Fq-Linear Calculus over Function Fields

arXiv:math/9807075v1 [math.NT] 15 Jul 1998

Fq -Linear Calculus over Function Fields Anatoly N. Kochubei Institute of Mathematics, Ukrainian National Academy of Sciences, Tereshchenkivska 3, Kiev, 252601 Ukraine E-mail: [email protected]

Copyright Notice: This work has been submitted to Academic Press for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible.

1

Abstract We define analogues of higher derivatives for Fq -linear functions over the field of formal Laurent series with coefficients in Fq . This results in a formula for Taylor coefficients of a Fq -linear holomorphic function, a definition of classes of Fq -linear smooth functions which are characterized in terms of coefficients of their Fourier-Carlitz expansions. A Volkenborn-type integration theory for Fq -linear functions is developed; in particular, an integral representation of the Carlitz logarithm is obtained.

Key words: Fq -linear function; Carlitz basis; Carlitz logarithm; Volkenborn integral; difference operator; Bargmann-Fock representation.

2

1

INTRODUCTION

It is well known that any non-discrete, locally compact field of characteristic p is isomorphic to the field K of formal Laurent series with coefficients from the Galois field Fq , q = pγ , γ ∈ Z+ . The foundations of analysis over K were laid in a series of papers by Carlitz (see, in particular, [3-5]), Wagner [20, 21], Goss [6-8], and Thakur [17, 18]. An interesting property of many functions introduced in the above works as analogues of classical elementary and special functions, is their Fq -linearity. Recall that a function f : K → K c (where K c is a completion of an algebraic closure of K) is called Fq -linear if f (t1 + t2 ) = f (t1 ) + f (t2 ) and f (αt) = αf (t) for any t, t1 , t2 ∈ K, α ∈ Fq . In a similar way one can define a notion of a Fq -linear function on any Fq -subspace of K, and also a notion of a Fq -linear operator on a vector space over K. Tools of classical calculus are not sufficient to study behavior of Fq -linear functions. For example, if such a function f is differentiable then f ′ (t) ≡ const, and all the higher derivatives vanish irrespective of possible properties of f . In particular, one cannot reconstruct the Taylor (n) coefficients of a holomorphic function – the classical formula contains the expression f n!(t) where both the numerator and denominator vanish. Thanks to Carlitz, we know the correct analogue of the factorial. A counterpart of a higher derivative is given in this paper. In fact, its main ingredient, the difference operator ∆(n) ,
n−1 ∆(n) f (t) = ∆(n−1) f (xt) − xq ∆(n−1) f (t), n ≥ 1; ∆(0) = id, where x is a prime element in K, was introduced by Carlitz and often used subsequently. Our approach is based on identities for the difference operator ∆ = ∆(1) , which emerge as a function field analogue of the Bargmann-Fock representation of the canonical commutation relations of quantum mechanics [9, 14]. Note that an analogue of the Schr¨odinger representation was obtained in [12]. We will also use ∆(n) in order to characterize “smoothness” of Fq -linear functions and obtain an exact correspondence between the degree of smoothness (or analyticity) and the decay rate for coefficients of Fourier-Carlitz expansions. Note that in the p-adic case (where smoothness is understood in a conventional sense) such a correspondence was found in [1, 2, 15]. Having a new “derivative”, we can introduce a kind of an antiderivative, and a Volkenborn type integral (see [15, 19] for the case of zero characteristic), which leads, in particular, to an integral representation of the Carlitz logarithm, establishing a direct connection between the latter and the Carlitz module operation.

2 2.1

PRELIMINARIES Carlitz Basis [3, 4, 6, 12, 20]

Denote by | · | the non-Archimedean absolute value on K; if z ∈ K, z=

∞ X i=n

ζi xi ,

n ∈ Z, ζi ∈ Fq , ζn 6= 0,

3

then |z| = q −n . It is well known that this valuation can be extended onto K c . Let O = {z ∈ K : |z| ≤ 1} be the ring of integers in K. The ring Fq [x] of polynomials (in the indeterminate x) with coefficients from Fq is dense in O. Let C0 (O, K c ) be the Banach space of all Fq -linear continuous functions O → K c , with the supremum norm k · k. Any function ϕ ∈ C0 (O, K c ) admits a unique representation as a uniformly convergent series ϕ=

∞ X

ci ∈ K c , ci → 0,

ci fi ,

i=0

satisfying the orthonormality condition kϕk = sup |ci |. i≥0

Here {fi } is the sequence of the normalized Carlitz Fq -linear polynomials defined as follows. Let e0 (t) = t, Y ei (t) = (t − m), i ≥ 1. (1) m∈Fq [x] deg m deg s if s ∈ Fq [x]). Since s−1 fi (s) = L−1 i τi (s) and |Li | = q , the function Cs (z) is differentiable with respect to s. The Carlitz logarithm logC (z) is defined as

logC (z) =

n ∞ X zq (−1)n , L n n=0

|z| < 1.

The Carlitz exponential is defined by the power series eC (z) =

n ∞ X zq

n=0

Dn

Theorem 4. (i) For any n = 0, 1, 2, . . . Z n tq dt = − O

14

,

|z| < 1.

1 . [n + 1]

(22)

(ii) For any n = 0, 1, 2, . . . Z

fn (t) dt =

O

(iii) If z ∈ K, |z| < 1, then

Z

(−1)n+1 . Ln+1

(23)

Cs (z) ds = logC (z) − z.

(24)

O

Proof. (i) We have seen that −

a



n

tq Dn



n−1

tq = , Dn−1

n ≥ 1.

It follows from the definition of the operator S that  qn  n+1 t tq − t S = Dn Dn+1 whence qn

S t S tq

n




xk (ii) We have




 tqn+1 − t Dnq  qn+1 t −t = , = Dn+1 [n + 1]




xk(q −1) − 1 1 = −→ − , [n + 1] [n + 1]

Z

′ fn (t) dt = (Sfn )′ (0) = fn+1 (0).

xk

n+1

k → ∞.

O ′

According to (1 ), the linear term in the expression for fi is i (−1)i i 0 (−1) t= t. Di Li

(25)

The differentiation yields (23). (iii) Applying the operator a− (with respect to the variable s) to the function Cs (z) we find that ∞ X i−1 a− C (z) = fi−1 (s)z q = Cs (z) s s i=1

whence

Ss Cs (z) = Cs (z) − C1 (z)s =

∞ X i=0

i

fi (s)z q − zs.

Fixing z and denoting ϕ(s) = Ss Cs (z) we obtain that ∞

ϕ(xn ) X fi (xn ) qi = z − z. xn xn i=0 15

(26)

It is seen from (1′ ) and (25) that (−1)i fi (xn ) −→ for n → ∞. xn Li On the other hand, t−1 fi (t) = Di−1 gqi −1 (t) = Di−1 Γqi −1 τi (t). As we know, |τi (t)| ≤ 1, |Di−1Γqi −1 | = q i , so that fi (xn ) i xn ≤ q

for all n. If z ∈ O, |z| < 1, then |z| ≤ q −1 , and the series in (26) converges uniformly with respect to n. Passing to the limit n → ∞ in (26) we come to (24). 2 Setting in (24) z = eC (t), |t| < 1, we get an identity for the Carlitz exponential: Z eC (st) ds = t − eC (t). O

On the other hand, the formula (24) implies a more general formula (conjectured by D.Goss). Corollary. If a ∈ O, z ∈ K, |z| < 1, then Z Csa (z) ds = a logC (z) − Ca (z).

(27)

O

Proof. Since it is shown easily that (for each fixed z) the mapping O → C01 (O, K c ) of the form a 7→ Csa (z) is continuous, it is sufficient to prove (27) for a = xn , n = 1, 2, . . . . Using (24), we find that Z n X n xn−k Cxqk−1 (z). (28) Csxn (z) ds = x (logC (z) − z) − k=1

O

Let t = logC (z). Then z = eC (t) (see [8, 12]). It follows from properties of eC [8] that n X k=1

n−k

x

Cxqk−1 (z)

=

n X k=1

xn−k (eC (xk t) − xeC (xk−1 t)) = eC (xn t) − xn eC (t) = Cxn (z) − xn z.

Substituting this into (28) we come to (27). 2 As Csa (z) = Cs (Ca (z)), equation (24) also implies that Z Csa (z) ds = logC (Ca (z)) − Ca (z) . O

Comparing this with (27) implies a logC (z) = logC (Ca (z)) which is precisely the functional equation of logC (z). 16

ACKNOWLEDGEMENTS The author is grateful to D.Goss for a lot of valuable remarks and suggestions, to D.Thakur for a helpful consultation, and to Z.Yang for pointing out an error in the first version of the paper. The work was supported in part by the Ukrainian Fund for Fundamental Research under Grant 1.4/62.

References [1] Y. Amice, Interpolation p-adique, Bull. Soc. Math. France 92 (1964), 117–180. [2] D. Barsky, Fonctions k-lipschitziennes sur anneau local et polynˆomes `a valeurs enti`eres, Bull. Soc. Math. France 101 (1973), 397–411. [3] L. Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), 137–168. [4] L. Carlitz, A set of polynomials, Duke Math. J. 6 (1940), 486–504. [5] L. Carlitz, Some special functions over GF (q, x), Duke Math. J. 27 (1960), 139–158. [6] D. Goss, Fourier series, measures and divided power series in the theory of function fields, K-Theory 1 (1989), 533–555. [7] D. Goss, A formal Mellin transform in the arithmetic of function fields, Trans. Amer. Math. Soc. 327 (1991), 567–582. [8] D. Goss, “Basic Structures of Function Field Arithmetic”, Springer, Berlin, 1996. [9] B. C. Hall, The Segal-Bargmann “coherent state” transform for compact Lie groups, J. Funct. Anal. 122 (1994), 103–151. [10] Y. Hellegouarch, Galois calculus and Carlitz exponentials, in: “The Arithmetic of Function Fields” (eds: D.Goss et al.), de Gruyter (1992), 33–50. [11] A. N. Kochubei, p-Adic commutation relations, J. Phys. A: Math. and Gen. 29 (1996), 6375–6378. [12] A. N. Kochubei, Harmonic oscillator in characteristic p, Lett. Math. Phys. (to appear). [13] R. Lidl and H. Niederreiter, “Finite Fields”, Addison-Wesley, Reading, 1983. [14] A. Perelomov, “Generalized Coherent States and Their Applications”, Springer, Berlin, 1986. [15] W. Schikhof, “Ultrametric Calculus”, Cambridge University Press, 1984. [16] A. Sudbery, “Quantum Mechanics and the Particles of Nature”, Cambridge University Press, 1986. 17

[17] D. Thakur, Gamma functions for function fields and Drinfeld modules, Ann. Math. 134 (1991), 25–64. [18] D. Thakur, Hypergeometric functions for function fields, Finite Fields and Their Appl. 1 (1995), 219–231. [19] A. Volkenborn, Ein p-adisches Integral und seine Anwendungen I, II, Manuscripta Math. 7 (1972), 341–373; ibid. 12 (1974), 17–46. [20] C. G. Wagner, Interpolation series for continuous functions on π-adic completions of GF (q, x), Acta Arithm. 17 (1971), 389–406. [21] C. G. Wagner, Linear operators in local fields of prime characteristic, J. Reine Angew. Math. 251 (1971), 153–160.

18