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p -adic Taylor Polynomials
S0019-3577(15)00127-5 http://dx.doi.org/10.1016/j.indag.2015.12.003 INDAG 370
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Indagationes Mathematicae
Received date: 8 December 2014 Accepted date: 21 December 2015 Please cite this article as: E. Nagel, p-adic Taylor polynomials, Indagationes Mathematicae (2016), http://dx.doi.org/10.1016/j.indag.2015.12.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
p-adic Taylor polynomials Enno Nagel
For a real number r ≥ 0, we define r-fold differentiability of a function on a p-adic vector space by the convergence of its Taylor polynomial expansion, and compare this differentiability definition with that by iterated divided differences, the textbook approach (from the 80’s) to define p-adic differentiability. This comparison applies to a recent definition of r-fold differentiability over a p-adic number field K that arises from the p-adic Langlands program over GL2 (K); yielding that this differentiability condition is equivalent to that via divided differences on K as vector space over Qp .
Contents Introduction non-Archimedean differential Calculus Fractional differential Calculus in p-adic Lie group representations Notations
2 2 5 6
1.
Fractional differentiability via iterated divided differences C ν -functions for a natural number ν C ρ -functions for ρ in [0, 1[ C r -functions for a real number r ≥ 0
6 7 8 9
2.
Taylor Polynomials Definition Differentiability of the Taylor polynomial coefficients The norm
9 9 10 13
1
3.
Comparison Necessity Sufficiency for functions of one variable Sufficiency for functions of many variables in Qp
14 14 18 19
4.
Fractional Differentiability from Representation Theory Definition Origin from Representation Theory Comparison
20 20 23 27
A.
Schikhof’s iterated divided differences 29 Definition of Schikhof’s iterated divided differences 30 Equivalence between Schikhof’s and our iterated divided differences 31
Introduction p-adic Calculus done right must use a differentiability definition stronger than that of ordinary Calculus to account for the total disconnectedness of the range. The prevalent definition (see [Sch84, Section 26ff.]), though conceptually sound, uses a differential whose number of variables doubles with each degree of differentiability and quickly becomes unwieldy. Since (fractional) p-adic differentiability has resurged in representation theory of p-adic Lie groups, interest arose in a different approach that keeps the number of variables at bay. We study a definition by the Taylor polynomial expansion on normed vector spaces, a function of two arguments (expansion point and offset), and compare this definition to those recently proposed in the representation theory of p-adic Lie groups. non-Archimedean differential Calculus Let us first discuss 1. how the differentiability condition over the real numbers falls short over the p-adic numbers and instead 2. is replaced over the p-adic numbers by a stronger differentiability condition via iterated divided differences, which however 3. in each iteration doubles the number of variables of the divided difference, but how a handier differentiability condition via the Taylor polynomial often comes to rescue.
2
Shortcomings of ordinary differentiability. Let K be a non-Archimedean field, that is, a field with an absolute value that is non-Archimedean, non-trivial and complete. The definition of differentiability over R transfers verbatim to K: A function : f X → K on an open subset X of K is (ordinarily) differentiable at a in X if there is f 0 (a) in K such that, for every sequence (xn ) in X, if xn → a then f (xn ) − f (a) → f 0 (a) xn − a Completeness and non-discreteness of K ensure that the derivative f 0 (a) exists in K and is unique. Still, this definition defies our intuition through the function f : Zp → Qp defined by X X an pn 7→ an p2n . (∗) n∈N
n∈N
It fulfills |f (·)| = |·|2 and so is differentiable with f 0 = 0. We diagnose though: • it is injective, but its derivative is 0, and • it is infinitely often differentiable but its Taylor polynomial expansion of degree greater than 1 does not converge. These pathologies occur because the intermediate-value theorem fails on the totally disconnected range K (in the wording that every continuous function f maps the smallest ball containing two points a and b onto that containing f (a) and f (b)). To compensate this failure, we define more strictly a function f : X → K on an open subset X of K as (non-Archimedeanly) differentiable if its divided difference f (x) − f (y) , f ]1[ (x, y) := x−y
defined for all distinct x and y in X, extends to a continuous function f [1] : X × X → K. For a function f : X → R the mean-value theorem proves the nonArchimedean and ordinary differentiability condition equivalent. Let us outline the first part of this article, where we expand this definition to functions of many variables by two different approaches and compare these:
3
Divided differences. In Section 1 we introduce the divided difference f [1] of a function f : X → E on an open subset X of a finite-dimensional vector space V with values in a Banach space E. It is a function f [1] : X × X → HomK (V, E). In particular it maps from a finite-dimensional vector space into a Banach space, like f does, and we define a differentiable function f as twice differentiable if the divided difference of f [1] is again differentiable. For applications in representation theory of p-adic Lie groups, it is important not to halt at iterated differentiability, but to further define r-fold differentiability for a real number r ≥ 0: We first define it for r < 1 by a strengthened Hölder condition (Definition 1.1). In general, we write r = ν + ρ for ν in N and ρ in [0, 1[, and ask the ν-th iterated divided difference f [ν] to be ρ-times differentiable. This non-Archimedean definition of differentiability preserves common facts in ordinary differential calculus ([Sch84, Section 27]) such as • invertibility of a function around a point where its derivative is invertible, • convergence of the Taylor polynomial expansion up to degree ν of a ν-times differentiable function ([Sch84, Chapter Ten]), and • completeness of the normed space of all differentiable functions, whereas the ordinary definition breaks each of these over K. (Indeed these facts follow with the non-Archimedean definition of differentiability straight from the definition whereas the ordinary definition demands a detour via the mean-value theorem or the fundamental theorem of calculus.) Taylor polynomial. Our differentiability definition by iterated divided differences leads to a sound differential calculus, yet with increasing degree of differentiability ν also to an exponential growth in the number of variables of the divided difference f [ν] . In Section 2, we define instead r-fold differentiability of a many-variable function f over K by convergence of the Taylor polynomial expansion of f , which is a function of two arguments (the expansion point and its offset), irrespective of the degree of differentiability. We then compare in Section 3 this differentiability definition to that by iterated divided differences: If f is differentiable then its Taylor polynomial expansion converges and vice versa, for f a one-variable function or a manyvariable function over Qp , if its Taylor polynomial expansion converges then f is differentiable.
4
Fractional differential Calculus in p-adic Lie group representations In the second half of this article (Section 4) we apply our results from the first half (Section 1, 2 and 3), to link the introduced notions of fractional differentiability to that used in the p-adic Langlands program. Fractional differentiability, that is, r-fold differentiability for a real number r ≥ 0, emerged from the p-adic Langlands program which links p-adic Galois and Lie group representations: Let K (the base field) be a finite extension of Qp and E (the coefficient field) a complete extension of Qp . The two-dimensional p-adic Langlands correspondence (see [Col14] or [Ber11]) puts a continuous action of the absolute Galois group Gal(K/K) of K on a 2-dimensional E-vector space in correspondence with a unitary continuous action by GL2 (K) on a, generally infinite-dimensional, E-Banach space Vb . The category of crystalline Galois actions is a full subcategory of that of all continuous actions of Gal(K/K) which is equivalent to a category of explicit linear algebraic objects, so called admissible filtered ϕ-modules, (see [ST02, Section 5]) and as such serves as a first test case for the p-adic Langlands correspondence. To a crystalline Galois action corresponds a unitary continuous GL2 (K)action on a Banach space Vb that is the topological completion of a vector space V of certain locally algebraic functions f : K → E. (It is the tensor product V = W ⊗ U of a vector space W of locally constant functions, whose choice comes from the local Langlands program, and a vector space U of Qp -algebraic functions, whose choice comes from the resemblance between certain parameters classifying all crystalline Galois actions and those classifying all irreducible algebraic actions (see [BS07, Introduction]).) The group action on Vb is unitary if Vb is the topological completion of V for a norm that is invariant under the group action. This norm is, for some real number r ≥ 0, a quotient norm of the norm k·kC r on the space of all r-times differentiable functions on K and Vb is a subquotient of the topological vector space C r (K) of all r-times differentiable functions on K. The real number r is determined by the action of GL2 (K) on V (read off from the action of a certain diagonal matrix on a generator of the module V over the group ring). Given V , it determines the topology of Vb (but the action of GL2 (K) on V only remotely). If r ≥ 0 is a nonintegral number, then r-fold differentiability was defined first in [BB10] for a function on Zp by a growth condition on its Mahler coefficients, its coefficients in terms of a distinguished orthogonal basis of the continuous functions on Zp .
5
In Section 4 we first recall Colmez’s equivalent approach in [Col10] via the Taylor polynomial expansion of a function on Zp that de Ieso in [DI13a] extended to functions on a finite extension oK of Zp . We then show how Colmez’s and de Ieso’s definition of a C r -function furnishes a Banach space, that of all C r -functions f : K → E, whose natural norm is invariant under the action of all triangular matrices in GL2 (K) on V . In Theorem 4.7 we answer the question raised at [DI13a, End of Section 1.1]: How does the definition of r-fold differentiability (in Section 1) via divided differences from non-Archimedean calculus compare to de Ieso’s definition (in Section 4) via Taylor polynomials from representation theory? Acknowledgments. Thanks to Christophe Breuil, Marco de Ieso and João Pedro dos Santos (who also proofread and offered clarifications) for enlightening discussions on p-adic differentiability and its role in representation theory. Notations We adopt conventional terminology in non-Archimedean Functional Analysis (see [PGS10, Section 2.1]): A non-Archimedean field is a field that has a nontrivial non-Archimedean absolute value and is complete. A non-Archimedean norm k·k on a vector space V is a norm that satisfies the strong triangle inequality, that is, kx + yk ≤ max{kxk, kyk} for all x and y in V . A nonArchimedean Banach space is a vector space that has a non-Archimedean norm k·k and is complete for it. Henceforth • K is always a non-Archimedean field, and • E is always a non-Archimedean K-Banach space. For a continuous function f on a compact set X that takes values in a normed space let kf ksup := sup{kf (x)k for all x ∈ X} denote its supremum norm; if X is not necessarily compact then this notation implies that this supremum exists.
1. Fractional differentiability via iterated divided differences In this section, we define r-fold differentiability classically via iterated divided differences. First we decompose r = ν + ρ ∈ R≥0 into its integer part ν ∈ N
6
and its fractional part ρ ∈ [0, 1[. Then we define ν-fold differentiability by iteratively building partial divided differences, and ρ-fold differentiability by a strengthened Hölder-continuity condition. Finally, an r-times differentiable function is a ν-times differentiable function such that each of its partial divided differences is ρ-times differentiable. C ν -functions for a natural number ν Differentiable functions. Let V be a finite-dimensional K-vector space, X a subset of V and f : X → E. Recall that a function f is differentiable at a in X if there is a linear map A : V → E such that for every > 0 there is a neighborhood U around a inside X such that f (x + h) = f (x) + Ah + R(x, h) with kR(x, h)k ≤ khk for all x + h,x in U . The following, equivalent, differentiability criterion requires a choice of coordinates on V but can be iterated. Let us fix a basis e1 , . . . ,ed of V by which V identifies with the d-fold direct sum K ⊕ · · · ⊕ K. Definition. Let X be a subset of V . The differential f ]1[ (x + h, x) of f at x + h, x in X with h ∈ K∗ d is the K-linear map A : V → E determined by A · hk ek = f (x + h1 e1 + · · · + hk−1 ek−1 + hk ek ) − f (x + h1 e1 + · · · + hk−1 ek−1 ) for all k = 1, . . . , d. The function f is a C 1 -function if f ]1[ extends to a continuous function f [1] : X × X → HomK (V, E). A subset X of V is accumulated if it is locally a product U = U1 × · · · × Ud of subsets U1 , . . . ,Ud of K without isolated points. For example, every open subset is accumulated. If X is accumulated then f uniquely determines the extended function f [1] because the domain of f ]1[ is dense inside X [1] . Remark. To put a topology on the set of all r-times differentiable functions, we would like to take the supremum of a differentiable function on a compact set. From this viewpoint the notion of an accumulated set suits us because a general, not necessarily locally compact, field can be covered by compact accumulated but not necessarily by compact open subsets. (For example Cp , the completion of the algebraic closure of Qp , has no nonempty open subset that is compact.)
7
C ν -functions for a natural number ν. Let X be an accumulated subset of V . Let f : X → E be a C 1 -function. Let us compare the domain and codomain of f [1] with that of f . The domain X [1] := X × X of f [1] is included in the finite-dimensional K-vector space V [1] = V × V with a canonical ordered basis, like the domain X of f , and the codomain E[1] := HomK (V, E) of f [1] is a KBanach space, like the codomain K of f . We can thus apply the differentiability condition to f [1] , and define iterated differentiability this way. Definition. Let ν in N. The function f : X → E is a C ν+1 -function • if f is a C ν -function, and • if X = X [ν] , V = V [ν] , E = E[ν] and f = f [ν] then f]1[ extends to a continuous function f[1] : X × X → HomK (V × V, E). Definition. Let X be compact without isolated points. The norm k·kC ν on C ν (X, E) is defined by kf kC ν := max{kf ksup , kf [1] ksup , . . . , kf [ν] ksup }. C ρ -functions for ρ in [0, 1[ Let ρ in [0, 1[. Roughly, ρ-fold differentiability is stricter Hölder-continuity. Let X be a subset of a finite-dimensional K-vector space and let E be a non-Archimedean K-Banach space. Definition 1.1. The function f : X → E is a C ρ -function if for every a in X and every ε > 0, there is a neighborhood U around a inside X such that kf (x) − f (y)k ≤ ε · kx − ykρ
for all x, y in U.
The fractional divided difference |f ]ρ[ | : OX × X → R≥0 of f is defined by |f ]ρ[ |(x, y) = kf (x) − f (y)k/kx − ykρ . The function f : X → E is a C ρ -function if and only if |f ]ρ[ | extends to a continuous function |f [ρ] | on all of X × X that vanishes on the diagonal of X × X. This extension is unique if X contains no isolated points. We define: Definition. Let X be compact without isolated points. The norm k·kC ρ on all C ρ -functions f : X → E is defined by kf kC ρ := max{kf ksup , k |f [ρ] | ksup }. It X is not necessarily compact then this notation implies that each supremum on the right-hand side, and thence their maximum, exists.
8
C r -functions for a real number r ≥ 0 We fix henceforth a real number r ≥ 0 and its decomposition r =ν+ρ into an
• integer part ν = brc ∈ N, and a • fractional part ρ = {r} ∈ [0, 1[.
Definition 1.2. Let X be an accumulated subset of V . The function f : X → E is a C r -function if f is a C ν -function and f [ν] is a C ρ function. Definition. Let X be an accumulated compact subset of V . The norm k·kC ν on C ν (X, E) is defined by kf kC ν := max{kf [0] ksup , . . . , kf [ν−1] ksup , kf [ν] kC ρ }.
2. Taylor Polynomials Let r ≥ 0 be a real number with integer part ν and fractional part ρ. We define r-fold differentiability of a function on a non-Archimedean vector space by the convergence of its Taylor polynomial expansion up to degree ν. Definition Let V be a K-vector space. Let SymnK (V, E) be all continuous symmetric K-multilinear maps M : V × · · · × V → E of n variables. These form a nonArchimedean K-Banach space by the operator norm kM k = sup{kM (x)k for all x with kxk ≤ 1}. That is, the supremum of M on the unit ball of V × · · · × V with respect to the product norm kv1 , . . . , vn k = max{kv1 k, . . . , kvn k}. The following definition generalizes that of onefold differentiability at the beginning of Section 1 to a higher differentiability degree r ≥ 0. Definition 2.1. Let X be an accumulated subset of V . The function f : X → E is a C rT -function if there are functions Dn f : X → Symn (V, E) for n = 0, 1, . . . , ν and Rv f : X × X → E such that X f (x + h) = Dn f (x)(h, . . . , h) + Rν f (x + h, x) n=0,...,ν
and for every a in X and ε > 0, there is a neighborhood U around a inside X such that kRν f (x + h, x)k ≤ εkhkr
9
for all x + h, x in U .
Differentiability of the Taylor polynomial coefficients Symmetric multilinear forms and homogeneous polynomials. A sum of symmetric multilinear maps is determined by its restriction onto the diagonal (if the characteristic of the field of definition is equal to 0 or greater than the maximal number of arguments of these maps) and this restriction becomes after a choice of basis a polynomial map. Given finitely many multilinear mappings, the following lemma shows how to recover each one of them by the restriction of their sum onto the diagonal. Lemma 2.2 (Polarization of an algebraic form). Let M 0 , . . . , M n be multilinear mappings of 0, . . . , n arguments in V respectively and the functions m0 , . . . , mn on V their restrictions onto the diagonal. Let m = m0 + · · · + mn . Then n!M n (x1 , . . . , xn ) = [∆x1 ◦ · · · ◦ ∆xn (m)](0), where, given x ∈ V , the difference operator ∆x on all functions on V is defined by m 7→ m(· + x) − m. Proof: Given nonzero x ∈ V , the difference operator ∆x diminishes the degree of m by 1. That is, if m corresponds to a sum M of multilinear mapping in up to n arguments then ∆x m corresponds to a sum of multilinear mappings in up to n − 1 arguments. Because, if N is a map of n arguments that is multilinear and symmetric then X n x ∆ N = N (· + x, . . . , · + x) − N = N (x, . . . , x, ·, . . . , ·) | {z } i i=1,...,n
i-times
with the highest term nN (x, ·, . . . , ·). We conclude iteratively that ∆x1 ◦ · · · ◦ ∆xn m is constant and, by the above equality, equal to n!M (x1 , . . . , xn ). We recover M n−1 by applying the above reasoning to the diagonal map mn−1 given by v 7→ m(v) − M n (v, . . . , v) = M 0 + M 1 (v) + · · · + M n−1 (v, . . . , v) and iteratively recover M n−2 , . . . , M 0 as well. Given δ > 0, let k·kδ be the norm on all polynomial maps f : V → E defined by kf kδ := sup{kf (x)k for all x with kxk ≤ δ}. Corollary 2.3. Let M 0 , . . . , M n be multilinear mappings on V of 0, . . . , n variables respectively with values in E. Let m0 , . . . , mn be their restrictions onto the diagonal and put m = m0 + · · · + mn . Then for every i = 0, . . . , n and δ in |K∗ |, kn! · · · i!M i k ≤ δ −i kmkδ .
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Proof: By Lemma 2.2, whose notation we adopt, kn!M n k ≤ kmk. So k(n − 1)!(n!M n−1 )k ≤ max{kn!mk, kn!M n k} ≤ max{kn!mk, kmk} = kmk. Iteratively kn! · · · i!M i k ≤ kmk for i = 0,. . . ,n. By multilinearity kM i kδ = δ i kM i k. Differentiability of the Taylor polynomial coefficients. Let f : X → E be a C rT -function. The estimate in Corollary 2.3 on the polynomial coefficients applies in particular, for fixed x in X, to each polynomial coefficient D0 f (x), . . . , Dn f (x). We prove via this estimate that the convergence of the Taylor polynomial remainder of f implies that of the Taylor polynomial of D1 f , . . . , Dn f . Proposition 2.4. Let either char K > ν or char K = 0. Let X be an accun mulated subset of V . If f ∈ C rT (X, E) then Dn f ∈ C r−n T (X, Sym (V, E)) for n = 0, . . . , ν. Proof: Let x, x + y, x + y + z ∈ X. Write out Rν f (x + y + z, x) − Rν f (x + y + z, x + y) X =(f (x + y + z) − Dn f (x)(y + z, . . . , y + z) n=0,...,ν
−(f (x + y + z) − =
X
n=0,...,ν
X
Dn f (x + y)(z, . . . , z)
n=0,...,ν
Dn f (x)(y + z, . . . , y + z) − Dn f (x + y)(z, . . . , z).
Fix x = x0 and abbreviate Γn = Dn f (x0 ). Recall that Γn is a multilinear mapping in n variables. Fix y = y0 . Then the mapping Γn (y0 + ·, . . . , y0 + ·) of n arguments is by multilinearity of Γn a sum of multilinear mappings of m arguments for m = 0, . . . , n. So X n n Γ (y0 + ·, . . . , y0 + ·) = Γn (y0 , . . . , y0 , ·, . . . , ·). | {z } m m=0,...,n m-times
11
We compute X
n=0,...,ν
Γn (y0 + ·, . . . , y0 + ·)
i + j i+j = Γ (y0 , . . . , y0 , ·, . . . , ·) | {z } i i,j with i+j=0,...,ν i-times X X X i + j Γjy0 = Γi+j (y0 , . . . , y0 , ·, . . . , ·) = | {z } i j=0,...,ν j=0,...,ν i=0,...,ν−j X
i-times
with Γjy0 =
P
i=0,...,ν−j
arguments. Together,
i+j i
i+j Γ (y0 , . . . , y0 , ·, . . . , ·) a multilinear mapping in j | {z } i-times
Rν f (x0 + y0 + ·, x0 ) − Rν f (x0 + y0 + ·, x0 + y0 ) X X i + n = Di+n f (x0 )(y0 , . . . , y0 , ·, . . . , ·) − Dn f (x0 + y0 )(·, . . . , ·) | {z } n n=0,...,ν i=0,...,ν−n i-times X n = Θ (x0 , y0 ) n=0,...,ν
with Θn (x0 , y0 ) for n = 0, . . . , ν the multilinear map of n arguments given by X i + n n Θ (x0 , y0 ) = Di+n f (x0 )(y0 , . . . , y0 , ·, . . . , ·)−Dn f (x0 +y0 )(·, . . . , ·). | {z } n i=0,...,ν−n i-times
Denote its restriction onto the diagonal by θn (x0 , y0 ) and put θ(x0 , y0 ) = θν (x0 , y0 ) + · · · + θ0 (x0 , y0 ).
By Corollary 2.3, if C(n) = 1/|ν!(ν − 1)! · · · n!| > 0 (well defined by our hypothesis on char K) then for every δ in |K∗ |, X i + n n kD f (x0 + y0 )(·, . . . , ·) − Di+n f (x0 )(y0 , . . . , y0 , ·, . . . , ·)k | {z } n i=0,...,ν−n
i-times
n
=kΘ (x0 , y0 )k ≤C(n)δ −n kθ(x0 , y0 )kδ =C(n)δ −n kRν f (x0 + y0 + ·, x0 ) − Rν f (x0 + y0 + ·, x0 + y0 )kδ ≤C(n)δ −n max{kRν f (x0 + y0 + ·, x0 )kδ , kRν f (x0 + y0 + ·, x0 + y0 )kδ }.
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n We ultimately want to prove that Dn f ∈ C r−n T (X, Sym (V, E)). Let a ∈ X and ε > 0. Let x + y, x in X and put δ := kyk. Then X i + n n kD f (x + y)(·, . . . , ·) − Di+n f (x)(y, . . . , y , ·, . . . , ·)k | {z } n i=0,...,ν−n
≤C(n)δ
−n
i-times
ν
max{kR f (x + y + ·, x)kδ , kRν f (x + y + ·, x + y)kδ }.
Since f is a C rT -function, there is an open ball U around a inside X such that kRν f (x + y, x)k ≤ εkykr for all x + y, x ∈ U . The ball of radius equal to δ around x + y is by the strong triangle inequality included in U . Thence kRν f (x + y + ·, x)kδ , kRν f (x + y + ·, x + y)kδ ≤ εkykr . By the canonical identification of Symi+n (V, E) with Symi (V, Symn (V, E)) via M 7→ [vv 7→ Mv = M (vv , ·)], we find Di+n f (x) to be a multilinear mapping in i arguments with values in Symn (V, E). We conclude X i + n n kD f (x + y) − Di+n f (x)(y, . . . , y )k | {z } n i=0,...,ν−n
≤C(n)δ
−n
i-times
r
r−n
εkyk = C(n)εkyk
.
That is, Dn f : X → Symn (V, E) is a C r−n functions T -function with respect to the i+n i n i n i n i+n D (D f ) : X → Sym (Sym (V, E), E) given by D (D f ) = n D f (x) for i = 0, . . . , ν − n. Corollary 2.5. Under the premises of Proposition 2.4 the functions D0 f , . . . , Dν f are uniquely determined and continuous. Proof: By Proposition 2.4 the functions D0 f, . . . , Dν f are in particular continuous. Therefore f is differentiable (by the convergence of its Taylor polynomial expansion) and its derivative D1 f is uniquely determined. Because D1 f is a 2 C r−1 T -function, by the same reasoning its derivative D f is uniquely determined, 2 ν and iteratively every function D f , . . . , D f is uniquely determined. The norm Let C rT (X, E) be the K-vector space of all C rT functions f : X → E. By Corollary 2.5 the functions D0 f, D1 f, . . . , Dν f are uniquely determined and differentiable of degree r, r − 1, . . . , ρ. Hence 1. in particular, the functions D0 f, D1 f, . . . , Dν f are continuous, and
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2. the remainder Rν f of its Taylor polynomial expansion converges as in Definition 2.1 if and only if the function (x, y) 7→ kRν f (x, y)k/kx − ykr
for distinct x, y ∈ X,
extends to a continuous function |∆r f | : X × X → R≥0 that vanishes on the diagonal. We conclude that the following norm on C rT (X, E) is well-defined. Definition 2.6. Let X be a compact accumulated subset of V . The norm k·kC r T on C rT (X, E) is defined by kf kC r := max{kD0 f ksup , . . . , kDν f k}∪{k |∆r f | ksup }. T
3. Comparison In this Section 3 we compare differentiability via divided differences in Section 1 with that via the Taylor polynomial in Section 2 on an open subset. We show that differentiability via divided differences implies that via the Taylor polynomial, and the inverse holds, that is, both differentiability conditions are equivalent, for functions of one variable and of many variables in Qp . Necessity Let in this subsection either char K > ν or equal to 0 and let X be an open subset of V . For a tuple p = (p1 , . . . , pd ) of entries in N and h = (h1 , . . . , hd ) of entries in K denote hp = hp11 · · · hpdd .
Lemma 3.1. Let f : X → E be a C r -function. Let x0 ∈ X such that the ball of radius around x0 is included in X. Let h in V such that kvk ≤ and for t in K with |t| ≤ 1 put F (t) = f (x0 + th). Then for every i = 0, . . . , ν, X Di F (t) = Dp f (x0 + th) · hp . (∗) p in Nd with p1 +···+pd =i
Proof: For a natural number i, let F (i) be the i-th Archimedean derivative of F , and for a tuple p = (p1 , . . . , pd ) of natural numbers, let f (p) be the Archimedean derivative of f that is taken p1 -times along the first, . . . , pd -times along the last coordinate. By [Sch84, Theorem 29.5] and by its multivariate version i! Di F = F (i) and p! Dp = f (p) . By our assumption on char K the factors i! and p1 ! · · · pd ! invert and we can replace the non-Archimedean by the Archimedean derivatives. Let I = {x ∈ K : |x| ≤ 1} and define T : I → X by T (t) = x0 + t · h. We obtain Equation (∗) by applying the Archimedean chain rule to F = f ◦ T .
14
Let V1 , . . . , Vn be vector spaces. The normed K-vector space Hom(V1 , Hom(V2 , . . . , Hom(Vn , E)) · · · ) is canonically isomorphic to MultK (V1 ×· · ·×Vn , E) := { all K-multilinear mappings m : V1 ×· · ·×Vn → E}. We identify E[ν] with Mult(V [ν−1] × · · · × V, E) by this isomorphism. Lemma 3.2. Let x0 in X. Let h in V , let I = {t ∈ K : |t| ≤ 1} and, for t in I, put T (t) = x0 + th. Let f : X → E and assume im T included in X. If f is r-times differentiable then f ◦ T is r-times differentiable, so for every t in I [ν] and λ in K[ν−1] × · · · × K[1] × K, h i (f ◦ T )[ν] (t) (λ) = f [ν] (T(t)) (λ · h) (3.1) where
• the function T : I [ν] → X [ν] is understood as T if ν = 0 and otherwise inductively for (t, t) in I [n+1] = I [n] × I [n] as (T (t), T (t)), and • the scalar product λ · h for λ ∈ K[ν−1] × · · · × K[1] × K is understood as λ · h if ν = 1 and otherwise inductively for (λ, µ) in K[n+1] = K[n] × K[n] as (λ · h, µ · h). Proof: If f is r-times differentiable then, because T is linear, in particular r-times differentiable, their composition f ◦ T is by [Nag11, Proposition 3.23] likewise r-times differentiable. If ν = 0 then Equation (3.1) holds. For ν > 0, by definition of the iterated divided difference and the chain rule for onefold divided differences, [ν] (f ◦ T )[ν+1] = (f ◦ T )[1] [ν] [ν] = (f [1] ◦ (T, T )) · T [1] = (f [1] ◦ (T, T )) · (·h) . By induction h h [ν] i [ν] i [1] [1] (f ◦ (T, T )) · (·h) (t) (λ) = f ◦ (T, T ) (t) (λ, ·h) [ν] = f [1] (T(t)) (λ · h, ·h) = f [ν+1] (T(t)) (λ, ·) · (h, h)
where the constant functions ·h respectively ·h send
15
• every t in I [1] to the mapping K → V given by scalar multiplication λ 7→ λ · h, respectively • every t in I [ν] to the mapping
K[ν−1] × · · · × K[1] → V [ν−1] × · · · × V [1]
given by component-wise scalar multiplication
(λν−1 , . . . , λ1 ) 7→ (λν−1 · h, . . . , λ1 · h). The function (h, h) is their tensor product on (K[ν−1] × · · · × K[1] ) × K.
Corollary 3.3. Let us keep the assumptions and notation of Lemma 3.2 above. Then, if X is moreover compact, [ν]
k(f ◦ T )[ν] kC ρ = kf| im T kC ρ /khkr . Proof: Because k·hk = khkν , Equation (∗) follows from Equation (3.1).
Lemma 3.4. Let f ∈ C r (X, E). Let I = {x ∈ K : |x| ≤ 1}. For x + h, x ∈ X define Tx,h : I → X by Tx,h (t) = x + th. For every a ∈ X and ε > 0, there is a neighborhood U around a inside X such that for all x + h, x in U , k(f ◦ Tx,h )<ν> kC ρ ≤ εkhkr . Proof: By Proposition A.2, k(f ◦ Tx,h )<ν> kC ρ = k(f ◦ Tx,h )[ν] kC ρ By assumption f ∈ C r (X, E), that is, f [ν] ∈ C ρ (X [ν] , E[ν] ). That is, for every a ∈ X and ε > 0 there is a neighborhood U around a inside X such that for all x + h, x ∈ U [ν] , kf [ν] (x + h) − f [ν] (x)k ≤ εkhkρ .
If x + h, x in U then im T is included in U [ν] . Thus, by Corollary 3.3, k(f ◦ Tx,h )[ν] kC ρ ≤ εkhkr . Proposition 3.5. Let f ∈ C r (X, E). There are functions Dp f : X → E for p in Nd with p1 + · · · + pd ≤ r and Rv f : X × X → E such that X f (x + h) = Dp f (x0 + th) · hp + Rν f (x + h, x) p in Nd with p1 +···+pd =i
and for every a ∈ X and ε > 0 there is a neighborhood U 3 a in X such that kRν f (x + h, x)k ≤ εkhkr
16
for all x + h, x ∈ U
Proof: Define T = Tx,h : I → V by T (t) = x + th for all t in I = {t ∈ K : |t| ≤ 1}. Define F = Fx,h : I → E by F = f ◦ T . Then there is, by [Nag11, Corollary 2.14], for all t ∈ I a Taylor-polynomial expansion X tν (F <ν> (t, 0, . . . , 0) − F <ν> (0, . . . , 0)) = F (t) − Di F (0)ti i=0,...,ν
for C r−i -functions Di F : X → E for i = 0, . . . , ν and a C ρ -function F <ν> : X <ν> → E. By Lemma 3.1, for t = 1, F <ν> (1, 0, . . . , 0) − F <ν> (0, 0, . . . , 0) X =F (1) − Di F (0) i=0,...,ν
=f (x + h) −
X
p in Nd with p1 +···+pd =i
Dp f (x0 + th) · hp .
For every ε > 0 and a ∈ X, there is by Lemma 3.4 a neighborhood U 3 a such that for all x + h, x ∈ U in particular <ν> <ν> kFx,h (1, 0, . . . , 0) − Fx,h (0, . . . , 0)k ≤ εkhkr .
We conclude kf (x + h) −
X
p in Nd with p1 +···+pd =i
Dp f (x0 + th) · hp k ≤ εkhkr .
Corollary 3.6. We have C r (X, E) ⊆ C rT (X, E). If X is a compact open subset of V then the inclusion C r (X, E) ,→ C rT (X, E) is a monomorphism of normed vector spaces. Proof: By Proposition 3.5, the inclusion holds with, in the notation of Definition 2.1, X Dn f (x)(h, . . . , h) := Dp f (x0 + th) · hp . p in Nd with p1 +···+pd =i
Let X be a compact accumulated subset of V . Given f in C r (X, E), we show kf kC r ≤ kf kC r , that is, T
(i) max{kD0 f ksup , . . . , kDν f k} ≤ kf kC r
and (ii) k |∆r f | ksup ≤ kf kC r .
(i) : Let n in {0, . . . , ν}. By definition kDn f ksup = max{kDp f ksup : p1 + · · · + pd = n} and kDp f ksup ≤ kf
ksup ≤ kf
17
(ii) : Let x + h, x in X. For t in {x ∈ K : |t| ≤ 1} let T (t) = x + th. Put F = f ◦ T . By Proposition 3.5, Rν f (x + h, x) = F <ν> (1, 0, . . . , 0) − F <ν> (0, 0, . . . , 0). By Proposition A.2, we find kF <ν> kC ρ = kF [ν] kC ρ . By Corollary 3.3, we conclude |∆r f |(x + h, x) = kF <ν> (1, 0, . . . , 0) − F <ν> (0, 0, . . . , 0)k/khkr [ν]
≤ kF <ν> kC ρ /khkr = kF [ν] kC ρ /khkr = kf| im T kC ρ .
Sufficiency for functions of one variable In one variable, every C rT -function is also a C r -function. The passage from many variables to one simplifies the definition of a C rT -function. Definition 3.7. Let X be an open subset of K. A function f : X → K is a C rT -function if there are functions D0 f ,. . . ,Dν f : X → K and Rν f : X × X → K such that X f (x + h) = Di f (x)hi + Rν f (x + h, x) for all x + h, x in X, i=0,...,ν
and for every a ∈ X and ε > 0 there is a neighborhood U around a inside X such that |Rν f (x + h, x)| ≤ ε|h|r for all x + h, x in U. Let f be a C rT -function and let us keep the notations of Definition 3.7. By Corollary 2.5 the functions D0 f , D1 f , . . . , Dν f : X → K and the function |∆r f | : OX × X → R≥0 defined by |∆r f |(x, y) := |Rν f (x, y)|/|x − y|r are uniquely determined by f . The functions D0 f , D1 f , . . . , Dν f are continuous and |∆r f | extends to a continuous function on X × X that vanishes on the diagonal of X × X. Hence the following definition is meaningful: Definition. Let C be a compact open subset of K. The norm k·kC r on C rT (C, E) T is defined by kf kC r := max{kD0 f ksup , . . . , kDν f ksup , k |∆r f | ksup }. T
Theorem 3.8 ([Nag11, Corollary 2.32]). If X is an open subset of K then C r (X, K) = C rT (X, K) as sets and if X is also compact then the identity mapping between sets id : C r (X, K) → C rT (X, K) is an isomorphism of locally convex K-algebras.
18
Sufficiency for functions of many variables in Qp Partial Taylor polynomial. Let k ∈ {1, . . . , d}. The following definition of a k C r·e T -function is that given in Definition 2.1 where now the vector h in V has a single nonzero entry in the k-th coordinate. Definition (2.1’). Let X be an open subset of Kd . The function f : X → E is a 0 1·ek k C r·e f , . . . , Dν·ek f : X → E T -function if there are continuous functions D f , D and Rν·ek f : X [ek ] → E on X [ek ] := {(x; t) ∈ X × K : x + t · ek ∈ X} such that X f (x + t · ek ) = Di·ek f (x)ti + Rν·ek f (x; t) i=0,...,ν
and for every a in X and every ε > 0, there is a neighborhood U inside X [ek ] around a such that |Rν·ek f (x; t)| ≤ ε|t|r
for all x + t · ek , x in U.
k Fix a coordinate index k ∈ {1, . . . , d}. Let f : X → E be a C r·e T -function and r·ek let us keep the notation of Definition 2.1’. The function |∆ f | is defined by
(x; t) 7→ |Rν·ek f (x; t)|/|t|r
for all (x; t) in X [ek ] with t in K∗
Continuity of D0 f, D1·ek f, . . . , Dν·ek f : X → K implies that of |∆|r·ek f on r·ek k X ]ek [ ; because f is a C r·e f | extends to a continuous function T -function, |∆ [ek ] on X , denoted likewise, that vanishes if t vanishes. The functions D0·ek , . . . , Dν·ek (and thus |∆r·ek f |) are uniquely determined by f by [Nag11, Lemma 3.52]. Definition. Let C be a compact open subset of Qdp . The norm k·kC r·ek on T C r·ek (C, E) is defined by kf kC r·ek ,C := max{kD0 f ksup , kD1·ek f ksup , . . . , kDν·ek f ksup } ∪ {k∆r·ek f ksup }. T
The following definition of a C rT -function is that of a C rT -function where, in the notation of Definition 2.1, the vector h in V has a single nonzero entry. Definition. Let X be an open subset of Qdp . The normed K-vector space C rT (X, E) is the initial normed K-vector space of C r,0,...,0 (X, E), . . . ,C 0,...,0,r (X, E). T T That is, • C rT (X, E) := C r,0,...,0 (X, E) ∩ . . . ∩ C 0,...,0,r (X, E) as K-vector space, T T • and its norm is given by k·kC r := max{k·kC r,0,...,0 , . . . , k·kC 0,...,0,r }. T
T
19
T
Theorem 3.9 ([Nag11, Corollary 3.60]). Let X be an open subset of Qdp . The inclusion map C r (X, E) ,→ C rT (X, E) is an isomorphism of locally convex Kvector spaces. This isomorphism results from the characterization of r-fold differentiability of a function by its Mahler coefficients, the coefficients of its expansion with respect to the Mahler basis, a canonical orthonormal basis of all continuous functions on Zp (that is dual to the Iwasawa isomorphism). See also [Glö13, Corollary 9.5] for a variant of this result. Total Taylor polynomial. Corollary 3.10 directly implies that r-fold differentiability via divided differences, as defined in Section 1, is equivalent to r-fold differentiability via the total Taylor polynomial, as defined in Section 2. Corollary 3.10. Let X be an open subset of Qdp . The inclusion map C r (X, E) ,→ C rT (X, E) is an isomorphism of locally convex K-vector spaces. Proof: Because a C r·e1 -function is the special case of a C rT -function where, in the notation of Definition 2.1, the vector h in Kd has a single nonzero entry, we have the continuous inclusions of locally convex K-vector spaces C r (X, E) ,→ C rT (X, E) ,→ C rT (X, E). Their composition is by Theorem 3.9 an isomorphism and thus the first inclusion C r (X, E) ,→ C rT (X, E) too.
4. Fractional Differentiability from Representation Theory We introduce de Ieso’s differentiability definition on a function f on oK , then motivate its origins from representation theory. Finally we show that de Ieso’s differentiability definition puts a convergence condition on the Qp -algebraic Taylor polynomial expansion of f and that this condition is equivalent to that of iterated differentiability via divided differences on oK as open subset of the Qp -vector space K. Definition Let r ≥ 0 be a real number. There is another notion of r-fold differentiability for functions on finite extensions o of Zp via Taylor polynomials by [DI13a]. It generalizes that initially introduced by Colmez in [Col10] on the domain Zp (and which makes in turn reference to the findings in [Sch78]).
20
The original definition by Colmez and de Ieso. We adopt the notations of [DI13a]. Let K be a finite extension of Qp with ring of integers o, and E be a finite field containing the normal closure of K. We denote by S the finite set of ring morphisms from K into E. We introduce the following standard multi-index notations over S. Let n ∈ NS . P • We put n = s∈S ns , and Q • given z ∈ o, we put zn = s∈S s(z)ns .
Definition 4.1 ([DI13a, Déf. 2.1]). The function f : o → E is a C rCdI -function, that is, an r-times differentiable function à la Colmez de Ieso, if there are bounded functions Di f : o → E for i ∈ NS with i ≤ r such that if we define Rν f : o × o → E by X Rν f (x, h) = f (x + h) − Di f (x)hi , i ∈NS with i≤r
then ∆r f (δ) := sup sup |Rν f (x0 , h)|/δ r x0 ∈o |h|≤δ
is a well-defined function ∆r f : R>0 → R≥0 which converges to 0 as δ does. De Ieso shows in [DI12, Section 2.2] that the functions Di f for i in NS are uniquely determined and in particular continuous. Thus Rν f : o × o → E is uniquely determined and it is continuous: on the diagonal by the condition on Rν f in Definition 4.1 and elsewhere by the continuity of all Di f for i in NS . Hence the following definition is meaningful. Definition. The norm k·kC r
CdI
kf kC r
CdI
r on C˜ CdI (o, E) is defined by
= max{kDi f ksup for all i in NS with i ≤ r} ∪ {k |∆r f | ksup }.
Compactness and uniform continuity. We give a variant of Definition 4.1 by a less uniform convergence condition on the Taylor polynomial expansion and show this convergence condition to be equivalent to that of Definition 4.1 by compactness of oE . Definition 4.2. A function f : o → E is a C rCdI -function if there are functions Di f : o → E for i ∈ NS with i ≤ r such that if we define Rbrc f : o × o → E by X Rν f (x, y) = f (x) − Di f (y)(x − y)i , i ∈NS with i≤r
21
then for every a included in o and ε > 0 there is a neighborhood U around a in o such that |Rν f (x, y)| ≤ ε|x − y|r
for all x, y ∈ U.
Proposition 4.3 (Adaption of [Nag11, Proposition 2.33]). Let f : o → E be a function. Then f is a C rCdI -function if and only if f is a C rCdI -function. Proof: Given a function f : o → E, it will suffice to show that the conditions on the functions Di f in Definition 4.1 respectively Definition 4.2 are equivalent. Recall by Definition 4.1 that f ∈ C rCdI (o, E) if ∆ r f (δ) → 0 for δ → 0. That is, for any ε > 0 there is a δ0 > 0 such that ˜ r f (δ0 ) := sup ∆r f (δ) < ε. ∆ 0<δ≤δ0
On the other hand f ∈ C rCdI (o, E) if |∆r f |(x + h, x) := |Rν f (x + h, x)|/|h|r , a priori defined for all different x + h, x in o, extends to a continuous function, denoted likewise, on all of o×o that vanishes on the diagonal. As o is a compact metric space, |∆r f | is continuous on o × o if and only if it is uniformly so. In particular on the diagonal, this says that for any ε > 0, there is δ > 0 such that ˜ r f (δ) := sup sup |∆r f |(x, y) < ε ∆ a∈o x,y∈B≤δ (a)
where B≤δ (a) is the ball of radius δ around a inside o. We have to prove ˜ r f (δ), that is, ˜ r f (δ) = ∆ ∆ sup
sup
a∈o x,y∈B≤δ (a)
|Rν f (x, y)|/|x − y|r = sup sup
sup
0<γ≤δ x∈o y with |y−x|≤γ
|Rν f (x, y)|/γ r . (∗)
We note that for any x, y ∈ o and δ > 0 we have |x − y| ≤ δ if and only if there is x0 ∈ o such that max{|x − x0 |, |y − x0 |} ≤ δ by the strong triangle inequality. Thus the left-hand side of (∗) equals sup
sup
x∈o y with |y−x|≤δ
|Rν f (x, y)|/|x − y|r .
(∗∗)
Furthermore we note that for any x, y ∈ o we have x 6= y and |x − y| ≤ δ if and only if |x − y| = γ for some 0 < γ ≤ δ. Thus sup
sup
x∈o y with |y−x|≤δ
|Rν f (x, y)|/|x − y|r = sup sup
sup
x∈o 0<γ≤δ y with |x−y|=γ
22
|Rν f (x, y)|/γ r
Now keeping x fixed, sup
sup
0<γ≤δ y∈o with |x−y|=γ
|Rν f (x, y)|/γ r = sup
sup
0<γ≤δ y∈o with |x−y|≤γ
|Rν f (x, y)|/γ r
as Rν f (x, y) = 0 for any y = x. But then sup
sup
x∈o y with |y−x|≤δ
|Rν f (x, y)|/|x − y|r = sup sup
sup
x∈o 0<γ≤δ y with |y−x|≤γ
|Rν f (x, y)|/γ r .
This is the claimed Equality (∗) after substituting the left-hand side by (∗∗). We define accordingly the norm k·kC r on all C rCdI -functions from o to E to CdI be that on C rCdI (X, E). Origin from Representation Theory We consider the case when K is Qp , and put G = GL2 (Qp ). Let • T = ∗ ∗ be all diagonal matrices in G, • N = 1 ∗1 respectively N0 = 1 Z1p all the unipotent respectively all integral unipotent matrices in G, and • P = T N = ∗ ∗∗ resp. P¯ = ∗∗ ∗ all upper respectively all lower triangular matrices in G.
The groups.
The representation. One way to build a E-linear action by GLn on a vector space, that is, a module over E[GLn ], is by inducing E-linear actions by smaller GLn1 , . . . , GLnr with n1 + · · · + nr = n on vector spaces W1 ,. . . ,Wr , as follows: Let W = W1 ⊗ · · · ⊗ Wr ; this is, an E[M ]-module for the subgroup M = GLn1 × · · · × GLnr of GLn . We extend its scalars to obtain the induced E[G]module indG H W := W ⊗E[M ] E[G]. In our case n = 2 = 1 + 1 the subgroup M consists of all diagonal matrices in G, that is, equals T , and T acts by a character. The representation V of GL2 (Qp ) that corresponds to crystalline representation of the absolute Galois group of Qp on a 2-dimensional vector space is the locally algebraic induced representation of (locally algebraic) character of T . The character χ : T → E∗ is the product χ = ψ · θ of • the algebraic character ψ = ψ1 ⊗ ψ2 : T → E∗ given by the ψ1 = ·l+k and ψ2 = ·l on Q∗p with l + k ≥ l ∈ Z, and
23
• the locally constant character θ = θ1 ⊗ θ2 : T → E∗ satisfying θ1 (Z∗p ) = θ2 (Z∗p ) = 1 (and thus determined by their values on p). Every character of T extends trivially to P¯ (and necessarily so, because its codomain E∗ is abelian and T is the commutator of P¯ ). The induced G-representation indG P¯ χ is explicitly given by the E-vector space indG pg) = χ(¯ p) · f (g) for all p¯ ∈ P¯ , g ∈ G} P¯ χ = {f : G → E : f (¯ on which G acts by right translation, that is, gf = f (·g). We put V := all locally algebraic functions in indG P¯ χ. The action by P on V . By the Iwasawa decomposition G = P K for the compact subgroup K = GL2 (ZP ) and the upper triangular matrices P . Thence every norm |·| on V that is invariant under P gives by kvk := inf{|kv| : k ∈ K} rise to a norm that is invariant under G and so it suffices to find a P -invariant norm on V . For this we study the P -stable subspace V (N ) := {all functions in V that vanish outside N P¯ }. Proposition. Let C lp≤k cpt (N, E) be all functions f : N → E that are locally polynomial of degree ≤ k and of compact support. The restriction mapping f 7→ f N is an isomorphism of E-vector spaces V (N ) −∼ → C lp≤k cpt (N, E) and the group P acts on C lp≤k (N, E) via the above isomorphism by • tf = χ(t)f (t ·) for all t in T with t · the right-conjugation by t on N , and • nf = f (n·) for all n in N . The invariant norm on N . We are looking for a P -invariant norm on V . Because P = T N we are looking for a norm k·k on C lp≤k cpt (N, E) that is • invariant under translation, and • fulfills ktf k = kf k for all t in T . The function k·k : C lp≤k cpt (N, E) → R≥0 ∪ {∞} given by kf k = sup{|pf (1)| : p in P }
24
• vanishes if and only if f vanishes, • fulfills the strong triangle inequality, and • because the support of f is compact, the supremum sup{|f (n)| : n ∈ N } under all translates in N exists, but is unbounded on the translates of f under T . The estimate X X kf0 k = k1N0 k = k 1nN0t k = k 1/χ(t)ntf0 k ≤ 1/|χ(t)|kf0 k n∈N0 /N0t
n∈N0 /N0t
shows that a necessary condition for the existence of a norm invariant under translation by T is |χ(t)| ≤ 1 for all t in T that stabilize N0 under rightconjugation. The norm kf k := sup{kR nt f (1, ·)kN0 : t ∈ T, n ∈ N }
(4.1)
shows that this condition is sufficient (where kR nt f (1, ·)kN0 is the supremum norm of the function n 7→ R nt f (1, n) on N0 ). This supremum is bounded, because f locally equals its Taylor polynomial. (The term Rf (n0 , n0 n) vanishes for n in a sufficiently small neighborhood t N0 of 1 or equivalently when |χ(t)| is sufficiently big.) It is 0 if and only if f is 0 because the support of f is compact. The invariant norm on Qp . Let us identify N with Qp and see how, under this identification, the just defined norm k·k corresponds to the norm k·kC r CdI by Colmez and de Ieso. First, we remark that the character |χ| : T → pZ is • trivial on T0 by definition, and • trivial on the center Z of GL2 (Qp ) by the necessary condition that |χ(t)| ≤ 1 for all t in T that stabilize N0 under right-conjugation. Thus |χ| is determined by its values on T /T0 Z. This group is isomorphic to p Z. Let t0 in T , for example 1 , be a preimage of 1 in Z under this group isomorphism. Let vE denote the valuation of E normalized by vE (p) = 1. We put r := vE (χ(t0 )) ≥ 0,
and conclude that r determines |χ| : T → pQ . Under the identification of N with Qp , • t = a d in T acts on N by tx := d/ax, and 25
• n=
1 b 1
in N acts on N by nx := b + x. n
Every neighborhood of 1 in N is of the form t0 N0 for some n in N and corresponds inside Qp to the ball B≤δ (0) around 0 of radius δ = p−n . We have |χ(tn0 )| = 1/δ r . We conclude that for t in T and n in N , kR nt f (1, ·)kN0 = kχ(t)Rf (n, ·)kN t = kRf (n, ·)kB≤δ /δ r . 0
Hence the norm k·k on V (N ) in Equation (4.1) identifies on C lp≤k cpt (Qp , E) with the norm kf k = sup sup sup |Rf (x, h)|/δ r . δ>0 x∈Qp |h|≤δ
This is the greatest norm on V (N ) invariant under P . However, if we consider only the Taylor polynomial R≤ν f of f up to degree ν, the resulting function kf k = sup sup sup |R≤ν f (x, h)|δ r δ>0 x∈Qp |h|≤δ
is again a P -invariant norm on C lp≤k cpt (Qp , E). If we restrict this norm k·k to all functions in C lp≤k cpt (Qp , E) that have support Zp , and identify these with C lp≤k (Zp , E) then this norm is equivalent to the norm on C lp≤k (Zp , E) given by kf k = max{kD0 f ksup , . . . , kDν f ksup } ∪ {sup sup sup |R≤k f (x, h)|/δ r }. δ>0 x∈Zp |h|≤δ
This is the norm whose completion of C lp≤k (Zp , E) yields the E-Banach space of C r -functions by Colmez and de Ieso. To construct the whole Banach space Vb de Ieso regards the functions in V as defined on G/P¯ = P1 (K) covered by two disjoint copies of oK . He obtains at r first the Banach subspace V C of C rCdI (oK , E) ⊕ C rCdI (oK , E) whose derivatives vanish in higher degrees that are determined by the highest degrees of locally Qp -polynomial functions in V ([DI13b, Section 4.2]). However • the norm of the first component of C rCdI (oK , E) ⊕ C rCdI (oK , E) is only fixed by those matrices in T that preserve N0 under right conjugation, whereas ¯0 . • that of its second component only by those matrices that preserve N To obtain a Banach space whose norm is invariant under all of T he quotients V so that both components identify. Consequently every function is invariant ¯0 under right conjugation, under all matrices in T that preserve either N0 or N that is, all of T . Cr
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Comparison Let d be the degree of the field extension K over Qp . We explain that the differentiability condition by Colmez and de Ieso is a convergence condition on the multivariate Taylor-polynomial up to total degree brc on d copies of Zp . Linear and polynomial maps over finite field extensions. Let σ : K → E be a field embedding. Let V be a K-vector space and let W be a E-vector space. A map A : V → W is σ-linear if it is additive and for every scalar λ in K it satisfies Aλ· = σ(λ)A·. Let σ1 , . . . , σN : K → E be field embeddings. A map M : V × · · · × V → W of N variables is (σ1 , . . . , σN )-multilinear if it is σ1 -linear in the first coordinate, . . . , σN -linear in the N -th coordinate. Proposition 4.4. We have a canonical bijection M Homσ (K, E) −∼ → HomQp (K, E). σ∈S
Proof: It is injective because by Dedekind’s Theorem, cf. [Lan02, Theorem VI.4.1], all Qp -algebra morphisms σ : K → E are linearly independent over E. It is a bijection because both sides have the same dimension over E. Alternatively, we have HomQp (K, E) = HomQp (K, Qp ) ⊗Qp E = K ⊗Qp E =
Y
σ∈S
E=
M
Homσ (K, E).
σ∈S
Here we choose a basis of K overQ Qp to identify K with its dual HomQp (K, Qp ) and the splitting K ⊗Qp E = σ∈S E comes from the Chinese remainder theorem. Corollary 4.5. We have a canonical isomorphism of finite-dimensional Evector spaces X ∼ MultN → MultN (σ1 ,...,σN ) (K × · · · × K, E) − Qp (K × · · · × K, E). σ∈S N
Proof: By the natural identification HomQp (K, HomQp (K, E)) = BilQp (K × K, E) with the Qp -billinear mappings on K×K, the bijection in Proposition 4.4 inductively carries over to a bijection between • all Qp -multilinear maps on K × · · · × K with values in E, and • all sums of all maps on K × · · · × K with values in E that are in every variable σ-linear for some σ in S.
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This bijection restricts to all symmetric maps. That is, we have a canonical isomorphism of finite-dimensional E-vector spaces X ∼ φ : Sym( MultN → SymN (σ1 ,...,σN ) (K × · · · × K, E)) − Qp (K × · · · × K, E). σ∈S N
Let us partition {1, . . . , N } into subsets nσ for σ in S. The K-vector space of all symmetric multilinear maps M : K × · · · × K → E of N variables that are σ-linear in nσ variables for each σ in S has as basis the functions Y Y x 7→ xn := σxn . σ∈S n∈nσ
By the Polarization Lemma 2.2, every multilinear symmetric map is determined by its restriction onto the diagonal. Hence the K-vector space of all homogeneous Qp -polynomial maps of degree N on K, that is, of all maps that are restrictions of Qp -multilinear maps M : K × · · · × K → E of N variables, has as basis the monomial functions Y x 7→ xn := σxnσ σ∈S
for all n in NS whose sum of entries equals N . We conclude: Corollary 4.6. The K-vector space of all Qp -polynomial functions on K has as basis the monomials Y x 7→ xn := σxnσ for n in NS . σ∈S
Comparison of C r - and C rCdI -functions. We show that, after fixing an identification of o with ZSp , the differentiability condition over o of Definition 1.2 via iterated divided differences is equivalent to that of Definition 4.2 via Taylor polynomials. Theorem 4.7. Let K be a finite extension of Qp and let o be its ring of integers that we regard as open subset of the Qp -vector space K. Every r-times differentiable function from o to E is a C rCdI -function and the inclusion map C r (o, E) ,→ C rCdI (o, E) is an isomorphism of normed E-algebras. Proof: We recollect all preceding characterizations of r-fold differentiability:
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1. Let us regard o as an open subset of the Qp -vector space K. By Corollary 3.10, the inclusion of normed E-algebras C r (o, E) ,→ C rT (o, E), that of C r -functions as in Definition A.1 via iterated divided differences included in that of C rT -functions as in Definition 2.1 via the convergence of their Taylor polynomial expansions, is an isomorphism. 2. Let f : o → E. Definition 2.1 says that f is a C rT -function if there is a family of Qp -polynomial maps (Tfx : x ∈ o) such that for every a in o and every > 0, there is a neighborhood U around a in o such that kf (x + h) − Tfx (h)k ≤ εkhkr for all x + h, x in U . The C rT -norm is defined as supremum of all these differences over all x + h, x in o and the coefficients of Tfx on all x in o. Because every Qp -polynomialQmap is by Corollary 4.6 a linear combination of monomial functions x 7→ σ∈S σxnσ for n in NS , it follows that f is a C rT -function as in Definition 2.1 if and only if f is a C rCdI -function as in Definition 4.2 and k·kC r = k·kC r . T
CdI
3. By Proposition 4.3 every function f : o → E is a C rCdI -function if and only if it is a C rCdI -function.
A. Schikhof’s iterated divided differences Let X be an accumulated subset of a finite dimensional K-vector space V and let E be a K-Banach space. We compare the n-th divided difference of a function f as given in Section 1, denoted by f [n] , with that by Schikhof, denoted by f
29
Definition of Schikhof’s iterated divided differences The symmetry of the differential allowed Schikhof to reduce for increasing degree of differentiability ν the exponential growth in the number of variables of the total differential f [ν] to a linear growth in the number of variables of certain partial differentials f
and X >n< := X1>n1 < × · · · × Xd>nd < .
Given x ∈ X
:=
k +1> k +1> f >n< (..; xk<0> , x<2> , . . . , x
Example. Let us consider a function of two variables f : X × Y → E for subsets X and Y of K. We have X >1,1< = {(x + u, x; y + v, y) : x + u, x ∈ X, y + v, y ∈ Y with u, v 6= 0} and f >1,1< is the first mixed partial difference quotient of f , that is, f >1,1< (x + u, x; y + v, y) [f (x + u, y + v) − f (x, y + v)] − [f (x + u, y) − f (x, y)] = . u·v A subset X of V is Cartesian if it is of the form X = X1 × · · · × Xd with X1 , . . . , Xd subsets of K and locally Cartesian if every x in X has a Cartesian neighborhood. Definition A.1. Let X be a subset of V and a in X.
30
• If X is Cartesian then the function f : X → E is C r at a = (a1 , . . . , ad ) if f >n< : X >n< → E is C ρ at ~a = (a1 , . . . , a1 ; . . . ; ad , . . . , ad ) in X
Y
f
f
n in Nd with n1 +···+nd =n
Definition. Let C be a compact Cartesian subset of V all of whose factors contain no isolated points. The norm k·kC r on C r (C, E) is defined by kf kC r = max{kf
Equivalence between Schikhof’s and our iterated divided differences Let V1 , . . . , Vn be vector spaces. The normed K-vector space Hom(V1 , Hom(V2 , . . . , Hom(Vn , E)) · · · ) ∗
is canonically isomorphic to V1∗ ⊗ · · · ⊗ Vn∗ ⊗ E. We identify E[ν] with V [ν−1] ⊗ · · · ⊗ V ∗ ⊗ E by this isomorphism (cf. Section 3).
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Proposition A.2. We have C r (X, E) = C r (X, E) as sets. If X is a compact Cartesian subset of V , all of whose factors contain no isolated points, then the inclusion map C r (X, E) ,→ C r (X, E) is an isomorphism of normed K-vector spaces. Proof: Let X = X1 × · · · × Xd where X1 , . . . , Xd are subsets of K. Let us fix n in N. We define mutually inverse homomorphisms between normed K-vector spaces Y φn : C 0 (X
and ψn : im φn →
Y
n∈Nd with n1 +···+nd =n
C 0 (X
such that, if f in C r (X, E) ∩ C r (X, E) then φn (f
and
ψn (f [n] ) = f
We define φn by F 7→
n∈Nd
X
with n1 +···+nd =n
ιn ◦ F ◦ pn
where pn : X [n] → X
and
∗
∗
ιn : E → V [n−1] ⊗ · · · ⊗ V [1] ⊗ V ∗ ⊗ E
are defined as follows: For n in N, let p0 and p1 be the projections from the product V [n+1] = V [n] × V [n] to its left and right factor. For i in {0, 1}n let pi denote the function pi1 ◦ · · · ◦ pin : V [n] → V.
Let p1 , . . . , pd be the projections from the d-fold product V = K × · · · × K onto its first, . . . , last factor. • Let us establish pn . Let e0 = (0, . . . , 0) in {0, 1}n and for i = 1, . . . , n, let ei = (. . . , 0, 1, 0, . . .) be the tuple in {0, 1}n whose sole nonzero entry is 1 at the i-th place. Let {N1 , . . . , Nd } be a partition of {1, . . . , n} into d families of n1 , . . . , nd elements. We let pn be the restriction of the function ((p1 ◦ pen )n∈N1 ∪{0} ; . . . ; (pd ◦ pen )n∈Nd ∪{0} ) to the domain X [n] and codomain X
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• Let us establish ιn . For n in N and i in {0, 1}n , let ιi : V ∗ → V [n] be the dual of pi . Let ι1 , . . . , ιd : K∗ → V ∗ be the duals of p1 , . . . ,pd . Let N denote all tuples in {1, . . . , d}{0,...,n−1} that have n1 entries equal to 1, . . . , nd entries equal to d. We identify K∗ ⊗ · · · ⊗ K∗ ⊗ E, where K appears n times, with E and define X X ιn = ιbn−1 ◦ ιNn−1 ⊗ · · · ⊗ ιb0 ◦ ιN0 ⊗ idE N ∈N bn−1 ∈{0,1}n−1
.. .
b0 ∈{0,1}0
We define ψn as follows. For n in N, let υ0 and υ1 be the inclusions of the left and right summand of V [n+1] = V [n] ⊕ V [n] . For i in {0, 1}n let υi denote υi1 ◦ · · · ◦ υin : V → V [n] . Let υ1 , . . . , υd be the inclusions of the first, . . . , last summand of V = K ⊕ ∗ · · · ⊕ K. For n in N and i in {0, 1}n , let πi : V [n] → V ∗ be the dual of υi . Let π1 , . . . , πd : V ∗ → K∗ be the duals of υ1 , . . . , υd . We fix a tuple N in {1, . . . , d}{0,...,n−1} that has n1 entries equal to 1, . . . , nd entries equal to d and let us fix n tuples bn−1 , . . . ,b0 in {0, 1}n , . . . ,{0, 1}0 . Let ∗ ∗ πn : V [n−1] ⊗ · · · ⊗ V [1] ⊗ V ∗ ⊗ E → E denote the function πNn−1 ◦ πbn−1 ⊗ · · · ⊗ πN0 ◦ πb0 ⊗ idE
where we identify (K∗ ⊗ · · · ⊗ K∗ ) ⊗ E with E. If F in im ψn then the function πn ◦ F factorizes over pn : X [n] → X
Let f : X → E. We prove by induction on ν that f is a C ν -function if and only if it is a C ν -function. For ν = 0 this holds because by definition f <ν> = f = f [ν] . Let ν in N. We assume by induction that f is a C ν−1 -function if and only if it is a C ν−1 -function. By definition f is a C ν -function if for every ν in Nd with ν1 + · · · + νd = ν, the function f >ν< extends to a C ρ -function f <ν> and by definition f is a C ν -function if, with F = f [ν−1] , the function F ]1[ extends to
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a C ρ -function on X [ν] . Thence f is C ν -function if and only if f is a C ν -function because φν (f <ν> ) = f [ν] and ψν (f [ν] ) = f <ν> . (∗) We conclude C r (X, E) = C r (X, E) as sets. ` Let X be an accumulated subset of V . Then X is the disjoint union U of Cartesian subsets of V , so a Y a Y U, E = C r (U, E) and C r U, E = C r (U, E). Cr U ∈U
U ∈U
Because C r (U, E) = C r (U, E) as sets for each U in U we conclude C r (X, E) = C r (X, E) as sets as well. Let X be compact Cartesian, that is, there are compact subsets X1 , . . . , Xd of K such that X = X1 × · · · × Xd . By definition of the norms of C r (X, E) and C r (X, E), the identity map of sets id : C r (X, E) → C r (X, E) is an isometric isomorphism between normed spaces because for every n = 0, . . . , ν the mappings φ0 , . . . , φν−1 and ψ0 , . . . , ψν−1 respectively φν and ψν are isometries for the C 0 -norm respectively C ρ -norm and satisfy Equation (∗).
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