Coleman’s theory of p-adic modular forms

Shu Sasaki^{∗} 17th September 2010

Abstract An introduction to p-adic modular forms, and in particular to Coleman’s theory of overconvergent modular forms in the nineties. This is thoroughly “pre-Coleman-Mazur”.

Contents

1 Introduction 2 2 Modular forms 3 3 Tate elliptic curves 4 4 Moduli space of elliptic curves 5 5 Katz modular forms 5 6 Overconvergent modular forms 6

6.1 Overconvergent modular forms of level N . . . . . . . . . . . . . . . . . . 6 6.2 Katz modular functions and Serre p-adic modular forms . . . . . . . . . . 7 6.3 Overconvergent modular forms of level Np . . . . . . . . . . . . . . . . . 8

7 Overconvergent modular forms are classical? 8 8 Coleman p-adic analytic families of modular forms 12 9 Appendix 17

9.1 Appendix 1 Crash course on rigid geometry . . . . . . . . . . . . . . . . 17 9.1.1 Tate’s classical approach . . . . . . . . . . . . . . . . . . . . . . . 17 9.1.2 Raynaud’s approach . . . . . . . . . . . . . . . . . . . . . . . . . 19

9.2 Appendix 2 Serre’s theory of p-adic Banach spaces . . . . . . . . . . . . . 20

∗Supported by EPSRC at King’s College London

1 Introduction

The theory of p-adic modular forms was initiated by Serre, Dwork, Katz in the seventies (the “Antwerp volumes”). Motivated to construct a p-adic L-function, Serre’s approach was to define “p-adic modular forms” to be simply limits of (q-expansions of) classical modular forms. In this optic, Serre observed that one could think of what had been known to be ‘weights” of modular forms as elements of Z_{p}^{×}, or more generally characters of Z_{p}^{×}.

Katz observed, on the other hand, that one could define modular forms of integer weight with p-adic coefficients to be sections of a line bundle “ω” over a moduli space over Z_{p} of elliptic curves, rather than “q-expansions with p-adic coefficients”. From this viewpoint, Serre’s p-adic modular forms of weight k ∈ Z were thought of as sections of ω^{⊗k} over the elliptic curves with ordinary reduction. While Serre’s approach vastly generalised classical modular forms, Katz’s analysis of elliptic curves with “not too supersingular” reduction led to finer understanding of the U_{p}-operator, and consequently to the definition of p-adic modular forms “with r-growth” (Coleman subsequently called them “r-overconvergent modular forms” in comparison with Serre’s “convergent” p-adic modular forms of q-expansions).

Hida approached to “p-adic modular forms” slightly differently (See Professor Ochiai’s notes, for example). Instead of “modular forms”, much of his emphasis was on algebras of Hecke operators, commonly referred to as “Hecke algebras”. Hida observed that one could define an idempotent “e” on algebras T(N) of Hecke operators acting on spaces S_{2}(Γ_{1}(Np^{∞})) of cusp forms whose q-expansions have p-adic coefficients, and noted that it singled out subalgebras in which U_{p} is a unit. The Hecke algebras T^{0}(N) = eT(N) of the type considered by Hida (which are torsion free algebras over Λ = Z[[(1 + pZ_{p})^{×}]] ≃ Z_{p}[[(u−1)]] (u = 1+p)) were rich in applications: for example, for a finite field extension L of the field of fractions of Λ, a Λ-algebra homomorphism F : T^{0}(N) → L, called a “Hida family” (note that, by a perfect duality S(Γ_{1}(Np^{∞})) × T(N) → Z_{p} defined by (f, T) ↦ c_{1}(f|T), one may think of this as a formal q-expansion F = ∑ F (T (n))q^{n} ∈ L[[q]]), gives rises to (the q-expansion in p-adic coefficients of) a classical cusp eigenform of weight k when specialising modulo the prime ideal in O_{L} above (u − (1 + p)^{k−2}) for k ∈ Z_{≥2}. This is a manifestation of Hida’s construction of p-adic families which “interpolate” classical modular forms, albeit they are “ordinary”.

Having had worked on non-archimedean analysis of (modular) curves, Coleman picked up the threads Katz had left and formulated a general framework of p-adic modular forms in which he coined the term “overconvergent modular forms”. In particular, he applied Serre’s theory of p-adic Banach spaces (which, strictly speaking, are vector spaces over non-archimedean fields; Coleman therefore had to generalise the theory slightly to incorporate Banach modules over rings) and defined an analogue of Hida’s idempotent “e”. This allowed him to construct families of finite slope overconvergent modular forms (by which we mean the (normalised) p-adic valuation of eigenvalues for U_{p} are finite; Hida’s ordinary forms are of slope 0).

In these notes, we shall introduce “work of Coleman” leading to “Coleman-Mazur”. We shall not make any reference to Coleman-Mazur (see Professor Yamagani’s notes, for example) nor to constructions p-adic families of modular forms on other algebraic groups (see papers written for example by Buzzard, Bellaïche, Chenevier, Emerton, Kassaei, Kisin, Loeffler, Stevens, Urban and Yamagami). Up to Section 5, we explain the Katz’ viewpoint of “algebraic” modular forms, which generalises classical modular forms over C. In Section 6, we introduce “overconvergent modular forms” (in which we freely use the language of rigid geometry; see Appendix 1), and state and sketch a proof of “overconvergent modular forms are classical if slope are small” (Theorem 3) in Section 7. This is arguably one of the most important theorems in p-adic modular forms. We also mention results about Galois representations associated to overconvergent modular forms. In Section 8, we explain how Coleman amalgamated the viewpoints of Serre, Katz and Hida to construct “Coleman families” of p-adic modular forms (Theorem 13). Serre’s theory of p-adic modular forms is briefly summarised in Appendix 2, which in fact is not exactly sufficient, as alluded above, to apply to Banach modules of overconvergent modular forms but it nevertheless should shed some light on key ingredients on the construction.

Acknowledgement I thank the organisers of the summer school, in particular, Professor Ochiai for the opportunity.

2 Modular forms

The group SL_{2}(Z) acts on the complex upper half plane 𝔥 = {z ∈ C|Im z > 0} by ( \begin{matrix}a & b \\ c & d \end{matrix}) z = \frac{az+b}{cz+d}. A subgroup Γ of SL_{2}(Z) is called a congruence subgroup if it contains the subgroup of matrices in SL_{2}(Z) which reduces to the identity matrix modulo N for some positive integer N.

Let k be an integer. For a holomorphic function f : 𝔥 → C, define (f|_{k}γ)(z) = (cz + d)^{−k}f(\frac{az+b}{cz+d}) for γ = ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ. We say f is of weight k and of level Γ if f|_{k}γ = f for every γ ∈ Γ. If ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) ∈ Γ, then f(z + b) = f(z) and f has a Fourier expansion f(z) = ∑ c_{n}(f)q^{n/b}, where q = e^{2πiz}, at infinity. If γ ∈ SL_{2}(Z), one checks that (f|_{k}γ)|_{k}γ′ = f|_{k}γ for γ′ ∈ γ^{−1}Γγ and therefore f|_{k}γ is of level γ^{−1}Γγ. We say that f is holomorphic (resp. vanishes) at the cusps if, for every γ ∈ SL_{2}(Z), f|_{k}γ has a Fourier expansion at infinity with c_{n}(f) = 0 for all n < 0 (resp. n ≤ 0).

A modular (resp. cusp) form of weight k and level Γ is a holomorphic function f : 𝔥 → C weight k and of level Γ which is holomorphic (resp. vanishes) at the cusps.

Let M_{k}(Γ) (resp. S_{k}(Γ)) denote the complex vector space of the modular (resp. cusp) forms of weight k and level Γ.

Example. Let k ≥ 4 be an even integer.

E_{k}(q) = \frac{ζ(1 − k)}{2} + ∑_{n=1}^{∞} σ_{k−1}(n)q^{n} = (\frac{2(2πi)^{k}}{(k − 1)!})^{−1} ∑_{(m,n)≠(0,0)}(mz + n)^{−k}

where q = e^{2πiz} and σ_{k−1}(n) = ∑_{0<d|n} d^{k−1} is a modular form of weight k and level SL_{2}(Z). Also,

E_{k}(q) − p^{k−1}E_{k}(q^{p}) = \frac{ζ(k − 1)}{2}(1 − p^{k−1}) + ∑ σ_{k−1}^{∗}(n)q^{n}

where σ_{k−1}^{∗}(n) = ∑_{0<d|n,(p,d)=1} d^{k−1} is a modular form of weight k and level Γ_{0}(p). In Hida theory and Coleman theory, this is an uber-example of modular forms which can be p-adic analytically interpolated with respect to weight.

For a positive integer M, let Γ_{1}(M) denote the subgroup of matrices in SL_{2}(Z) which reduce mod M to ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} ). The spaces M_{k}(Γ_{1}(M)) and S_{k}(Γ_{1}(M)) have a natural action of (Z/MZ)^{×} by \overline{d} ↦ ⟨\overline{d}⟩ where f|⟨\overline{d}⟩ = (cz + d)^{−k}f(\frac{az+b}{cz+d}) for any ( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(Z) such that c ≡ 0 and d ≡ \overline{d} modulo M.

For f ∈ M_{k}(Γ_{1}(M)), the operator T_{p} for a prime p which does not divide M is defined by

(f|T_{p})(z) = (1/p) ∑_{i=0}^{p−1} f (( \begin{matrix}1 & i \\ 0 & p \end{matrix} ) z) + p⟨p⟩f(pz).

and U_{p} for a prime p which divides M is defined by

(f|U_{p})(z) = (1/p) ∑_{i=0}^{p−1} f (( \begin{matrix}1 & i \\ 0 & p \end{matrix} ) z)

In terms of q-expansions, T_{p} (resp. U_{p}) sends f(q) = ∑_{n=0}^{∞} c_{n}(f)q^{n} to ∑_{n=0}^{∞} c_{np}(f)q^{n}+ ∑_{n=0}^{∞} c_{n}(f|⟨p⟩)q^{np} (resp. ∑_{n=0}^{∞} c_{np}(f)q^{n}). Define T (n) for a positive integer n by

∑_{n=1}^{∞} T (n)n^{−s} = ∏_{p|̸M}(1 − T_{p}p^{−s} + ⟨p⟩p^{1−2s})^{−1} ∏_{p|M}(1 − U_{p}p^{−s})^{−1}

3 Tate elliptic curves

An elliptic curve over C is a compact Riemann surface of of genus 1 and is isomorphic as a complex manifold to the complex torus (See [21] for example). By standard calculations, the elliptic curve C/(Z + zZ) with differential 2πidz is given as the plane cubic curve

E : Y^{2} = 4X^{3} − \frac{E_{4}(q)}{12}X + \frac{E_{6}(q)}{216}

defined over Z[1/6]((q)) with differential dX/Y such that C/(Z + zZ) ≃ E(C) by X = ℘(z) and Y = ℘′(z), where ℘ is the Weierstrass ℘-function (See [33]). The map z ↦ e^{2πiz} defines an isomorphism C/(Z + zZ) ≃ C^{×}/q^{Z} and therefore gives rise to an analytic isomorphism G_{m}(C)/q^{Z} ≃ E(C).

If we change the coordinate by X = x + \frac{1}{12} and Y = x + 2y, the equation defines a curve

Tate(q) : y^{2} + xy = x^{3} + c_{2}(q)x + c_{3}(q)

over Z((q)) with canonical differential dx/(2y + x), where

c_{2}(q) = −5\frac{E_{4} − 1}{240} = −5 ∑_{n≥1} σ_{3}(n)q^{n}

c_{3}(q) = (−5\frac{E_{4} − 1}{240} − 7\frac{E_{6} − 1}{−504}) /12 = ∑_{n≥1} \frac{5σ_{3}(n) − 7σ_{5}(n)}{12}q^{n}

which we shall call the Tate curve Tate(q) over Z((q)). It is a theorem of Tate [34] that there is a p-adic analytic isomorphism G_{m}(Q̄_{p})/q^{Z} ≃ Tate(q)(Q̄_{p}).

4 Moduli space of elliptic curves

Let Y_{1}(N) denote the quotient Γ_{1}(N)\𝔥. By adjoining a finite set of cusps, corresponding to the orbits of Q ∪ {∞} by the action of Γ_{1}(N), one can compactify Y_{1}(N) and get a compact Riemann surface X_{1}(N). By GAGA (Serre’s “Geometrique Algebrique et Geometrie Analytique”), it corresponds to a smooth projective algebraic curve X_{1}(N)_{C} over C such that X_{1}(N)_{C}(C) ≃ X_{1}(N).

Define an action of Γ_{1}(N) on 𝔥 × C by ( \begin{matrix}a & b \\ c & d \end{matrix}) (z, z′) = (\frac{az+b}{cz+d}, (cz + d)^{k}z′) for z ∈ 𝔥 and z′ ∈ C. The projection Γ_{1}(N)\(𝔥 × C) → Y_{1}(N) is naturally a line bundle over Y_{1}(N). One can check that this extends to the cusps and defines a line bundle ℒ on the compactification X_{1}(N). The sections of the line bundle ℒ over X_{1}(N) can be identified with M_{k}(Γ_{1}(N)).

One may think of Γ_{1}(N)\𝔥 as the set of isomorphism classes of elliptic curves over C with a point of order N by the orbit Γ_{1}(N)z ↦ (C/(zZ + Z), 1/N + (zZ + Z)). Define an action of Z × Z on 𝔥 × C by (z, z′)(m, n) = (z, z′ + mz + n) for m, n ∈ Z. Then the quotient (𝔥 × C)/(Z × Z) defies a “family of elliptic curves” over 𝔥. The quotient E_{1}(N) =^{def} Γ_{1}(N)\(𝔥 × C)/(Z × Z) then may thought of as the “universal” elliptic curve over Y_{1}(N).

Let T^{∗} be the cotangent space of C at the origin. Let Γ_{1}(N) act on 𝔥 × T^{∗} by

( \begin{matrix}a & b \\ c & d \end{matrix}) (z, dz^{∗}) = (\frac{az+b}{cz+d}, (cz + d)dz^{∗}) for dz^{∗} ∈ T^{∗} and the projection Γ_{1}(N)\(𝔥 × T^{∗}) →

Y_{1}(N) defines the relative cotangent bundle of E_{1}(N) over Y_{1}(N). It then follows that the line bundle ℒ is identified with the restriction along the zero section of the cotangent bundle. In the next section, we shall define modular forms with integral coefficients in this optic; this follows from the standard fact that Y_{1}(N) (and X_{1}(N)) not only is algebraic but indeed has models over Z[1/N].

5 Katz modular forms

One can define “algebraic modular forms”. Let N ≥ 5 and Y_{1}(N)_{Z[1/N]} be the smooth curve over Z[1/N] which represents the functor which sends a Z[1/N]-scheme S to the set of isomorphism classes of (E, P) where E is an elliptic curve over S and P ∈ E(S) = Hom(S, E) be a point of exact order N. There is a universal elliptic curve E_{1}(N) over Y_{1}(N)_{Z[1/N]} and we let ω denote the pull-back along the zero section of the relative sheaf Ω_{E_{1}(N)/Y_{1}(N)_{Z[1/N]}} of differentials on E_{1}(N).

Let X_{1}(N)_{Z[1/N]} be its compactification. One can “toroidal”-compactify Y_{1}(N)_{Z[1/N]} ([9]), or one can think of X_{1}(N)_{Z[1/N]} as a moduli space of generalised elliptic curves ([17]). Then the invertible sheaf ω over Y_{1}(N)_{Z[1/N]} canonically extends to an invertible sheaf over X_{1}(N)_{Z[1/N]} which we shall again denote by ω.

For a Z[1/N]-algebra A, a modular form over A of weight k and level Γ_{1}(N) is a section of ω^{⊗k} over X_{1}(N)_{Z[1/N]} × A and let M_{k}(Γ_{1}(N); A) denote the A-module of the sections H^{0}(X_{1}(N) ×_{Z[1/N]} A, ω^{⊗k}). Note that M_{k}(Γ_{1}(N); C) = M_{k}(Γ_{1}(N)).

If N ≤ 4, the functor is no longer representable; in which case one can work with X_{1}(M) with N|M and M ≥ 5 and define an element of the Γ_{1}(N)/Γ_{1}(M)-invariants of H^{0}(X_{1}(M), ω^{⊗k}) to be a modular form of weight k and level Γ_{1}(N).

There is a map Spec(Z((q))⊗_{Z}Z[1/N]) → X_{1}(N)_{Z[1/N]} corresponding to (Tate(q), i^{can}) where i^{can} comes from the embedding μ_{N} ↪ G_{m}. This map extends to a cusp, called ∞, and gives Spec(Z[[q]] ⊗_{Z} Z[1/N]) → X_{1}(N)_{Z[1/N]}. For a Z[1/N]-algebra A, if f ∈ M_{k}(Γ_{1}(N); A), its pull-back to Spec(Z[[q]] ⊗_{Z} Z[1/N] ⊗_{Z[1/N]} A) is of the form

(∑_{n=0}^{∞} c_{n}(f)q^{n})(ω^{can})^{⊗k}

where ω^{can} is the canonical differential on Tate(q), and we refer to ∑_{n≥0} c_{n}(f)q^{n} as the q-expansion of f at the cusp ∞. A different choice of embedding i : μ_{N} ↪ Tate(q) correspond to another cusp c and one can define the q-expansion of f at c similarly. We then let S_{k}(Γ_{1}(N); A) denote the submodule of f ∈ M_{k}(Γ_{1}(N); A) whose q-expansions at all cusps have constant term zero.

For f ∈ M_{k}(Γ_{1}(N); A), the operator T_{p} for a prime p which does not divide N is defined by

(f|T_{p})((E, P)) = (1/p) ∑_{D⊂E[p]} f((E/D, (P + D)/D)),

where the sum is over the (p + 1) finite flat subgroup schemes of E[p] of order p, and U_{p} for a prime p which divides N is defined by

(f|U_{p})((E, P)) = (1/p) ∑_{D⊂E[p],D∩⟨P ⟩={0}} f((E/D, (P + D)/D))

where the sum is over the finite flat subgroup schemes of E[p] of order p which has only the trivial intersection with the subgroup ⟨P ⟩ generated by P.

6 Overconvergent modular forms

Henceforth, we shall assume the reader is “aware” of some definitions and facts about rigid geometry. See Appendix 1, or [4] and [3].

6.1 Overconvergent modular forms of level N

Assume N is a prime to p. We henceforth denote X_{1}(N)_{Z[1/N]} (resp. Y_{1}(N)_{Z[1/N]}) by X (resp. Y) for brevity. Let C_{p} be the completion of an algebraic closure \overline{Q}_{p} of Q_{p}, and W denote the Witt vectors W (\overline{F}_{p}) of \overline{F}_{p}. Let K be any finite extension of the field of fractions of W. Let X^{an} denote the rigid space (X×_{Z[1/N]})^{an} over K.

The curve (X ×_{Z[1/N]} W) ×_{W} \overline{F}_{p} has a finite number of supersingular points. Let SS denote the admissible open subset of points in X^{an} which specialise to supersingular points. For x ∈ SS(\overline{F}_{p}), since X × W is smooth, one can choose a parameter T_{x} so that O_{X×W,x}^{∧} ≃ W [[T_{x}]].

For r ∈ p^{Q} and 1 ≥ r > 1/p, let SS[0, r) (resp. SS[0, r] for 1 > r ≥ 1/p) denote the set of points P in one of the residue discs such that the normalised (|p| = 1/p) p-adic norm |T_{x}(P)| ∈ [0, r) (resp. ∈ [0, r]), and we define X_{≥r} (resp. X_{>r}) to be the complement X^{an} −SS[0, r) (resp. X^{an} −SS[0, r]). They are admissible open subsets and in particular X_{≥r} is a connected affinoid subdomain. One can check the connectedness as in Section 2 of [7]. Alternatively, one observes that the reduction of X_{≥r} contains the ordinary locus and X_{≥r} therefore is defined as the preimage of a connected subvariety covered by affine varieties which intersect with the ordinary locus (which is connected and is dense in the closed fibre). It is therefore connected by Proposition 1.3.3 in [3].

When the Eisenstein series E_{p−1}, lifting the Hasse invarinat, is a true modular form, one may use E_{p−1} ∈ H^{0}(X ×W, ω^{⊗(p−1)}), instead of the parameter T_{x} on the supersingualr residue disc of x, to define overcoonvergence; the reduction mod p has a simple zero at x and think of O_{X_{1}(N)×W,x} as the direct limit of functions O_{U}(U) on neighbourhoods U of x on which E_{p−1} is trivialised.

For r as above, we call an element of M_{k}^{≥r}(N) =^{def} H^{0}(X_{≥r}, ω^{an,⊗k}) an r-overconvergent modular form of weight k and level Γ_{1}(N), and the union M_{k}(N) of H^{0}(X_{≥r}, ω^{an,⊗k}) is the space of overconvergent modular forms of weight k and level Γ_{1}(N). By definition, X_{≥r} contains the cusp ∞ and we let S_{k}(N) denote the subspace of overconvegent modular forms whose q-expansions at ∞ have constant term zero. By q-expansion principle and the connectedness of X_{≥r}, this suffices. We call it the space of overconvergent cusp forms of weight k and level Γ_{1}(N).

The sections H^{0}(X_{≥1}, ω^{an,⊗k}) is the space of p-adic (or convergent) modular forms and may be identified with Katz modular functions [24]

6.2 Katz modular functions and Serre p-adic modular forms

For an integer N ≥ 3 prime to p and an integer r, let Y_{1}(Np^{m})^{0} be the moduli space over Z_{p} of “trivialised” elliptic curves E over p-adically complete and separated Z_{p}-algebra equipped with an embedding i_{N} : μ_{N} ↪ E and an isomorphism i_{p^{∞}} : Ĝ_{m} → Ê. Every isomorphism class of (E, i_{N}, i_{p^{∞}}) gives rise to the elliptic curve E equipped with an embedding i_{N} × (i_{p^{∞}}|_{μ_{p^{m}}}) : μ_{Np^{m}} ↪ E by restriction, and therefore Y_{1}(Np^{m})^{0} may be thought of as the inverse limit (with respect to m ≥ r) of the direct limit (with respect to n) of Y_{1}(Np^{m})_{Z[1/N]}×(Z/p^{n}Z). In this optic, it is easy to see that Y_{1}(Np^{m})^{0} = Y_{1}(N)^{0} and we call an element of the coordinate ring M_{1}^{mer}(N)^{0} of Y_{1}(N)^{0} a Katz p-adic meromorphic modular function of (tame) level N over Z_{p}.

Let Tate(q)^{∧} be the Tate elliptic curve over the punctured disc Z_{p}((q))^{∧}. Then there is a map Spec(Z_{p}((q))^{∧}) → Y_{1}(N)^{0} corresponding to Tate(q)^{∧} with canonical structures and the pull-back of Y_{1}(N)^{0} to Spec(Z_{p}((q))^{∧}) defines a q-expansion map M_{1}^{mer}(N)^{0} → Z_{p}((q))^{∧}.

We let M_{1}(N)^{0} be the subring of elements f ∈ M_{1}^{mer}(N)^{0} such that f(q) ∈ Z_{p}[[q]] and S_{1}(N)^{0} be the subring of those f(q) ∈ qZ_{p}[[q]]. We call an element of M_{1}(N)^{0} (resp. S_{1}(N)^{0}) a Katz p-adic (holomorphic) modular functions of (tame) level N (resp. Kaz p-adic cuspidal modular functions of tame level N).

We have a natural action of Z_{p}^{×} × (Z/NZ)^{×} ≃ lim_{←}(Z/Np^{m}Z)^{×} on M_{1}^{mer}(N)^{0} (and therefore on M_{1}(N)^{0} and S_{1}(N)^{0}) by (⟨x⟩f)(E, i_{N}, i_{p^{∞}}) = f(E, x^{p}i_{N}, x_{p}^{−1}i_{p^{∞}}) for x^{p} ∈ (Z/NZ)^{×} and x_{p} ∈ Z_{p}^{×}.

Given a character κ : Z_{p}^{×} → Z_{p}^{×}, if f ∈ M_{1}(N)^{0} satisfies ⟨x_{p}, x^{p}⟩f = κ(x_{p})f, we say f is of weight κ. If κ(x) = x^{k} for k ∈ Z, we say f is of weight k.

Proposition 1 For k ∈ Z, the Katz p-adic modular functions of weight k and level N defined over K are exactly M_{k}^{≥1}(N).

Proof. This is Proposition I.3.5 in [20].

Let κ be a continuous character Z_{p}^{×} → Z_{p}^{×}. For a p-adically complete and separated Z_{p}-algebra A, a series f(q) ∈ A[[q]] is called a Serre p-adic modular form of weight κ and level N defined over A if there exists a sequence of classical modular form f_{n} of weight k_{n} ∈ Z and of level N defined over A such that the q-expansions of f_{n} converge to f(q) in A[[q]] and κ(x) ≡ x^{k_{n}} mod p^{n} for all x ∈ Z_{p}^{×}.

Proposition 2 A series f(q) is a Serre modular form of arithmetic weight κ if and only if it is the q-expansion of a Katz p-adic modular function of weight κ.

Proof. This is A1.6 in [24]. See also Proposition I.3.11 in [20].

6.3 Overconvergent modular forms of level Np

Let p be prime to N. Let X_{1}(p)_{Z[1/N]} (resp. X_{0}(p)_{Z[1/N]}) be the (smooth) compactification of the fine moduli space Y_{1}(p)_{Z[1/N]} (resp. Y_{0}(p)_{Z[1/N]}) of elliptic curves E equipped with a point P of exact order N and a point Q of order p (resp. a finite flat subgroup scheme C of E[p] of order p). If we let π_{1} : Y_{1}(p)_{Z[1/N]} → Y_{Z[1/N]} and denote the “forgetful” map which takes (E, P, Q) to (E, P), it also extends to the cusps and we shall denote it again by π_{1}. Similarly for π_{1} : X_{0}(p)_{Z[1/N]} → X_{Z[1/N]}.

For p^{−p/(p+1)} ≤ r ≤ 1, we define X_{1}(p)_{≥r} (resp. X_{0}(p)_{≥r}) to be the admissible open subset of X_{1}(p)^{an} (resp. X_{0}(p)^{an}) defined to be the connected component of π_{1}^{−1}(X_{≥r}) containing the cusp ∞. An (r-) overconvergent modular forms of weight k and level Γ_{1}(Np) (resp. Γ_{1}(N) ∩ Γ_{0}(p)) is then an element of H^{0}(X_{1}(p)_{≥r}, ω^{⊗k}) (resp. H^{0}(X_{0}(p)_{≥r}, ω^{⊗k}))

We let M_{k}(Np) (resp. M_{k}(N; p)) denote the space of overconvergent modular forms of weight k and level Γ_{1}(Np) (resp. Γ_{1}(N) ∩ Γ_{0}(p)). As in the level N case, by definition, X_{1}(p)_{≥r} and X_{0}(p)_{≥r} both contain the cusp ∞, and we let S_{k}(Np) (resp. S_{k}(N; p)) denote the space of overconvergent modular forms of weight k and level Γ_{1}(Np) (resp. Γ_{1}(N) ∩ Γ_{0}(p))

7 Overconvergent modular forms are classical?

Theorem 3 An overconvergent modular form of weight k and level Γ_{1}(N), which is an eigenform for U_{p}, is a classical modular form if the slope is strictly less than k − 1. If its slope is precisely k −1, unless it is in the image of the theta operator θ^{k−1}, it is a classical modular form.

Coleman proves [11] this by “Hodge theory (de Rham cohomology) of modular curves”. We shall sketch the proof.

We have a locally free O_{X×_{Z[1/N]}K}-module R^{1}f_{∗}Ω_{E/X×_{Z[1/N]}K}(logC) of rank 2 where E is the semiabelian scheme f : E → X ×_{Z[1/N]} K whose restriction to the non-cuspidal locus is the universal elliptic curve E_{1}(N) over Y ×_{Z[1/N]} K, and C is the inverse image by f of the cuspidal divisor X − Y. For brevity, for k ≥ 2, we denote by ℋ_{k−2} the locally free O_{X×_{Z[1/N]}K}-module Sym^{k−2}R^{1}f_{∗}Ω_{ℰ /X×_{Z[1/N]}K}(logC) of rank k − 1. It is a result of Faltings that ℋ_{k−2} has a canonical descending filtration, called Hodge filtration, (Fil^{i}ℋ_{k−2})_{i∈Z} such that Fil^{i}ℋ_{k−2} = ℋ_{k−2} for i ≤ 0, Fil^{i}ℋ_{k−2} is a locally free O_{X×_{Z[1/N]}K}- module of rank k − 1 − i for 1 ≤ i ≤ k − 2 and Fil^{i}ℋ_{k−2} = 0 for i ≥ k − 1 and (Fil^{i}ℋ_{r})(Fil^{j}ℋ_{s}) = Fil^{i+j}ℋ_{r+s}. Let ∇ denote the Gauss-Manin connection

∇_{k−2} : ℋ_{k−2} → ℋ_{k−2} ⊗ Ω_{X×_{Z[1/N]}K}(logC),

and let ℋ_{k−2} ⊗ Ω_{X×_{Z[1/N]}K}^{·}(logC) denote the de Rham complex of O_{X×_{Z[1/N]}K} modules with logarithmic singularities

ℋ_{k−2} →^{∇} ℋ_{k−2} ⊗ Ω_{X×_{Z[1/N]}K}(logC).

In [11], Coleman defines operators: “Verschiebung”

V : ℋ_{k−2}(X_{>p^{−1/(p+1)}}) → ℋ_{k−2}(X_{>p^{−p/(p+1)}}),

“Frobenius”

F : ℋ_{k−2}(X_{>p^{−p/(p+1)}}) → ℋ_{k−2}(X_{>p^{−1/(p+1)}}),

and the theta operator

θ^{k−1} : ω^{⊗2−k} → ω^{⊗k}.

By comparison theorem, Theorem 2.4 in [2] (See also [13]), it follows that the algebraic de Rham cohomolgy of rigid spaces H^{1}(X_{>p^{−p/(p+1)}}, ℋ_{k−2} ⊗ Ω_{X×_{Z[1/N]}K}^{·}(logC)) and H^{1}(X_{>p^{−1/(p+1)}}, ℋ_{k−2} ⊗ Ω_{X×_{Z[1/N]}K}^{·}(logC)) are both isomorphic to the algebraic de Rham cohomology of a Zariski scheme H^{1}(X_{1}(N), ℋ_{k−2} ⊗ Ω_{X×_{Z[1/N]}K}^{·}(log(C ∪ ss)) where by ss we mean a finite set of points in SS ⊂ X^{an}, one of which lies in each supersingular discs in SS. Denote H^{1}(X_{>p^{−p/(p+1)}}, ℋ_{k−2}⊗Ω_{X×_{Z[1/N]}K}^{·}(logC)) simply by H. One also checks that, by the degeneration of the E_{1} spectral sequence for the de Rham cohomology, the hypercohomology is isomorphic to (ℋ_{k−2}⊗Ω_{X×_{Z[1/N]}K}^{1}(log(C))(X_{>p^{−p/(p+1)}})/(∇ℋ_{k−2})(X_{>p^{−p/(p+1)}}).

Let H_{par} denote the image of H^{1}(X, ℋ_{k−2} ⊗Ω_{X×_{Z[1/N]}K}^{·}(logC)⊗ℐ_{C}) in H^{1}(X, ℋ_{k−2} ⊗ Ω_{X×_{Z[1/N]}K}^{·}(logC)) where ℐ_{C} denotes the ideal sheaf corresponding to the cuspidal divisor. Unravelling the definition, one can check that H_{par} is isomorphic to the subspace of the classes in H which are trivial on the cuspidal residue discs and the supersingular annuli in W_{1}.

For each supersingular annuli with a choice of orientation and a choice of parameter, one can define a residue map (Lemma 2.1 in [13]). We then let S_{k}(N)^{0} (resp. S_{k}(Γ_{1}(N)∩ Γ_{0}(p))^{0}) be the subspace of overconvegent (resp. classical) cusp forms of weight k and level Γ_{1}(N) (resp. Γ_{1}(N) ∩ Γ_{0}(p)) with trivial residues on SS.

Lemma 4 M_{k}(N)/θ^{k−1}M_{2−k}(N) (resp. S_{k}(N)^{0}/θ^{k−1}M_{2−k}(N)) is isomorphic to H (resp. H_{par}) and the action of U_{p} on the quotient corresponds to the action of V on H by the isomorphism.

Let π_{1} : X_{0}(p)^{an} → X^{an} be the forgetful map which takes (E, P, Q) to (E, P). By Katz-Lubin theory of canonical subgroups [24] according to which, for r > p^{−p/(p+1)}, one can write down a section of π_{1} which sends (E, P) to (E, P, H_{1}(E)) where H_{1}(E) is the canonical subgroup of E and which gives rise to an isomorphism from X_{≥r} to X_{0}(p)_{≥r}. We then have M_{k}(Γ_{1}(N) ∩ Γ_{0}(p)) ⊂ H^{0}(π_{1}^{−1}(X_{>p^{−p/(p+1)}}), ω_{X_{0}(p)^{an}}^{⊗k}). Furthermore,

H^{0}(π_{1}^{−1}X_{>p^{−p/(p+1)}}, ω_{X_{0}(p)^{an}}^{⊗k}) ↪ H^{0}(X_{>p^{−p/(p+1)}}, ω_{X^{an}}^{⊗k}) ↪ H^{0}(X_{>p^{−p/(p+1)}}, ℋ_{k−2} ⊗ Ω_{X}^{1}(logC))

as ω^{⊗k−2} = Fil^{k−2}ℋ_{k−2} ↪ ℋ_{k−2} and the Kodaira-Spencer isomorphism ω^{⊗2} ≃ Ω_{X}^{1}(logC). By calculating θ^{k−1} at cusps, one checks (Corollary 6.3.1 in [11]) that, if the (normalised) p-adic valuation of an eigenvalue of U_{p} (resp. V) on M_{k}(Γ_{1}(N) ∩ Γ_{0}(p)) (resp. H) is fixed and is strictly less than k − 1, the induced map of the subspace of the cusp forms in S_{k}(Γ_{1}(N)∩Γ_{0}(p)) of the slope to the subspace of classes in H_{par} of the slope is an injection. One then checks that it also is a surjection by dimension-counting (Lemma 6.5 in [11]). Finally, deduce the subspace of modular forms of the fixed slope maps isomorphically to the subspace of classes in H of the slope by calculating the dimensions of their quotients by the cusp forms and the parabolic cohomology classes with the slope respectively in terms of the degrees of SS and the cuspidal divisor (which turn out equal).

Remark. Kassaei in [25] reproves the theorem by a very different approach. Given an overconvergent eigenform f of weight k ≥ 2 and level Γ_{1}(Np) with U_{p} acting with nonzero eigenvalue α (in which case it is said to be of finite slope), [25] observes (Proposition 4.3 in [25]) that if it is defined over the complement in X_{1}(p)^{an} of the component of the ordinary locus containing 0, g =^{def} f − (p^{k−1}/α)f|F extends further to the component. He then observes that the infinite sum g + (p^{k−1}/α)g|F + (p^{k−1}/α)^{2}g|F^{2} + ⋯ converges and, if the slope v_{p}(α) is strictly less than k − 1, one can glue it with f.

His approach has since been generalised by the author [29] to the case of (Hilbert) modular forms on moduli spaces of Hilbert-Blumenthal abelian varieties. Similarly, V.Pilloni in his thesis proves an analogue for (Siegel) modular forms on GSp_{4} over Q.

Let K be as above and O be its ring of integers. Suppose we have an overconvergent modular form of weight k ∈ Z_{≥1} and level Γ_{1}(N) in M_{k}^{≥r}(N) for some p^{−p/(p+1)} < r < 1, and suppose that it is an eigenform with its all eigenvalues defined in K and that it is of finite slope, i.e., the U_{p} eigenvalue is nonzero. It is certainly a convergent modular form, in other words, an element of M_{k}^{≥1}(N). Think of it as a Katz p-adic modular function, it corresponds, by duality (Theorem 5.3 in [23]), to a homomorphism T(N) → O where T(N) is the “full” Heck algebra lim_{←}(T_{2}(Np^{m}) ⊗_{Z} Z_{p}) where T_{2}(Np^{m}) is the Hecke algebra generated over Z by T_{p} for all prime p and ⟨q⟩ for the primes q not dividing Np acting on S_{2}(Np^{m}). It then follow from a result of Hida (Theorem II in [22]), which associates a Galois pseudo-representation G_{Q} → GL_{2}(T(N)_{𝔪}) to the corresponding maximal ideal 𝔪 ⊂ T(N), that there exists a semi-simple Galois representation, obtained by composition with T(N)_{𝔪} → O, unramified outside Np, associated to the Katz p-adic modular function and therefore the overconvergnet modular form.

Galoios representations associated to overconvergent modular forms are characterised by

Theorem 5 (Kisin [27]) Galois representations associated to overconvergent eigenforms of finite slope are trianguline in the sense of Colmez [16].

Perhaps the best reference for this is in [2].

We have criteria for determining classical overconvergent modular forms in terms of their associated Galois representations. For overconvergent modular forms of weight k ≥ 2 and of finite slope,

Theorem 6 (Kisin [27]) A finite slope overconvergent eigenform of weight k ≥ 2 whose Galois representation is potentially semistable at p is either a classical modular form or has “critical” slope k − 1 and it is in the image of θ^{k−1} (In particular, it is not even overconvergent; it is convergent (Corollary 10, [14]) ).

This is Theorem 6.6 in [27]. Kisin defines (Theorem 6.3 in [27]) p-adic (crystalline) periods V_{f} → (B_{cris}^{+} ˆ⊗_{Q_{p}}K)^{φ=a_{p}(f)} of Galois representations G_{Q} → GL(V_{f}) associated to overconvergent eigenforms f by p-adically interpolating periods of classical modular forms. This follows from observing that the locus of classical modular forms is Zariski dense in the Coleman-Mazur eigencurve [15], which in fact is an immediate consequence [10] of the theorem of Coleman [11]. As a result, one can read slope from the action of Frobenius on periods, and relate “small slope” and “local Galois representations (at p) associated to overconvergent eigenforms”; remarkably, “p-adic Hodge theory” forces overconvergent eigenforms of finite slope to have slope in [0, k − 1]. Then the theorem follows from the theorem of Coleman above.

For overconvergent modular forms of weight 1 which correspond to Artin representations, we have

Theorem 7 (Buzzard-Taylor [8], Buzzard [6]) Let L be a finite extension of Q_{p} with ring of integers O and maximal ideal λ. Suppose that p ≥ 5. Suppose also that f is an overconvergent eigenform of weight one defined over O and its associated Galois representation ρ_{f} : G_{Q} → GL_{2}(O) satisfies the following condition.

1. ρ_{f} ramifies at only finitely many primes

2. the restriction ρ_{f}|_{D_{p}} to the decomposition group D_{p} at p is a direct sum of characters α_{p} and β_{p} : D_{p} → O^{×} such that α_{p}(I_{p}) and β_{p}(I_{p}) are finite and (α_{p}/β_{p}) is nontrivial modulo λ.

3. the reduction (ρ_{f} mod λ) is absolutely irreducible

Then f is a classical weight one cuspidal eigenform.

Remarks. 1. When ρ_{f} is unramified at p, this is a result of Buzzard and Taylor [8].

2. The theorem holds for p = 2 albeit with more conditions for the main result of Dickinson [18] to work, and has applications [7] to the strong Artin conjecture for odd icosahedral Gal(Q̄/Q) → GL_{2}(C).

3. The forthcoming work [30] of the author generalises the theorem to the Hilbert case (p ≥ 5 and p = 2), and proves many new cases of the strong Artin conjecture for totally odd two-dimensional icosahedral Artin representations of the absolute Galois group of a totally real field.

8 Coleman p-adic analytic families of modular forms

In this section, we construct “Coleman families of overconvergent p-adic modular forms” (Theorem 13).

Firstly, we shall summarise the section. Following Serre, Coleman defines “weights” to be characters Z_{p}^{×} → C_{p}^{×} and calls the rigid space W over Q parameterising the characters weight space. There are (p − 1) connected components all isomorphic to the unit ball B; its restriction (which amounts to specifying the action of Diamond operators) to the subgroup μ_{p−1} = (Z/pZ)^{×} of torsion elements in Z_{p}^{×} determines it. We tacitly assume this and work always with B.

Given a character κ : Z_{p}^{×} → C_{p}^{×}, an overconvergent modular form of weight κ and level Γ_{1}(Np) is roughly defined to be the Eisenstein series E_{κ} of weight κ, which has long been known to exist, times a rigid analytic function defined on an admissible open subset of the rigid generic fibre of X_{1}(Np) which may be thought of as an overconvergent modular form of weight zero. The definition of the action of U_{p} is slightly more tricky; essentially the action of U_{p} on E_{κ} forces us to multiply forms by the “fudge factor” E_{1}(q)/E_{1}(q^{p}). Note that it is critical to the construction that the Eisenstein series E_{1} of weight 1 does not vanish on the ordinary locus and therefore on its tubular neighbourhood.

A “family, of weights parameterised by an admissible open subset U ⊂ B”, is then defined roughly to be the set M(U) of overconvergent modular forms of weight κ for κ ∈ U; one might just as well think of this as a “sheaf” of Banach modules of overconvergent modular forms. Let M denote the union of all M(U) for U contained in the ball of radius p^{(p−2)/(p−1)}.

The algebra T ⊂ End(M) of Hecke operators acting on M turns out “no good” for applications of Coleman’s generalisation of Serre’s theory of Banach spaces (by which one wants to single out where U_{p} acts with finite slope, analogous to the Hida idempotent), and Coleman considers that, given a character κ, a submodule M^{α}(B_{κ,≤r}) of families over B_{κ,≤r} ⊂ B_{<p^{(p−2)/(p−1)}} centred at κ of a “small enough” radius r < p^{(p−2)/(p−1)} and of a fixed slope α < ∞, and works instead with the algebra T_{B_{κ,≤r}}^{α} of Hecke operators acting on M^{α}(B_{κ,≤r}). The Coleman family is then a homomorphism F : SpecT_{B_{κ,≤r}}^{α} → B_{κ,≤r} (See Theorem 13); the pull-back along a point of B_{κ,≤r} corresponds by duality to an overconvergent modular form of weight (character) corresponding to the point. By the theorem of Coleman, “overconvergent modular forms of small slope are classical”, if a character is an“integer weight k” and if it satisfies α < k − 1, then the specialisation gives rise to a classical modular form and one may think of F as “interpolation by overconvergent modular forms of a classical modular form”

We shall freely use facts (if not all of them) about Serre’s theory of p-adic Banach spaces and notations defined as in Appendix 2.

For brevity, let p be an odd prime. The units Z_{p}^{×} of the p-adic integers Z_{p} is isomorphic to μ_{p−1} × (1 + pZ_{p})^{×}, where μ_{p−1} is the maximal torsion subgroup of Z_{p}^{×} and denote the projection onto μ_{p−1} by x ↦ ⟨x⟩_{μ_{p−1}}. This is just x ↦ lim_{n→∞} x^{p^{n}}. Define ⟨x⟩_{(1+pZ_{p})^{×}} to be x⟨x⟩_{μ_{p−1}}^{−1}.

Let W (resp. B) be the rigid generic fibre over Q_{p} of the formal Z_{p}-scheme SpfZ_{p}[Z_{p}^{×}] (resp. SpfZ_{p}[(1 + pZ_{p})^{×}]). One checks that the C_{p}-valued points W(C_{p}) equals the continuous homomorphisms Z_{p}^{×} → C_{p}^{×}. It is a union of (p − 1) components W^{(i)} (1 ≤ i ≤ p − 1) each of which is a unit ball isomorphic to B. The C_{p}-valued points W^{(i)}(C_{p}) are thought of as the set of characters Z_{p}^{×} → C_{p}^{×} whose restrictions to μ_{p−1} ⊂ Z_{p}^{×} send x ∈ μ_{p−1} to x^{i}.

Let B_{<p^{(p−2)/(p−1)}} ⊂ B be the ball centred at 0 of B of radius < p^{1−1/(p−1)} = p^{(p−2)/(p−1)}. The constraint on the radius is because ⟨x⟩_{(1+pZ_{p})^{×}}^{w}, defined to be exp(w⟨x⟩_{(1+pZ_{p})^{×}}) is then convergent if w ∈ B_{<p^{(p−2)/(p−1)}} for x ∈ Z_{p}^{×} (note that |w| < p^{(p−2)/(p−1)} and (⟨x⟩_{(1+pZ_{p})^{×}} − 1) ∈ pZ_{p} and therefore |w⟨x⟩_{(1+pZ_{p})^{×}}| = |w((⟨x⟩_{(1+pZ_{p})^{×}} − 1) + 1)| < p^{(p−2)/(p−1)}p^{−1} = p^{−1/(p−1)}). We then let A be B_{<p^{(p−2)/(p−1)}} × (Z/(p − 1)Z). Coleman calls the elements of A accessible (the term coined in 1.4 of [15]).

The C_{p}-valued points of A may be thought of as the set of characters

x ↦ ⟨x⟩_{μ_{p−1}}^{i}⟨x⟩_{(1+pZ_{p})^{×}}^{w}

for 1 ≤ i ≤ p − 1 and w ∈ B_{<p^{(p−2)/(p−1)}} and therefore A embeds into W.

We shall define overconvergent modular forms of accessible weights using well-known p-adic properties of Eisenstein series. We recall these firstly.

Let ζ^{∗} be the “family (over W) of the p-adic L-functions L_{p}” defined such that if a character κ ∈ W(C_{p}) is the character x ↦ ⟨x⟩_{(1+pZ_{p})^{×}}^{w}χ(x) for w ∈ C_{p} with |w| < p^{(p−2)/(p−1)} and a character χ of finite order, ζ^{∗}(κ) = L_{p}(1 − w, χ). In view of Theorem 12.2 in [35], it may be defined more explicitly. See B.1 in [12].

Definition For κ ∈ W, define

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

where σ_{κ}^{∗}(n) = ∑_{0<d|n,(n,p)=1} κ(d)d^{−1}.

This gives an example of overconvergent modular forms.

Proposition 8 If κ is the character x ↦ ⟨x⟩_{(1+pZ_{p})^{×}}^{k}χ(x) for k ∈ Z and a character χ : Z_{p}^{×} → C_{p}^{×} of finite order, then G_{κ} is the q-expansion of an overconvergent modular form of weight k of level Γ_{1}(lcm{p, condχ}) and Z/lcm{p, condχ}Z acts by x ↦ χ(x)⟨x⟩_{μ_{p−1}}^{−k}. Moreover if k ≥ 1, it is (the q-expansion of) a classical modular form.

Proof. Firstly, it follows from Theorem 4.5.1 in [26] that it is the q-expansion of p-adic modular forms. Then one checks that its U_{p}-eigenvalue is 1. Finally deduce that it is overconvergent by Proposition II 3.22 in [20]. It follows from Corollary 7.2.14 in [28] that it is classical. Note that, since ⟨x⟩_{(1+pZ_{p})^{×}}^{k}χ(x) = x^{k}(⟨x⟩_{μ_{p−1}}^{−1})^{k}χ(x), the nebentypus character is defined by the character x ↦ (⟨x⟩_{μ_{p−1}}^{−1})^{k} of μ_{p−1} and the character χ of μ_{p−1} × (1 + pZ_{p})^{×}/(1 + condχZ_{p})^{×}.

It is more useful to have an example of overconvergent modular forms with constant term 1.

Definition For κ ∈ W

E_{κ}(q) = (ζ^{∗}(κ)/2)^{−1}G_{κ}(q).

and let E_{1}(q) be E_{x↦⟨x⟩_{(1+pZp)^{×}}}(q) a classical modular form of weight 1 and of level Γ_{1}(p) and (Z/pZ)^{×} acts by x ↦ ⟨x⟩_{μ_{p−1}}^{−1}.

Given an explicit example, E_{κ}, of overconvergent modular forms “of weight κ” which “generalises” an overconvergent modular form of weight k ∈ Z, Coleman defines an overconvergent modular form of weight κ to be “one”(by which we mean a q-expansion) which, when divided by the overconvergent modular form E_{κ} “of weight κ”, gives an overconvergent modular form of weight 0. The definition critically depends on E_{κ}; in particular, on the fact that we have an explicit overconvergent modular form whose weight “varies p-adically” and which does not vanish on a tubular neighbourhood of the ordinary locus containing the cusp ∞.

Definition We say a series F (q) = ∑_{n=0}^{∞} c_{n}q^{n} ∈ C_{p}[[q]] is the q-expansion of an overconvergent modular form of level Γ_{1}(Np) of weight κ = (w, i) ∈ B_{<p^{(p−2)/(p−1)}} × (Z/(p − 1)Z) = A if F (q)/E_{1}(q)^{w} is the q-expansion of an overconvergent function in O(X_{1}(p)_{≥r}) for some r ∈ (p^{−1/(p+1)}, 1) such that F |⟨x⟩ = x^{i}F for x ∈ μ_{p−1} ⊂ (Z/NpZ)^{×}.

We let M_{κ}(Np) denote the space of overconvergent modular forms of weight κ and level Γ_{1}(Np).

Definition For an admissible open subset U ⊂ B_{<p^{(p−2)/(p−1)}}, we say a formal qexpansion ∑_{n=0}^{∞} c_{n}(w)q^{n} ∈ O(U)[[q]] with coefficients c_{n}(w) rigid analytic functions on U is a family over U of overconvergent modular (resp. cusp) forms of level Γ_{1}(Np) if, for any w ∈ U, its quotient by E_{1}(q)^{w} is the q-expansion of an overconvergent function on X_{1}(p)_{≥r} (resp. which vanishes at the cusps of X_{1}(p)_{≥r} ).

Let M(N) (resp. S(N)) be the O(B_{<p^{(p−2)/(p−1)}})-module of all families over U ⊂ B_{p^{(p−2)/(p−1)}} of overconvergent modular (resp. cusp) forms, M(N)^{(i)} (resp. S(N)^{(i)}) for i mod (p − 1) be the sub O(B_{<p^{(p−2)/(p−1)}})-module of families of overconvergent modular (resp. cusp) forms F such that F |⟨x⟩ = x^{i}F for x ∈ μ_{p−1}.

Let k be an integer. Suppose E is an overconvergent modular form in M_{k}^{≥r}(Np) for some p^{−1/(p+1)} < r < 1 and suppose it does not vanish on X_{1}(p)_{≥r}. Let e denote the overconvergent function E(q)/E(q^{p}). One can then define the action of U_{p} on M_{k}^{≥r}(Np) by the composite

M_{k}^{≥r}(Np) →^{E^{−1}} O(X_{1}(p)_{≥r}) →^{U_{p}∘e} O(X_{1}(p)_{≥r^{1/p}}) ↪ O(X_{1}(p)_{≥r}) →^{E} M_{k}^{≥r}(Np).

This follow from the observation of Coleman in B3 of [12]. Following this observation, we now define the action of U_{p} on overconvergent modular forms of weight κ and on families of overconvergent modular forms.

Lemma 9 There exists a rigid function e on ⋃_{r∈(p^{−1/(p+1)},1]} X_{1}(p)_{≥r} with q-expansion E_{1}(q)/E_{1}(q^{p}). For any ϵ ∈ R such that |p| < ϵ, there exists r ∈ (p^{−1/(p+1)}, 1] such that e is defined on X_{1}(p)_{≥r} and |e − 1| ≤ ϵ.

By the lemma, for w ∈ B_{<p^{(p−2)/(p−1)}}, there exists r such that |e − 1| < p^{−1/(p−1)}|w^{−1}|. Note that e^{w} =^{def} exp(w log(1 + (e − 1))) converges if and only if |w||log(1 + (e − 1))| ≤ p^{−1/(p−1)}, which by lemma 5.5 in [35] is equivalent to |w||e − 1| ≤ p^{−1/(p−1)}.

For κ = (w, i) ∈ A, we then define the operator U by the composite

M_{κ}^{≥r}(Np) →^{E_{1}^{−w}} O(X_{1}(p)_{≥r}) →^{U_{p}∘e^{w}} O(X_{1}(p)_{≥r^{1/p}}) ↪ O(X_{1}(p)_{≥r}) →^{E_{1}^{w}} M_{κ}^{≥r}(Np)

where U_{p} is the operator on the weight 0 overconvergent modular forms of level Γ_{1}(Np) defined by (f|U_{p})(E, P, Q) = 1/p ∑_{⟨Q⟩≠D⊂E[p]} f(E/D, (P + D)/D, (Q + D)/D).

Let t_{1} < t_{2} < ⋯ be an increasing sequence in |C_{p}| ∩ [1, p^{(p−2)/(p−1)}) such that for every n ∈ Z_{≥1}, it follows from the lemma that there exists r_{n} ∈ p^{Q} ∩ (p^{−1/(p+1)}, 1] such that e is defined on X_{1}(p)_{≥r_{n}} and |e − 1| ≤ p^{−1/(p−1)}t_{n}^{−1}. The {B_{≤t_{n}} × X_{1}(p)_{≥r_{n}}}_{n∈Z_{≥1}} then define an admissible covering of B_{<p^{(p−2)/(p−1)}} × X_{1}(p)_{>p^{−1/(p+1)}}. This follows from Maximum Modulus Principle (See Proposition 5 in 9.1.4 of [4]). For every n ∈ Z_{≥1}, we may extend U as above “O(B_{≤t_{n}})-linearly” to a continuous O(B_{≤t_{n}})-linear operator U on O(B_{≤t_{n}} × X_{1}(p)_{≥r_{n}}) which, thought of as a family of operators over B_{≤t_{n}}, specialises to U_{p} ∘ e^{w} at w ∈ B_{≤t_{n}}.

Lemma 10 O(B_{≤t_{n}} ×X_{1}(p)_{≥r_{n}}) is orthonormisable over O(B_{≤t_{n}}) and U acts completely continuously.

Proof. This is A.5.2 and A.5.1 in [12].

Lemma 11 The restriction of the series det(1 − T U|O(B_{≤t_{n+1}} × X_{1}(p)_{≥r_{n+1}})), which is analytic on B_{≤t_{n+1}} × C_{p}, to B_{≤t_{n}} × C_{p} is the characteristic series det(1 − T U|O(B_{≤t_{n}} × X_{1}(p)_{≥r_{n}}). In fact, there is a unique rigid analytic function P (w, T) on B_{<p^{(p−2)/(p−1)}} ×C_{p} which is a power series in T with coefficients rigid analytic functions on B_{<p^{(p−2)/(p−1)}}, which converges for |w| < p^{(p−2)/(p−1)} and which interpolates the characteristic series of U_{p} on r-overconvergent moduar forms of weight k ∈ Z for p^{−p/(p+1)} < 1.

Proof. This is Theorem B3.2 in [12].

The lemma applies to the subspace of “cuspidal” rigid functions in O(B_{<p^{(p−2)/(p−1)}} × X_{1}(p)_{>p^{−1/(p+1)}}) which vanish at the cusps and on which the μ_{p−1} acts by x ↦ x^{i} for some 1 ≤ i ≤ p−1, and we shall denote by P_{i}^{0}(w, T) the corresponding rigid analytic functions on B_{<p^{(p−2)/(p−1)}} × C_{p}.

For x ∈ (Z/NZ)^{×} × Z_{p}^{×}, define an action, “Diamond operator in families”, of ⟨x⟩ on M(N) (and also on S(N)) by (F |⟨x⟩)_{w}(q) = ⟨x⟩_{(1+pZ_{p})^{×}}^{w}E_{1}^{w}(q) ((F_{w}E_{1}^{−w})|⟨x⟩_{(Z/NpZ)^{×}}) (q) for w ∈ B_{<p^{(p−2)/(p−1)}}, where by ⟨x⟩_{(Z/NpZ)^{×}}, we mean the composite of the projection to (Z/NpZ)^{×} followed by the “standard” Diamond operator acting on level Np forms. When k ∈ Z, (F |⟨x⟩)_{k} = x^{k}F_{k}|⟨x⟩.

For a prime l, let u_{l} be the operator on O(B_{<p^{(p−2)/(p−1)}})[[q]] sending ∑ a_{n}q^{n} to ∑ a_{nl}q^{n}.

Lemma 12 For each prime l, there is a unique continuous operator T (q) on M(N) such that

• (F |T (l))_{w} = E_{1}^{w}((F_{w}E_{1}^{−w})|U) for l = p

• (F |T (l))(q) = u_{l}(F (q)) for l|N

• (F (T (l))(q) = u_{l}(F (q)) + l^{−1}(F |⟨l⟩)(q^{l}) for l not dividing Np.

Proof. This is Lemma B5.1 in [12].

Let T := T_{K} be the O(B_{<p^{(p−2)/(p−1)}})-algebra generated over O(B_{<p^{(p−2)/(p−1)}}) by the operators ⟨x⟩ for x ∈ (Z/NpZ)^{×} and the T (l). Define T (n) in T for the positive integers n by

∑_{n≥1} T (n)n^{−s} = ∏_{l|̸Np}(1 − T (l)l^{−s} + ⟨l⟩l^{−1−2s})^{−1} ∏_{l|Np}(1 − T (l)l^{−s})^{−1}

where the products are over primes l.

Fix a non-zero rational number α ∈ Q, 1 ≤ i ≤ p − 1, and κ ∈ B_{<p^{(p−2)/(p−1)}}. We shall construct a slope α family of overconvergent modular forms interpolating an overconvergent modular form of weight κ. When α = 0, this is a “Hida family”.

Let r ∈ p^{Q} ∩ (0, p^{(p−2)/(p−1)}) such that the set of closed points P in the zero locus of P_{i}^{0}(w, T) such that v_{p}(T (P)) = −α is finite of degree d over the affinoid disc B_{κ,≤r} ⊂ B_{<p^{(p−2)/(p−1)}} centered at κ of radius ≤ r. As the geometry of the ‘variety” cut out by the characteristic power series is ‘reasonable”, when cut out further by “slope”, one can choose a small enough neighbourhood of κ in the weight space, so that it is finite over it. This formally follows from A.5.5.2 in [12].

We apply the main theorem of Appendix 2. Let Q, a polynomial of degree d in O(B_{κ,≤r}), be the factor of P_{i}^{0}(w, T) corresponding to the degree d map. Then N(Q), with respect to the action of U on S(N)^{(i)}, is projective O(B_{κ,≤r})-module of rank d (This is analogous in Hida theory to singling out “ordinary forms” by the Hida idempotent on the space of modular forms over Z_{p}). As O(B_{κ,≤r}) is an affinoid, it is in fact a free module. Let T^{Col} := T_{κ}^{α}(N) be the image of T⊗O(B_{κ,≤r}) in End_{O}(B_{κ,≤r})(N(Q)), which is a free O(B_{κ,≤r})-module of finite rank.

Theorem 13 (Construction of Coleman families) Let x be a closed point SpL → SpT^{Col} corresponding to T^{Col} → L for a finite extension L ⊂ C_{p} of K. Set F_{x}(q) be the formal qexpansion whose n-th coefficient in L is the image of T (n) by x. If k is an integer k > α+1 and (x ↦ x^{k}) ∈ B_{κ,≤r}(K), the map (SpT^{Col} ×_{SpO(B_{κ,≤r}),x} SpK)(L) to L[[q]] sending x to F_{x}(q) defines a bijection onto the set of q-expansions of classical cusp eigenforms over L of weight k and level Γ_{1}(Np), μ_{p−1} acts by x ↦ x^{i}, and slope α.

9 Appendix

9.1 Appendix 1 Crash course on rigid geometry

9.1.1 Tate’s classical approach

Let K be a field. A map | | : K → R_{≥0} is a non-archimedean absolute value if it satisfied the following conditions.

|a| = 0 if and only if a = 0 |ab| = |a||b| for a, b ∈ K |a + b| ≤ max{|a|, |b|} It is discrete if |K^{×}| is discrete in R_{≥0}. We say K with a non-archimedean absolute value | | is complete if every Cauchy

sequence with respect to | | converges in K.

Exercise If a, b ∈ K and |a| ≠ |b|, then |a + b| = max{|a|, |b|}.

Exercise A series ∑_{ν} a_{ν} with a_{ν} ∈ K for all n is a Cauchy sequence if and only if lim_{ν→∞} |a_{ν}| = 0. In particular if K is complete, ∑_{ν} a_{ν} converges if and only if lim_{ν→∞} |a_{ν}| = 0.

Exercise K comes equipped with the metric topology, which is totally disconnected, i.e., any subset in K consisting of more than one point is not connected.

Fix an algebraic closure K̄ of K.

Definition Let B^{n}(K̄) = {(x_{1}, …, x_{n}) ∈ (K̄)^{n}| |x_{i}| ≤ 1} f = ∑_{ν∈N^{n}} c_{ν}X^{ν} := ∑_{ν=(ν_{1},...,ν_{n})∈N^{n}} X_{1}^{ν_{1}} ⋯ X_{n}^{ν_{n}} ∈ K[[T]] := K[[X_{1}, …, X_{n}]] con-

verges on B^{n}(K̄) if and only if lim_{ν→∞} |c_{ν}| = 0

Definition The K-algebra T_{n} = K⟨X_{1}, …, X_{n}⟩ of all power series in K[[X_{1}, …, X_{n}]] which converges on B^{n}(K̄) is the Tate algebra of strictly convergent power series.

Definition The Gauss norm on T_{n} is defined by |f| = max_{ν}|c_{ν}| for f = ∑_{ν} c_{ν}. Then T_{n} is complete with respect to the Gauss norm, and therefore it is a Banach

space over K.

If 𝔪 is a maximal ideal of T_{n}, then the field T_{n}/𝔪 is a finite field extension of K. The map from B^{n}(K̄) to the set MaxT_{n} of all maximal ideals in T_{n} sending x ∈ B^{n}(K̄)

to 𝔪_{x} = {f ∈ T_{n}|f(x) = 0}; the induced map Gal(K̄/K)\B^{n}(K̄) → MaxT_{n} is bijective. The T_{n} is noetherian, i.e, evey ideal in T_{n} is finitely generated.

Definition A K-algebra A is an affinoid K-algebra if there is an epimorphism α : T_{n} → A for some n ∈ N.

An affinoid K-algebra A is noetherian and jacobson.

Definition For an affinoid K-algebra α : T_{n} → A, define the residue norm | |_{α} on A to be |α(f)|_{α} = inf_{g∈kerα} |f − g|. It is a K-algebra norm with respect to which A is a Banach K-algebra.

Definition Let MaxA denote the set of all maximal ideals in A.

Definition Define the sup semi-norm | |_{sup} on A, i.e., a norm except for the condition |f|_{sup} = 0 ⇒ f = 0) by |f|_{sup} = sup_{x∈MaxA}|f(x)|_{A/x} where by f(x) we mean the image of f by A → A/x ↪ K̄. If A is reduced, | |_{sup} is a norm equivalent to any residue norm.

Let T_{n} → A be an affinoid K-algebra. If T_{n} = Gal(K̄/K)\B^{n}(K̄) has the affine n sub-K̄-space topology and MaxA ↪ MaxT_{n} induces the topology, called the canonical topology on MaxA. One can define the topology more explicitly.

Proposition 14 The canonical topology on X = MaxA is finer than Zariski topology. IN particular, the affinoid subdomains of X form a basis for the canonical topology.

Proof. This is Theorem 3 and Corollary 4 in 7.2.5 in [4].

Definition Let A be an affinoid K-algebra and X = MaxA. A subset U ⊂ X is called an affinoid subdomain if there exists X′ = MaxA′ → X which maps into U and which is universal with respect to the property.

For f_{1}, …, f_{n}, g ∈ A with no common zeros on X, the subset X(f/g) of points x ∈ X such that |f_{1}(x)| ≤ |g(x)|, …, |f_{n}(x)| ≤ |g(x)| is called a rational subdomain. It is an affinoid subdomain. The corresponding affinoid A⟨X_{1}, …, X_{n}⟩/(f_{1} − gX_{1}, …, f_{n} − gX_{n}) will simply be denoted by A⟨f/g⟩.

One equips X = MaxA with Grothendieck topology. The Grothendieck topology in which “admissible open” subsets are affinoid subdmains and “admissible open” coverings are finite coverings by affinoid subdomains is called the weak topology. The weak topology induces the canonical topology.

On X = MaxA with the weak topology, one defines a presheaf O_{X} by O_{X}(X(f/g)) = A⟨f/g⟩ for a rational subdomain X(f/g) = MaxA⟨f/g⟩. By Tate’s acyclicity theorem (Theorem 1 in 8.2.1.in [4]), it in fact defines a sheaf.

Formally one can strengthen the weak topology on X = MaxA to be strong topology which satisfies (if not uniquely characterise it) the following conditions.

G0. ∅ and X are admissible open subsets of X.

G1. Let U ⊂ X be an admissible open subset and V be a set. Assume there exists an admissible covering {U_{i}}_{i∈I} of U such that V ∩ U_{i} is admissible open in X for all i ∈ I. Then V is admissible open in X.

G2. Let {U_{i}}_{i∈I} be a covering of an admissible open subset U ⊂ X such that U_{i} is admissible open in X for all i ∈ I. Assume that the covering has a refinement. Then it is an admissible covering of U.

And the sheaf O_{X} extends uniquely to a sheaf with respect to the strong topology, which we shall denote again by O_{X}.

Definition An affinoid variety over K is a pair X, O_{X} of a set X = MaxA for an affinoid K-algebra A equipped with strong topology and a sheaf O_{X} with respect to it.

The homomorphisms of affinoid K-algebras A → B correspond to the morphisms of affinoid varieties SpB → SpA.

Definition A rigid analytic variety over K is a set X with a Grotendieck topology satisfying G0-G2 and a sheaf O_{X} of K-algebras such that there is an admissible covering X = ⋃_{i∈I} X_{i} where each (X_{i}, O_{X}|X_{i}) is an isomorphic to an affinoid variety.

Rigid analytic varieties are locally ringed spaces with Grothendieck topology (stronger than Zariki topology) obtained by “gluing” affinoid spaces. More formally,

Proposition 15 Let X be a set with a covering X = ∪_{i∈I}X_{i} such that each X_{i} comes equipped with a Grothendieck topology satisfying the properties G0-G2 and is “compatible”. Then there exists a Grothendieck topology on X uniquely characterised by the following properties.

1. X_{i} is admissible open in X and it induces the Grothendieck topology on X_{i} by restriction.

2. It satisfies the properties G0-G2.

3. {X_{i}}_{i∈I} is an admissible covering of X.

There is a functor “an” from the category of schemes X over K locally of finite type to the category of rigid analytic varieties X^{an} over K. For example, (SpecA)^{an} is SpA.

9.1.2 Raynaud’s approach

Let R be the valuation ring with a uniformiser π of a complete non-archimedean field K with absolute value | |. Raynuad’s construction holds with greater generality but fr our applications, this suffices. An R-algebra A is of topologically finite type if A is of the form A = R⟨X_{1}, …, X_{n}⟩/𝔞 for some ideal 𝔞.

Lemma 16 Any R-algebra of topologically finite type is (π)-adically complete and separated.

For a R-algebra A topologically of finite type, a formal scheme SpfA is a locally ringed space (𝔛, O_{𝔛}) where 𝔛 is a set Spec(A ⊗_{R} R/(π)) with Zariski topology and a sheaf O_{𝔛} such that O_{𝔛}(𝔛 − V (f)) = lim_{←n}(A ⊗_{R} R/(π^{n+1}))[f^{−1}] for f ∈ A. A formal scheme over R locally of topologically finite type is a locally ringed space which is locally SpfA for a R-algebra A of topologically finite type.

There is a functor rig from the category of formal R-schemes 𝔛 locally of topologically finite type to the category of rigid spaces 𝔛^{rig} over K. For example, (SpfA)^{rig} = Sp(A ⊗_{R} K). The functor induces an equivalence of categories between the category of quasicompact admissible formal schemes over R, localised by admissible formal blow-ups and the category of quasi-compact quasi-separated rigid spaces over K.

Let X be a scheme over R locally of finite type. There are two rigid analytic varieties over K “associated to X”. Firstly, let (X ×_{R} K)^{an} be the rigid space over K associated to the “Zariski” generic fibre X ×_{R} K. Secondly, we let 𝔛^{rig} be the rigid space over K associated to the formal completion 𝔛 along the closed fibre X ×_{R} (R/π). Then there is a canonical morphism of rigid analytic varieties 𝔛^{rig} → (X × R)^{an}. It is an open immersion if X is separated of finite type, and an isomorphism if X is proper.

There is a morphism of topoi sp : (𝔛)^{rig} → 𝔛 ×_{R} (R/π). We shall give its definition as a map of underlying sets as Berthelot does. Let {SpfA} be a finite open covering of 𝔛. Then {Sp(A⊗_{R} K)} defines an open affinoid covering of 𝔛^{rig}. If x ∈ 𝔛^{rig}, x ∈ Sp(A⊗_{R} K) for some A. It then corresponds to a K-algebra homomorphism A → K(x) which sends A into the valuation ring R(x) of K(x). It then induces A/(π) → k(x) =^{def} R(x)/𝔪_{R(x)} and define a closed point sp(x) of 𝔛 ×_{R} (R/π).

Proposition 17 Let X be a scheme over R locally of finite type. Let Z be a closed subscheme of X×_{R}R/(π) and 𝔛_{Z} be the formal completion of X along Z. Then sp^{−1}(Z) ≃ (𝔛_{Z})^{rig}.

9.2 Appendix 2 Serre’s theory of p-adic Banach spaces

Let K be a field with a non-trivial non-archimedean norm.

A Banach space over K is a vector space V over K equipped with a norm | | : V → R_{≥0} which satisfies |λv| = |λ||v| for λ ∈ K and v ∈ V and |v + w| ≤ max{|v|, |w|}, and V is complete with respect to the induced metric.

For example, let I be any set. Define V to be the set of functions f : I → K defined on I such that f(i) → 0 as i → ∞, i.e., for any ϵ ∈ R_{>0}, there are only finitely many i ∈ I satisfy |f(i)| ≥ ϵ. V is then a Banach space over K with respect to the norm |f| = max_{i∈I} |f(i)|.

If i ∈ I, denote by e_{i} ∈ V the function e_{i}(i) = 1 and e_{i}(j) = 0 for all j ≠ i, and |e_{i}| = 1 with respect to the norm. Then the e_{i} form a basis of V. More formally,

Definition Let V be a Banach space over K. An orthonormal basis for V over K is a collection {e_{i}}_{i∈I} of elements in V such that |e_{i}| = 1 for all i ∈ I and for all v ∈ V, there exists a unique collection λ_{i} ∈ K (i ∈ I) such that lim_{i→∞} λ_{i} = 0, v = ∑_{i∈I} λ_{i}v_{i} and |v| = max_{i∈I} |λ_{i}|.

Let V and W be Banach spaces over K. Define L(V, W) to be the set of continuous linear maps V → W. Then it is a Banach space over K with respect to the norm |u| = sup_{0≠x∈V} \frac{|u(v)|_{W}}{|v|_{V}} ∈ R_{<∞} for u ∈ V.

Suppose V (resp. W) is a Banach space over K with an orthonormal basis {e_{i}}_{i∈I} (resp. {f_{j}}_{j∈J}). Then a continuous linear map u : V → W is determined by the “matrix” (a_{ji}) defined by u(e_{i}) = ∑_{j∈J} a_{ji}f_{j} for i ∈ I.

Conversely, what does a collection of a_{ji} ∈ K gives rise to a continuous linear map u : V → W? It turns out that it suffices to demand that 1) for all i ∈ I, lim_{j→∞J} a_{ji} = 0,i.e., “the entries of the matrix tend to 0 down the column” and 2) there exists a constant C such that |a_{ji}| ≤ C for every i ∈ I and j ∈ J.

Definition Let V and W be Banach spaces over K and L(V, W) be the Banach space defined above. We say u : V → W is finite rank if the image of u is finite-dimensional.

Let F(V, W) ⊂ L(V, W) be the subspace of finite rank linear maps.

Definition The elements of the closure of F in L are called compact (or completely continuous) linear maps.

It is an elementary exercise to check that u : V → W is compact if and only if sup_{i,j} |a_{ji}| < ∞ and lim_{j→∞} sup_{i∈I} |a_{ji}| = 0.

Definition Let V be a Banach space over K with an orthonormal basis {e_{i}}_{i∈I} and u be a compact operator acting on V. Then define the characteristic power series in t of u on V to be

CP S(t) = det(1 − tu|V)

More explicitly, it is a series ∑_{m≥0} c_{m}t^{m} where

c_{m} = (−1)^{m} ∑_{S⊂I,card(S)=m} ∑_{σ:S→S} sgn(σ) ∏_{i∈S} a_{iσ(i)}

where (a_{ji}) is the matrix representing u with respect to {e_{i}}_{i∈I}.

Proposition 18 Let V be a Banach space over K and u be a completely continuous operator on V. Let a ∈ K be a zero of CP S(t) of order d. Then V decomposes uniquely as a direct sum V = N(a) ⊕ F (a) of closed u-invariant subspaces such that dimN(a) = d and 1 − au is nilpotent (resp. invertible) on N(a) (resp. F (a)).

Proof This is Theorem 12 in [32]

Proposition 19 Let K, V, u, a be as above. If we let Q(t) = ∏(1 − bt) where b runs through the conjugates of a with multiplicity. Then V decomposes as a direct sum V = N(Q) ⊕ F (Q) such that Q(u) is nilpotent (resp. invertible) on N(Q) (resp. F (Q)).

Proof One checks that u′ = 1 − Q(u) is a completely continuous operator on V. And then observe that a = 1 is a zero of det(1 − tu′|V) and apply the preceding proposition to u′ and a = 1.

Coleman generalises the Serre’s theory of p-adic Banach spaces over non-archimedean fields to Banach modules over Banach algebras.

A commutative ring A with a unit complete and separated with respect to a nontrivial ultrametric norm | | is called a Banach algebra. We say a Banach algebra A is semi-simple if the intersection of the maximal ideals in A is 0 and the residue norm on the quotient of A by any maximal ideal is multiplicative.

A module M over a Banach algebra A which is equipped with a ultrametric norm with respect to which it is complete and which satisfies |am| ≤ |a||m| is called a Banach module.

Coleman proves

Proposition 20 (Proposition A4.2 in [12]) Let M be a Banach module over a Banach algebra A and u be a completely continuous operator on M. Let a ∈ A be a zero of CP S(t) =^{def} det(1 − tu|M) of order d. Then M decomposes as a direct sum M = N(a) ⊕ F (a) of u-invariant closed A-submodules such that (1 − au)^{d}N(a) = 0 and 1 − au is invertible on F (a).

Furthermore,

Theorem 21 (Proposition A4.3 and A4.5) Let A, M, u, a as above. If CP S(t) ∈ A{{t}} factorises as the product of relatively prime factors Q(t) and S(t) ∈ A{{T }} where Q(t) is a polynomial in t of degree d with Q(0) = 1 and the leading coefficient a unit in A. We then have a decomposition M = N(Q) ⊕ F (Q) such that Q^{∗}(u) =^{def} u^{d}Q(u^{−1}) annihilates N(Q) and is invertible on F (Q).

Furthermore if A is semisimple, N(Q) is projective A-module of rank d.

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