Coleman s theory of p-adic modular forms

Transcription

1 Coleman s theory of p-adic modular forms Shu Sasaki 9th February 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 Overconvergent modular forms of level N Katz modular functions and Serre p-adic modular forms Overconvergent modular forms of level Np Overconvergent modular forms are classical? 8 8 Coleman p-adic analytic families of modular forms 12 9 Appendix Appendix 1 Crash course on rigid geometry Tate s classical approach Raynaud s approach Appendix 2 Serre s theory of p-adic Banach spaces Supported by EPSRC at King s College London 1

2 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 2

3 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 ( a c d b ) z = az+b. A subgroup Γ of SL cz+d 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( az+b) for γ = ( cz+d a c d b ) Γ. We say f is of weight k and of level Γ if f kγ = f for every γ Γ. If ( b ) Γ, then f(z + b) = f(z) and f has a Fourier expansion f(z) = cn (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) = where q = e 2πiz SL 2 (Z). Also, ζ(1 k) 2 + ( ) 2(2πi) σ k 1 (n)q n k 1 = (k 1)! n=1 (m,n) (0,0) (mz + n) k and σ k 1 (n) = 0<d n dk 1 is a modular form of weight k and level E k (q) p k 1 E k (q p ) = ζ(k 1) (1 p k 1 ) + σ 2 k 1(n)q n 3

4 where σk 1 (n) = 0<d n,(p,d)=1 dk 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 ( 0 1 ). The spaces M k (Γ 1 (M)) and S k (Γ 1 (M)) have a natural action of (Z/MZ) by d d where f d = (cz + d) k f( az+b ) for any ( a b cz+d c d ) SL 2(Z) such that c 0 and d 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 p 1 (f T p )(z) = (1/p) i=0 and U p for a prime p which divides M is defined by f (( 1 i 0 p ) z ) + p p f(pz). p 1 (f U p )(z) = (1/p) f (( ) ) 1 i 0 p 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 (1 U p p s ) 1 i=0 n=1 T (n)n s = p M(1 T p p s + p p 1 2s ) 1 p M 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 E 4(q) 12 X + 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 + 1 and Y = x + 2y, the equation defines a 12 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 3 (q) = c 2 (q) = 5 E 4 1 = 5 σ 3 (n)q n 240 n 1 ( 5 E E ) /12 = n 1 5σ 3 (n) 7σ 5 (n) q n 12 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 ( ˉQ p )/q Z Tate(q)( ˉQ p ). 4

5 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 ( a c d b ) (z, z ) = ( az+b, (cz + cz+d 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 ( a c d b ) (z, dz ) = ( az+b, (cz + cz+d 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 E1 (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 ) 5

6 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 ( c n (f)q n )(ω can ) k n=0 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) f((e/d, (P + D)/D)), D E[p] 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) f((e/d, (P + D)/D)) D E[p],D P ={0} 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 Q p of Q p, and W denote the Witt vectors W (F p ) of 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 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(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 6

7 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 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 X1 (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 r def k (N) = 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 : Ĝm 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 μp 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 mer 1 (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 mer 1 (N) 0 Z p ((q)). We let M 1 (N) 0 be the subring of elements f M mer 1 (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 mer 1 (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 1 p 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 1 k (N). 7

8 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 kn 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 N p 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 π1 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, 8

9 (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 Frobenius and the theta operator V : H k 2 (X >p 1/(p+1)) H k 2 (X >p p/(p+1)), F : H k 2 (X >p p/(p+1)) H k 2 (X >p 1/(p+1)), 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 1 X Z[1/N] K (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 9

10 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 (π1 1 (X >p p/(p+1)), ω k X 0 (p) ). Furthermore, an H 0 (π1 1 X >p p/(p+1), ω k X 0 (p) ) H 0 (X an >p p/(p+1), ω k X ) H 0 (X an >p p/(p+1), H k 2 Ω 1 X (logc)) as ω k 2 = Fil k 2 H k 2 H k 2 and the Kodaira-Spencer isomorphism ω 2 Ω 1 X (logc). By calculating θ k 1 at cusps, one checks (Corollary 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 r k (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 1 k (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]. 10

11 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 Qp K) ϕ=ap(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 Dp 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( ˉQ/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. 11

12 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 α B κ, r of Hecke operators acting on M α (B κ, r ). The Coleman family is then a homomorphism F : SpecT α B κ, r 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 μp 1. This is just x lim n x pn. Define x (1+pZp) to be x x 1 μ p 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 12

13 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 w (1+pZ p ), defined to be exp(w x (1+pZp) ) 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+pZp ) 1) pz p and therefore w x (1+pZp) = 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 i μ p 1 x w (1+pZ p) 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 w (1+pZ p ) 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) = ζ (κ) 2 where σ κ(n) = 0<d n,(n,p)=1 κ(d)d 1. + n 1 σ κ(n)q n This gives an example of overconvergent modular forms. Proposition 8 If κ is the character x x k (1+pZ p ) 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 k μ p 1. Moreover if k 1, it is (the q-expansion of) a classical modular form. Proof. Firstly, it follows from Theorem 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 in [28] that it is classical. Note that, since x k (1+pZ p) χ(x) = x k ( x 1 μ p 1 ) k χ(x), the nebentypus character is defined by the character x ( x 1 μ p 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. 13

14 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 1 μ p 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 nq 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 q- expansion 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 r k (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 r k (Np) by the composite M r k E 1 (Np) O(X 1 (p) r ) Up e O(X 1 (p) r 1/p) O(X 1 (p) r ) E M r (Np). k 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 ɛ R such that p < ɛ, there exists r (p 1/(p+1), 1] such that e is defined on X 1 (p) r and e 1 ɛ. 14

15 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 r κ (Np) E w 1 O(X 1 (p) r ) U p e w O(X 1 (p) r 1/p) O(X 1 (p) r ) Ew 1 M 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) rn and e 1 p 1/(p 1) t 1 n. The {B tn X 1 (p) rn } 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 of [4]). For every n Z 1, we may extend U as above O(B tn )-linearly to a continuous O(B tn )-linear operator U on O(B tn X 1 (p) rn ) which, thought of as a family of operators over B tn, specialises to U p e w at w B tn. Lemma 10 O(B tn X 1 (p) rn ) is orthonormisable over O(B tn ) 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 tn+1 X 1 (p) rn+1 )), which is analytic on B tn+1 C p, to B tn C p is the characteristic series det(1 T U O(B tn X 1 (p) rn ). 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 Pi 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 w (1+pZ p) E w 1 (q) ( (F w E1 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. κ 15

16 Lemma 12 For each prime l, there is a unique continuous operator T (q) on M(N) such that (F T (l)) w = E w 1 ((F w E w 1 ) 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 (1 T (l)l s ) 1 n 1 T (n)n s = l Np(1 T (l)l s + l l 1 2s ) 1 l Np 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 Pi 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 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 Pi 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 α κ(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 q- expansion 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(Bκ, 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. 16

17 9 Appendix 9.1 Appendix 1 Crash course on rigid geometry 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 ˉK of K. Definition Let B n ( ˉK) = {(x 1,..., x n ) ( ˉK) n x i 1} f = ν N c n ν X ν := ν=(ν 1,...,ν n) N X ν n 1 1 X ν n n K[[T ]] := K[[X 1,..., X n ]] converges on B n ( ˉK) 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 ( ˉK) 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 ( ˉK) to the set MaxT n of all maximal ideals in T n sending x B n ( ˉK) to m x = {f T n f(x) = 0}; the induced map Gal( ˉK/K)\B n ( ˉK) 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. 17

18 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 ˉK. 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( ˉK/K)\B n ( ˉK) has the affine n sub- ˉK-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 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 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. 18

19 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 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. 19