}(s, T) in place of S(ϵ) and P_{ϵ}^{0}(s, T) if we suppose ϵ is not trivial. When r = 1 all we needed to know is that our form F was not constant which is true in this case.

Now allow r to be arbitrary. Shrinking B, if necessary, there exists an H ∈ N^{∗}(Q) =: B_{Q} ⊗ N(Q) such that N^{∗}(Q) = B_{Q}H and

uF = XH.

Suppose

E^{s}(q)H(q) = ∑_{n≥0} b_{n}q^{n}

where the b_{n} ∈ B_{Q}. Let Y_{Q} be the rigid space sitting over B whose ring of functions is B_{Q}. If k ∈ Z, and y is a point of Y_{Q} above k′, ∑_{n≥0} b_{n}(y)q^{n} is the q-expansion of an overconvergent modular form F_{k′} of weight k′, character ϵτ^{−k} and slope α. If k ≥ α + 1 then F_{k′} is classical. In fact, because B_{Q} is étale over B, F_{k′} is an eigenform. In this way we get a weak version of an “R-family” in the sense [GM-F] where R = B_{Q}.

Recall, W = B × Z/ρZ. It will sometimes be convenient for us to to replace our base B with W. Indeed, as we will see in §11, the ring Λ embeds naturally into the ring of rigid analytic functions on W. First, we identify Z/ρZ with D̂ via i ∈ Z/ρZ ↦ τ^{i}. Then we may view A(W) as A(B)[D] and, if t < |π/q| write A(W(t)) for A(B[0, t])[D] = A(B[0, t] × D̂). For (t, v) ∈ T , we make M(t, v) into a Banach A(W(t))-module as follows: If f = ∑_{d∈D} f_{d}d is an element of A(W(t)) where f_{d} ∈ A(B[0, t]) and G is an element of M(t, v) we set

fG = ∑_{d∈D} f_{d}G|⟨d⟩.

Henceforth, we will write f as ∑_{d∈D} f_{d}⟨d⟩. Now for ϵ ∈ D̂, let e_{ϵ} ∈ (1/ρ)Z_{p}[D] be the element (1/ρ) ∑_{d∈D} ϵ^{−1}(d)⟨d⟩. Then any element m in an A(W(t)) module equals ∑_{ϵ∈D̂} m_{ϵ} where m_{ϵ} = e_{ϵ}m. We put “new” norms | |^{∗} on M(t, v), for (t, v) ∈ T , as follows: Suppose H is in M(t, v). Then we set

|H|^{∗} = Max_{ϵ∈D̂}{|H_{ϵ}|_{Z_{t}(v)}}.

When p is odd | |^{∗} equals the spectral norm and is equivalent to it in general. Moreover, M(t, v) is a Banach module over A(W(t)) with respect to | |^{∗}.

If B is a Z_{p} algebra and ϵ ∈ D̂, we also let ϵ denote the unique B-module homomorphism from B[D] to B which takes ⟨d⟩ to ϵ(d) ∈ Z_{p}. Lemma 8.2. With respect to the norm | |^{∗}, M(t, v) is orthnormalizable over A(W(t)). Moreover, on M(t, v), U is a completely continuous A(W(t))-operator. There is a series Q_{N}(T) ∈ A(W)[[T]] whose restriction to A(W(t)) is the characteristic series for this operator. It is characterized by the identities:

ϵ(Q_{N}(T)) = P_{ϵ}(s, T),

for all ϵ ∈ D̂.

Proof. For each 1 ≤ i ≤ ρ, let v_{i,1}, …, v_{i,n}, … be an orthonormal basis for M(t, v, τ^{i}) over A(B[0, t]) (with respect to the spectral norms). Then the set w_{1}, …, w_{n}, …, where

w_{n} = v_{1,n} + ⋯ + v_{ρ,n}

is an orthonormal basis for M(t, v) over A(W(t)).

The fact that U is an operator over A(W(t)) follows from the fact that e|⟨d⟩ = e for all d ∈ D, Complete continuity follows immediately from the facts that the operator is completely continuous over A(B) and that | |^{∗} is equivalent to the spectral norm. The proof of the existence of Q_{N}(T) follows the same lines as the proof of the existence of P (s, T) in §8. Finally, the last assertion follows from elementary linear algebra.

By the q-expansion over W of an element F ∈ A(V), we mean the series ∑_{n} a_{n}q^{n} ∈ A(W)[[q]] where a_{n} = ∑_{d∈D} a_{n,d}⟨d⟩, a_{n,d} ∈ A(B) and ∑ a_{n,d}q^{n} is the q-expansion of F |⟨d^{−1}⟩.

As in §6, using (1), we may think of W as a subspace of C containing the image of Z. When k = (s, i) ∈ 2(Z_{p} × Z/ρZ) ⊂ W, k ≠ 0, G_{k}(q) is the q-expansion of a Serre modular form of weight k [S-MZp §1.6].

9. Non-integral Weight Recall, B = B(0, |π/q|), ρ = LCM(p − 1, 2), D = (Z/qZ)^{∗} and W = B × Z/ρZ.

In this section, we will let ⟨⟨a⟩⟩ = a/τ(a), for a ∈ Z_{p}^{∗}.

We will give a working definition of overconvergent forms of non-integral weight and of overconvergent families of modular forms.

As in §6, Z_{1}(Nq) denotes the affinoid subdomain of X_{1}(Nq) which is the connected component of the ordinary locus containing the cusp ∞. (In the notation of §7, this is also X_{Nq}(0).) Definition. We say F (q) = ∑_{n=0}^{∞} a_{n}q^{n}, a_{n} ∈ K, is the q-expansion of an overconvergent form of level Γ_{1}(Nq) and weight k = (s, i) ∈ W over K if F (q)/E(q)^{s} is the q-expansion of an overconvergent function on Z_{1}(Nq) in X_{1}(Nq) of character τ^{i} for the action of D.

If U is an admissable open subspace of B we also say that F_{s}(q) = ∑_{n=0}^{∞} a_{n}(s)q^{n}, a_{n} ∈ A(U), is a family of overconvergent forms over U of level Γ_{1}(Nq) if F_{s}(q)/E(q)^{s} is the q-expansion of an overconvergent function on U × Z_{1}(Nq) over U. We say this family has type i ∈ Z/ρZ if H has character τ^{i} for the action of D. We say this family is an eigenvector for U with eigenvalue f(s) ∈ A(U) if U(H) = f(s)H.

For k ∈ W, let M_{k}^{†}(N) denote the vector space over K of weight k overconvergent modular forms of level Γ_{1}(N), let M^{†}(N) denote the A(B) module of families of overconvergent forms over B of level Γ_{1}(N) and, for i ∈ Z/ρZ, M^{†}(N, i) the subspace of those of type i.

Clearly, if F (q) is an overconvergent form of weight k and G(q) is an over convergent form of weight j, F (q)G(q) is an overconvergent form of weight k + j. Also,

M^{†}(N) = ⨁_{i∈Z/ρZ} M^{†}(N, i)

and if A^{†}(N) denotes the A(B) algebra of overconvergent functions on B × Z_{1}(Nq) over B, then A^{†}(N) is isomorphic to M^{†}(N) as an A(B) module. For k = (s, i) ∈ W we have a natural homomorphisms

M^{†}(N) → M^{†}(N, i) → M_{k}^{†}(N),

where the first arrow is the projection. Theorem 9.1. Suppose i ∈ 2Z/ρZ. Then G_{s,i} is an overconvergent family of eigenforms over B if i ≠ 0 and over B − {0} if i = 0 of level Γ_{1}(q) and type i with eigenvalue 1 for U.

Proof. First we observe that there are ρ/2 cusps C in Z_{1}(q). For c ∈ C, let ]c[ denote the residue disk in Z_{1}(p) containing c. We may regard q as a parameter on the residue disk ]∞[ of the cusp ∞. Fix (t, v) ∈ T , t > 1, and let A = A(Z_{t}(v)). Let I_{C} ⊂ A be the ideal of B[0, t] × C. The homomorphism h: A → A/I_{C}A := B is respected by U and by the diamond operators.

Fix i ∈ 2Z/ρZ. We will work on the τ^{i} eigensubspace of A for the action of D, A_{i}, which maps onto the τ^{i} eigensubspace of B, B_{i}, and this latter is free of rank one over A(B[0, t]). We also set P_{i}(s, T) = P_{τ^{i}}(s, T). The following diagram commutes:

\begin{matrix}A_{i} & →^{h} & B_{i} \\ ↓^{U} & & ↓^{id} \\ A_{i} & →^{h} & B_{i} \end{matrix}.

It follows that 1 − T divides P_{i}(s, T). Restricting s to an integer k at least 2 and using (8.2) and [C-CO. Thm. 8.1] we see that (1−T)^{2} does not divide P_{i}(s, T) since every U_{(k)}-eigenvector in M_{k}(τ^{i−k}) with eigenvalue 1 is a classical modular form of weight k and character τ^{i−k} and the dimension of these is one. Let X ⊆ B[0, 1] be an affinoid in the complement of the zero locus S of ΔP_{i}(s, T). Then our Riesz theory, Theorem 4.3, tells us that the eigenspace of U over X with eigenvalue 1 is locally free of rank one. In fact, using the map h and the above commutative diagram, we see this module is free spanned by Π(H) := F where where Π is the Riesz theory projector onto the eigenvalue 1 subspace and H is any function in A_{i} which maps to the element of B_{i} which is 1 along B[0, t] × ∞. We may suppose that an infinite set of integers at least 2 are contained in X. Then for any such integer k, we know F_{k}(q) = E_{k,i}(q)/E(q)^{k} since in this case we know E(q)^{k}F_{k}(q) must be the q-expansion of a classical modular form and the q-expansions of F_{k} and E_{k,i}/E^{k} have the same constant term, 1. Since the q-expansion coefficients of F must be analytic on X, we see that F_{k}(q) = E_{s,i}(q)/E(q)^{s} for all s ∈ X such that L_{p}(1 − s, τ^{i}) ≠ 0. Since this true for any affinoid X in B[0, t] − S we conclude that L_{p}(1−s, τ^{i})F_{s} =: H_{s} is an overconvergent analytic function on (B[0, t]−S)×Z_{1}(p). But the q-expansion of H_{s} clearly extends to B[0, t]×]C(∞)[ when i ≠ 0 and to (B[0, t] − {0})×]C(∞)[ when i = 0. Hence as G_{s,i}(q) = H_{s}(q)E(q)^{s}, |s| ≤ t, G_{s,i}(q) a family of forms over B[0, t] when i ≠ 0 and over B[0, t] − {0} when i = 0. Since this is true for any t such that t < |π/q| the theorem follows.

Corollary 9.1.1. For (s, i) ∈ W, (s, 2i) ≠ (0, 0), there exists an overconvergent form of type (s, i) with q-expansion G_{s,2i}(q).

Corollary 9.1.2. For each i ∈ 2Z/ρZ there exists an overconvergent function F_{i} on B × Z_{1}(q) such that

F_{i}(s, q) = G_{s,i}(q)/G_{s,i}(q^{p}).

We also see that E_{s,i}(q) is a family of overconvergent forms over the complement in B of the zeroes of L_{p}(1 − s, τ^{i}). So when i = 0, it is a family of overconvergent forms over all of B. In particular, we can replace E^{s}(q) with E_{s,0} in our definition of overconvergent forms of non-integral weight and of families of overconvergent forms.

Remark. We could now upgrade our Fredholm theory by using the function F_{0} in place of e^{s}. Let T^{∗} denote the subset of T consisting of pairs (t, v) such that F_{0} converges on Z_{t}(v). (The set T^{∗} also projects onto Q ∩ [1, |π/q|).) Let V^{∗} denote the rigid subspace of V admissablly covered by the affinoids Z_{t}(v) where (t, v) ∈ T^{∗}. Let U^{∗} be the operator h ↦ U(hF_{0}) on A(V^{∗}). It is a completely continuous operator on A(Z_{t}(v)) for each (t, v) ∈ T^{∗}. It also sits in a commutative diagram

A(V^{∗}) ⟶^{·f} A(V^{∗})
↓^{U}  ↓_{U^{∗}}
A(V^{∗}) ⟶^{·f} A(V^{∗})

where f is the function with q-expansion E(q)^{s}/E_{s,0}(q) which is a unit using Theorem 9.1 and the fact that E_{s,0}(q)/E(q)^{s} is congruent to 1. It follows that the characteristic power series of U^{∗} is the same as that of U. The reason why this is an improvement, is that the q-expansion coefficients of F_{0} are Iwasawa functions, so that U^{∗} preserves the submodule of A(V^{∗}) consisting of element whose q-expansions are Iwasawa functions. This will be used, in a subsequent article, to give a conceptual proof of the fact, proven in the appendix, that the coefficients of Q_{N}(T) are Iwasawa functions.

Suppose now that N = 1.

For i ∈ 2Z/ρZ we have an overconvergent function E_{(i)} defined on V^{∗} away from the fibers above the zeroes of L_{p}(1 − s, τ^{i}) such that

E_{(i)}(s, q) = E_{(s,i)}(q)/E_{(s,0)}(q).

It follows that E_{(i)}|⟨d⟩ = τ(d)^{i}E_{(i)} for d ∈ Z_{p}^{∗}.

Theorem 9.3. Suppose i ∈ 2Z/ρZ. Suppose D is an overconvergent function on V which satisfies

D(∞) = 1 and D|⟨d⟩ = τ^{i}(d)D

for d ∈ Z_{p}^{∗}, then away from the zeroes of ΔP_{τ^{i}}(s, 1) and L_{p}(1 − s, τ^{i}),

\frac{−UR_{τ^{i}}((s, 1), U)}{ΔP_{τ^{i}}(s, 1)}D(s) = E_{(i)}(s).  (3)

Proof. We know for k an integer at least 2 and i ∈ 2Z/ρZ, the U_{(k)}-eigensubspace of M_{k}(τ^{i−k}) with eigenvalue 1 is one dimensional and spanned by G_{k,i}. It follows, in particular, that

UE_{(i)} = E_{(i)}.

Thus 1 − T divides P_{τ^{i}}(s, T) and since the aforementioned eigenspaces have dimension one, 1 − T divides P_{τ^{i}}(s, T) simply. Our Riesz theory, the uniqueness of analytic continuation and the fact that the two sides of (3) agree at the cusps now implies that equation (1) holds whenever both sides are defined.

Remark. Overconvergent functions like D certainly exist, for example, we can take D to be the function (which is “constant in the s direction”) E_{m,i}/E_{m,0} where m is an integer at least 1.

Since, the polar divisor of E_{(i)} is the divisor of L_{p}(1−s, τ^{i}), equation (3) implies;

Corollary 9.3.1. For i ∈ 2Z/ρZ, L_{p}(1 − s, τ^{i}) divides ΔP_{τ^{i}}(s, 1).

This and our Riesz theory implies that if k ∈ Z_{p} and ord_{k}L_{p}(s, τ^{i}) = n, then the generalized eigenspace with eigenvalue 1 for the U operator acting on convergent modular forms of Serre weight (k, k + i) is at least n + 1.

Questions. Does ΔP_{τ^{i}}(s, 1) divide L_{p}(1 − s, τ^{i})? Suppose (k, i) ∈ W. Are there ever any eigenvectors of level Γ_{1}(q) for U in the space of convergent modular forms of type (k, i) with eigenvalue 1 not in the span of G_{k,i}?

We also deduce from the proof of the theorem;

Corollary 9.3.2. For each i ∈ 2Z/ρZ, E_{s,i}(q) is the q-expansion of an overconvergent family of eigenforms of type i on the complement of the zero locus of L_{p}(1−s, τ^{i}) in B.

10. Hecke Operators and R-families In this section we eshew the notion of “radius of overconvergence.” For d ∈ Z_{p}^{∗},

we will now use ⟨⟨d⟩⟩ to denote d/τ(d). Recall, M^{†}(N) is the A(B) module of families of overconvergent forms.

Now we define an action of Hecke on M^{†}(N). First, if l ∈ (Z/NqZ)^{∗} we define

(F |⟨l⟩^{∗})_{s}(q) = E^{s}(q)(\frac{F_{s}}{E^{s}} |⟨l⟩)(q).

for s ∈ B. When k ∈ Z, (F |⟨l⟩^{∗})_{k} = τ^{k}(l)F_{k}|⟨l⟩ so that, in this case,

⟨⟨l⟩⟩^{k}(F |⟨l⟩^{∗})_{k} = l^{k}F_{k}|⟨l⟩.  (0)

For prime l, let ψ_{l} be the operator on A(B)[[q]]

ψ_{l}(∑_{n} a_{n}q^{n}) = ∑_{n} a_{nl}q^{n}.

Then, if l|Np there is a unique continuous operator U(l) on M^{†}(N) such that, for F ∈ M^{†}(N),

F |U(l)(q) = ψ_{l}(F (q)).

When l = p,

F |U(p) = E^{s}U(\frac{F}{E^{s}} ).

When l|N one may verify this by first showing, using a correspondence in the ususal way, that for g ∈ A^{†}(N), there is an element in A^{†}(N) with q-expansion ψ_{l}(g(q)). Then, observe that

ψ_{l}(F (q)) = E^{s}(q)ψ_{l}((\frac{F}{E^{s}} e_{l}^{s})(q)).

More generally:

If n and M are relatively prime positive integers, we let Γ(M; n) denote the congruence subgroup Γ_{1}(M) ∩ Γ_{0}(n) of SL_{2}(Z) and X(M; n) the modular curve of this level over K. We can repeat all of our previous constructions and definitions on this level and we will use obvious extensions of our previous notations.

If F is modular form on X_{1}(N) and l is a prime not dividing N, we let F |V_{l} denote the modular form on X(N; l) such that

F |V_{l}(Y, α, C, ω) = F (Y/C, l^{−1}ν ∘ α, ν̌^{∗}ω)

where Y is an elliptic curve, α: μ_{N} ↪ Y is an injective homomorphism, C is a cyclic subgroup of Y of order l, ω is a non-vanishing differential on Y, ν: E → E/C is the natural isogeny and ν̌ is its dual. Then

F |V_{l}(q) = F (q^{l}).

Since E(q)/E(q^{l}) is congruent to 1 modulo q, and both E and E|V_{l} have weight (1, 0), there is an element e_{l}^{s} ∈ A^{†}(N; l) whose q-expansion is (E(q)/E(q^{l}))^{s} and which is invariant under the action of D. Lemma 10.2. For each prime number l not dividing Np there is a unique continuous operator T (l) on M^{†}(N) such that for F ∈ M^{†}(N),

(F |T (l))(q) = ψ_{l}(F (q)) + l^{−1}⟨⟨l⟩⟩^{s}(F |⟨l⟩^{∗})(q^{l}).  (1)

Proof. If G_{s}(q) is the right hand side of (1),

\frac{G_{s}(q)}{E^{s}(q)} = ψ_{l}(\frac{F}{E^{s}} e_{l}^{s})(q) + l^{−1}⟨⟨l⟩⟩^{s}e_{l}^{−s}(\frac{F}{E^{s}} |⟨l⟩)(q^{l})

which, by the previous discussion, is the q-expansion of a function in A^{†}(N; l). Moreover, when k is a integer, G_{k} is clearly of level Γ_{1}(N) since the specialization of (1), in this case, is the classical formula for the l-th Hecke operator acting on the overconvergent modular form F_{k} of weight k. Now consider, the function in A^{†}(N; l), T r(G_{s}/E^{s}) − (l + 1)G_{s}/E^{s}, where T r is the trace map from level Γ(N; l) to level Γ_{1}(N). By what we have said, it is zero when s is an integer. It follows that it is zero for all s since it is an analytic function. This implies the lemma as T r(G_{s}/E^{s}) is of level Γ_{1}(N).

Remark. Our proof implies that T (l) acts on forms which converge on some strict neighborhood of Z_{1}(Nq) ⊗ B for each prime l. With a little more care we can show that one can use the same neighborhood for all l (at least when p ≠ 2). The key fact needed to prove this is: If R is a ring of characteristic p and A is the Hasse invariant form over R, then if E/R is an elliptic curve, ω generates H^{0}(E, Ω_{E/R}^{1}) and γ: E → γE is an isogeny of degree prime to p,

A(E, ω) = A(γE, γ̌^{∗}ω)

where γ̌: γE → E is the dual isogeny to γ.

Let T =: T_{K} denote the A(B)-algebra generated over A(B) by ⟨d⟩^{∗} for d ∈ D, T (l) for primes l such that (l, Np) = 1 and U(l) for primes l such that l|Np. Similarly, if L is an extension of K in C_{p} and k ∈ W(L) we may define operators ⟨d⟩_{k}^{∗} for d ∈ (Z/NqZ)^{∗}, T_{k}(l) for primes l such that (l, Np) = 1 and U_{k}(l) for primes l dividing Np on M_{L,k}^{†}(N). We let T_{L,k} denote the L-algebra generated by these over L. We define additional operators S(n), for n > 0 ∈ Z in T_{K} by the formal identity:

∑_{n≥1} \frac{S(n)}{n^{t}} = ∏_{l|Np}(1 − U(l)l^{−t})^{−1} ∏_{(l,Np)=1}(1 − T (l)l^{−t} + ⟨⟨l⟩⟩^{s}⟨l⟩^{∗}l^{−1−2t})^{−1},

where the products are over primes l and when k ∈ W(L), we define S_{k}(n) in T_{L,k}, similarly. When, k ∈ Z, it follows from (0) that ⟨d⟩_{k}^{∗} = τ^{k}(d)⟨d⟩ and hence T_{k} is the usual Hecke algebra acting on overconvergent weight k modular forms of level Γ_{1}(Nq) (see [G, Chapt. II]).

It is clear that we have a natural homomorphism, h ↦ h_{k}, from T onto T_{k} for k ∈ W(L) which takes ⟨d⟩^{∗} to ⟨d⟩_{k}^{∗} and S(n) to S_{k}(n). Also;

Corollary 10.2.1. If k ∈ W(L), h ∈ T and F ∈ M^{†}(N) then

(hF)_{k} = h_{k}F_{k}.

R-families

Before we proceed, we point out that if α ∈ Q, α ≠ 0, the slope α subspace of M_{k}(N) is canonically isomorphic to the slope α subspace of M_{k}^{†}(N) and we identify the two. For a rigid space U ⊆ B and an element F ∈ M^{†}(N)_{U}, we let a_{n}(F) ∈ A(U) denote the coefficient of q^{n} in its q-expansion .

Suppose α is a rational number and k_{0} and i are integers such that 0 ≤ i < ρ and k_{0} > α + 1. Suppose 0 < r < |π/q| and r ∈ |K| such that the slope α affinoid in the zero locus of P_{i}^{0}(s, T) is finite of degree d over the affinoid disk B over K of radius r around k_{0}. (We know d = d^{0}(k_{0}, α, τ^{i−k}).) This disk exists by Corollary 5.3.1. Let A = A(B). Then, by Theorem 4.3, the A-module N = (M^{†}(N, i)_{B})_{α} is projective of rank d over A. Since A is a PID, this module is, in fact, free. Let R denote the image of T ⊗ A in End_{A}(N). It follows that R is also free of finite rank. In particular, R is the ring of rigid analytic functions on an affinoid X(R) with a finite morphism to B.

We have an A-bilinear pairing

\begin{aligned}⟨ , ⟩: R × N & → A \\ ⟨h, m⟩ & = a_{1}(hm). \end{aligned}

Similarly, if k ∈ B(L), we have an L-bilinear pairing ⟨ , ⟩_{k} from R_{k} × N_{k} to L. (In our previous terminology, we are actually working over the point (k, i) of W(L).)

Proposition 10.3. The pairing ⟨ , ⟩ is perfect.

Proof. First, arguing exactly as in the proof of [H-LE, Thm. 5.3.1], we see that if h ∈ R, ⟨h, m⟩ = 0 for all m ∈ N implies h = 0 and ⟨h, m⟩ = 0 for all h ∈ R implies m = 0. The key point is that if F ∈ N, ⟨S(n), F ⟩ equals the n-th q-expansion coefficient of F.

Now if k ∈ B(C_{p}), the same argument yields the same conclusion for the pairing ⟨ , ⟩_{k}: T_{k} × N_{k} → C_{p}, but since this is a pairing over a field, it follows that ⟨ , ⟩_{k} is perfect.

Since A is a PID, it suffices to check that the homomorphism,

\begin{aligned}γ: R & → Hom_{A}(N, A) \\ γ(h)(n) & = ⟨h, n⟩, \end{aligned}

is an isomorphism. By Corollary 10.2.1, if h ∈ R and m ∈ N, the restriction of ⟨h, m⟩ to k is ⟨h_{k}, m_{k}⟩_{k}. Since N is free, Hom_{A}(N, A)_{k} = Hom_{C_{p}}(N_{k}, C_{p}). Thus γ_{k} is an isomorphism for all k ∈ B(C_{p}). This implies γ is an isomorphism and the propositon follows.

Corollary 10.3.1. The degree of X(R) → B is d^{0}(k_{0}, α, τ^{i−k}).

Theorem 10.4. Suppose L ⊂ C_{p} is a finite extension of K and k is an integer such that k ∈ B(K) and k > α + 1. Then the mapping from X(R)_{k}(L) to L[[q]], x ∈ X(R)_{k}(L) ↦ F_{x}(q), where

F_{x}(q) = ∑_{n≥1} η_{x}(S(n))q^{n}

and η_{x}: R → L is the homomorphism corresponding to x, is a bijection onto the set of q−expansions of classical cusp forms on X_{1}(Nq) over L of weight (k, i) and slope α.

Proof. After extending scalars we may suppose L = K. First, suppose x ∈ X(R)_{k}(K). Then it follows from the proposition and the freeness of R that there is an m ∈ N such that (⟨h, m⟩)_{k} = η_{x}(h). This equals a_{1}(h_{k}m_{k}) by Corollary 10.2.1. Since η_{x} is homomorphism m_{k} is an eigenform. It, also, follows that F_{x}(q) is the q-expansion of m_{k} and since k > α + 1 that m_{k} is classical.

Now suppose F (q) = ∑_{n≥1} a_{n}q^{n} is the q-expansion of a weight k cuspidal eigenform on X_{1}(Nq) of weight (k, i) and slope α. It follows that F (q) ∈ N_{k}. Hence gives rise to a K-linear map η: T_{k} → K, η(h) = ⟨h, F (q)⟩_{k}. Since F is an eigenform η is a ring homomorhism so corresponds to a point x ∈ X(R)_{k}(K). Finally, since ⟨S(n), F (q)⟩ = a_{n}, F_{x}(q) = F (q).

11. Higher Level In this section, we will explain how our series P_{N}(T) also “controls” forms on

X_{1}(Np^{n}), (N, p) = 1 for n ≥ 1 (the proofs will appear in a future article).

If γ ∈ Z_{p}^{∗}, let [γ] denote the corresponding element in the completed group ring Λ of Z_{p}^{∗} over Z_{p}. Then there exists a unique injective homomorphism α from Λ into A^{0}(W) such that, for γ ∈ Z_{p}^{∗},

α([γ])(s, i) = ⟨⟨γ⟩⟩^{s}τ(γ)^{i}.

(In other words, [γ] goes to the element ⟨⟨γ⟩⟩^{s}⟨γ⟩ in A(B)[D] = A(W).) It follows that, for λ ∈ Λ, α(λ) is bounded on W and

|λ| = |α(λ)|_{W},

where | | is the absolute value on Λ described in §1.

We will show that

Theorem 11.1. The series P_{N}(T) lies in Λ[[T]].

We, actually, give one proof of this in the appendix using explicit formulas but it is also possible to give a proof of this using the Fredholm theory developed earlier.

The space of overconvergent forms of level Np^{n} together with the U-operator on this space is defined in §7. For κ ∈ C and F (T) = ∑_{n≥0} B_{n}T^{n} ∈ Λ[[T]] we set κ(F)(T) = ∑_{n≥0} κ(B_{n})T^{n}. We can map Z_{p}^{∗} onto (Z/p^{n}Z)^{∗} which is naturally a direct factor of (Z/Np^{n}Z)^{∗}. Hence, we may regard characters on Z_{p}^{∗} of conductor p^{n} as characters on (Z/Np^{n}Z)^{∗}. We will also prove;

Theorem 11.2. If κ(x) = χ(x)⟨x⟩^{k} where k is an integer, χ: Z_{p}^{∗} → C_{p}^{∗} is a character of finite order and p^{n} = LCM(q, f_{χ}), then κ(P_{N})(T) is the characteristic series of the U-operator acting on overconvergent modular forms of level Np^{n}, weight k and character χ.

12. A Refined Conjecture. Let ϵ ∈ D̂ and Y_{ϵ} ⊂ B × C_{p} be the zero locus of P_{ϵ}^{0}(s, T). For r ∈ |C_{p}| let Y_{r,ϵ} be the affinoid subspace of Y where |T | ≤ r^{−1}. We know Y_{r,ϵ} is locally finite over B. Then Conjecture 1 of Gouvêa-Mazur [GM-FM] would follow from the statement: Conjecture 12.1. If r ≠ 0, the space Y_{r,ϵ} is finite over any affinoid disk of radius r contained in B.

Indeed, suppose k and k′ are integers such that k − k′ ≡ 0 mod p^{n}, then this statement implies

∑_{α≤n} d^{0}(k, ϵ, α) = ∑_{α≤n} d^{0}(k′, ϵ, α),

since these numbers are the degrees of Y_{p^{n}} over the points k and k′ (there are only a finite number of non-zero terms in these sums). So if α ∈ Q, k and k′ are strictly greater than α + 1, it follows from Theorem C that,

d^{0}(k, ϵ, α) = d^{0}(k′, ϵ, α)

as long as k ≡ k′ mod p^{m} for an integer m ≥ α.

In fact, as we saw in §8 and §10, much of Conjectures 2 and 3 of [GM-F] would also follow.

Let Z be the zero locus of P_{N}^{0}(s, T) and Z_{r} the affinoid subspace of Z determined by the inequality |T | ≤ r^{−1}. Then Z_{r} is the union, over ϵ ∈ D̂, of the spaces Y_{r,ϵ} so must also be finite over B[a, r] for a ∈ B. One can say something about the putative degree of Z_{r} over B[a, r] in some cases. Suppose a is an integer k > 2 and r = p^{1−k}. Then, using [C-C0, ,Lem. 8.7] and the obvious generalization of [C-CO, Cor. 7.2.3] to level Gamma_{1}(Np) we see that if Z_{r} is finite over B[a, r], it must have degree D_{k} = d_{k} + C + e where d_{k} is the dimension of space of classical cusp forms of weight k, C is the number of cusps on the Igusa curve, and e is the rank of the p-torsion on the Jacobian of the Igusa curve. If P_{N}^{0}(s, T) = ∑ a_{n}(s)T^{n}, then one can even predict assuming the above finiteness what v(a_{n}(s)) is when s ∈ B[a, r] and n = D_{k}.

Appendix: Formulas.

Fix a positive integer N prime to p. Let Q_{N}^{0}(T) be the charateristic power series of U on level Γ_{1}(Nq) overconvergent cusp forms whose coefficients are in A(W), as in §9. Let Y be the component of the ordinary non-cuspidal locus in the reduction of X_{1}(Nq) containing ∞ and, for x a closed point of Y, a(x) ∈ Z_{p} the unit root of Frobenius on the fiber above x. Using the Monsky-Reich trace formula, generalizing results of Dwork [D1], Katz [K] and Adolphson [A], one obtains: Theorem I.1. Suppose N ≥ 4. Then, for s ∈ W,

Q_{N}^{0}(s, T) = ∏_{r≥0} ∏_{x∈Y} (1 − a(x)^{s−2}(ā(x)/a(x))^{r}T^{deg(x)})^{−1}

where the second product is over closed points of Y and ā(x) is the complex conjugate of a(x) (which is an imaginary quadratic number). (Recall, for s = (t, i) ∈ W and β ∈ Z_{p}^{∗}, β^{s} = ⟨⟨β⟩⟩^{t}τ^{i}(β).) Corollary I.1.1. The coefficients of Q_{N}(T) as a series in T lie in the Iwasawa algebra Z_{p}[[Z_{p}^{∗}]]. This answers a question of [GM-CS].

Now for an order O in a number field, let h(O) denote the class number of and w(O) denote the number of roots of unity in O. If γ is an algebraic integer, let O_{γ} be the set of orders in Q(γ) containing γ. Finally, for m an integer, let W_{p,m} denote the finite set of γ ∈ Q_{p} such that Q(γ) is an imaginary quadratic field, γ is an algebraic integer and

Norm_{Q(γ)/Q}(γ) = p^{m} and v(γ) = 0.  (1)

Then, another version of the above theorem is: Theorem I.2. Suppose N ≥ 4. Then

T \frac{d}{dT} Q_{N}^{0}(k, T)/Q_{N}^{0}(k, T) = ∑_{m≥1} A_{m}(k)T^{m}

where

A_{m}(k) = ∑_{γ∈W_{p,m}} ∑_{O∈O_{γ}} h(O)B_{N}(O, γ) · \frac{γ^{k}}{γ^{2} − p^{m}}

where B_{N}(O, γ) is the number of elements of O/NO of order N fixed under multiplication by γ.

Also, using Hijikata’s application of the Eichler-Selberg trace formula [Hj] Koike [Ko] proved “in level 1,” Theorem I.3. (Koike) Let N = 1 and k ∈ W be “even”. Then

T \frac{d}{dT} Q_{1}^{0}(k, T)/Q_{1}^{0}(k, T) = ∑_{m≥1} A_{m}(k)T^{m}

where

A_{m}(k) = 1 + ∑_{γ∈W_{p,m}} ∑_{O∈O_{γ}} \frac{h(O)}{w(O)} · \frac{γ^{k}}{γ^{2} − p^{m}} .

We will now use these formulas to prove the existence of non-classical overconvergent eigenforms.

Definition. We say an overconvergent modular form of level Γ_{1}(Nq) and integral weight k is a new form if its image in M_{k}(d) is zero under any of the degeneracy “trace” maps for any proper divisor d of N and old if it is a sum of elements in the images of M_{k}(d) where d runs over the proper divisors of N.

We note that the image of a classical modular form is new in this sense if and only if it is new “away from p.”

We denote the space of new forms of weight k by M_{k}^{new}(N) and of old forms by M_{k}^{old}(N). We see that

M_{k}(N) = M_{k}^{old}(N) ⊕ M_{k}^{new}(N)  (2)

as T_{k} modules.

Proposition I.4. Let k be an integer. Then there exist weight k level Γ_{1}(Nq) overconvergent new forms of arbitrarily large slope. Proof. We must show the characteristic power series G_{N}(k, T) of U acting on the space of weight k overconvergent new forms of level Γ_{1}(Nq) is not a polynomial. Let

\frac{T d}{dT}G_{N}(k, T)/G_{N}(k, T) = ∑_{m≥1} D_{m}(k)T^{m}.

It suffices to show that the numbers D_{m}(k) are algebraic and are not all defined over a finite extension of Q.

For a positive integer n let f(n) be the number of distinct prime divisors of n and

t(n) = \{ \begin{matrix}(−2)^{f(n)} & if n is square free \\ 0 & otherwise. \end{matrix}

Using (2) and the two linearly disjoint degeneracy maps from forms of level Γ_{1}(Mq) to forms of level Γ_{1}(Mlq) for primes l and positive integers M prime to p, one can show

G_{N}(k, T) =: ∏_{d|N} P_{N/d}^{0}(k, T)^{t(N/d)}.

We will complete the proof only in the case in which N = l^{t} where l is an odd prime and l^{t−1} ≥ 5.

Suppose M is any integer at least 5. Let K ⊂ Q_{p} be a quadratic field of discrimimant D less than −M. Also, suppose for simplicity of exposition that D ≢ 1 mod 4. Then there exists an m ∈ N and an element γ ∈ K ∩ W_{p,m} such that γ ≡ 1 mod NO_{K} where O_{K} is the maximal order of K. In fact, since N ≥ 5, −γ is the only other element of K ∩ W_{p,m}. Then Theorem I.2 implies D_{m}(k) = α + β where

α = ∑_{O∈O_{γ}} h(O)(B_{N}(O, γ) − 2B_{N/l}(O, γ))\frac{γ^{k}}{γ^{2} − p^{m}}

and β is a sum of elements contained in quadratic fields different from K (B_{N}(O, −γ) = B_{N/l}(O, −γ) = 0 for all O ∈ O_{γ}). The corollary will follow from the claim: K = Q(α). It is easy to see that K = Q(γ^{k}/(γ^{2} − p^{m})) and B_{N}(O_{K}, γ) = N^{2} − 3(N/l)^{2} + 2(N/l^{2})^{2} > 0. Thus all we need verify is that

C(O) =: B_{N}(O, γ) − 2B_{N/l}(O, γ) ≥ 0

for all O ∈ O_{γ}. We first observe that the numbers B_{H}(O, γ) only depend on the power of l dividing [O_{K} : O]. Therefore suppose K has discriminant D and O = Z[l^{s}\sqrt{D}] is in O_{γ}. Also suppose γ = 1 + α where α = N(a + bl^{r}\sqrt{D}) where a, b ∈ Z and (l, b) = 1. It follows that r ≥ 0 and t + r ≥ s. Suppose x ∈ O and x has order N modulo NO. Let x = c + dl^{s}\sqrt{D}. Then

αx ≡ Nbcl^{r}\sqrt{D} mod NO.

Hence, γx ≡ x mod (N/l)O if and only if cl^{r} ≡ 0 mod l^{s−1} (here when s = 0 we require no condition on c) and γx ≡ x mod NO if and only if cl^{r} ≡ 0 mod l^{s}. Suppose first that r < s − 1, then in either case l|c so (d, l) = 1. Thus,

\begin{aligned}C(O) & = l^{t−(s−r)}(N − N/l) − 2l^{t−1−((s−1)−r)}(N/l − N/l^{2}) \\ & = l^{r−s}N(N − 3N/l + 2N/l^{2}) > 0. \end{aligned}

Suppose now r = s − 1, then c may be arbitrary in the first case and l|c in the second. Thus,

\begin{aligned}C(O) & = (N/l)(N − N/l) − 2((N/l)^{2} − (N/l^{2})^{2}) \\ & = N^{2}/l − 3(N/l)^{2} + 2(N/l^{2})^{2} > 0, \end{aligned}

because l > 2. Suppose finally that r > s − 1. Then,

C(O) = N^{2} − 3(N/l)^{2} + 2(N/l^{2})^{2} > 0.

This establishes the claim. Remarks. (1) One may deduce that the field generated by the coefficients of Q_{N}^{0}(k, T), for any k ∈ Z, equals the compositum of all the imaginary quadratic fields in Q_{p} in which p splits. (2) We expect that the same methods can be used to prove that there exist overconvergent forms of level Γ_{1}(Np^{n}), weight k and character χ of arbitrarily large slope. (3) This proof gives no information on the distribution of the weight k slopes.

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