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P-adic oscillatory integrals and Newton Polyhedra.

Resumen

En este articulo damos una estimativa asintotica para integrales oscilantes p-adicas que dependen de dos parametros. Estas integrales son soluciones de ecuaciones seudodiferencialesp-adicas del tipo Schrodinger.

Palabras clave: Cuerpos p-adicos, integrales oscilatorias, funciones zeta locales de Igusa, poliedros de Newton, ecuaciones seudodiferenciales.

Abstract

In this paper we give an asymptotic estimate forp-adic oscilatory integrals depending of two parameters. These integrals are solutions of Schrodinger-type pseudo-differential equations.

Key words: p-adie fields, oscillatory integrals, Igusa local zeta function, Newton polyhedra, pseudo-differential equations.

1. Introduction

Let K be p-adic field, i.e. a finite extension of [Q.sub.p], [R.sub.K] the ring of integers of K, [P.sub.K] the maximal ideal of [R.sub.K], and [bar.K] = [R.sub.K]/[P.sub.K] the residue field of K. The cardinality of the residue field of K is denoted by q, thus [bar.K] = [F.sub.q]. For z [member of] K, [upsilon](z) [member of] Z [intersectoin] {+[infinity]} denotes the valuation of z, [[absolute value of z].sub.K] = [q.sup.-[upsilon](z)] and ac z = z[[pi].sup.-[upsilon](z)] where [pi] is a fixed uniformizing parameter for [R.sub.K]

Let [PSI] denote a standard additive character of K, thus, for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where Tr denotes the trace.

Let [phi]([xi])[member of] [R.sub.K][[xi]], [xi] = ([[xi].sub.1],..., [[xi].sub.n]), = be a non-constant polynomial, t [member of] K, with [upsilon](t) < 0, x = ([x.sub.1],..., [x.sub.n]) [member of] [K.sup.n], with [upsilon]([x.sub.i]) < 0, i = 1,..., n, and

m = min{[upsilon]([x.sub.1]),..., [upsilon]([x.sub.n]), [upsilon](t)} .

We put <x, [member of]> = [[SIGMA].sub.i][x.sub.i][[xi].sub.i], for x, [xi] [member of] [K.sup.N]. To these data we associate the following parametric exponential sum

J (x,t, [phi],K) = J(x,t)

= [q.sup.-mn] [summation over ([xi] mod [[pi].sup.m])] [PSI](t[phi]([xi]) + <x, [xi]>). (1-1)

The exponential sum J (x, t) can be expressed as an integral of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

where is the Haar measure of [K.sup.n] normalized so that the volume of [R.sup.n.sub.K] is 1. A more general type of oscillatory integrals is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-3)

where h is a Bruhat-Schwartz function, i.e. a locally constant function with compact support. Integrals of the form [[integral].sub.K] [PSI](a[xi] + b[xi]) h ([xi]) are called Gaussian ones. These integrals have explicitly calculated in several cases, and they appear in certain p-adic quantum models [12, Chap. 1, Sect. V, and Chap. 3].

The integrals I (x, t) are the non-archimedean counterpart of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

where [??] is the Fourier transform of f. Consider the Schrodinger-type equation

[partial derivative]u/[partial derivative]t = I[phi](D)u, u(x,0) = f(x), (1.5)

here [phi] (D) is a pseudo-differential operator having symbol [phi] ([xi]). Then function u (x, t) is a solution for the initial value problem (1.5) (see e.g. [10, Chap. VII, VIII]).

As a consequence of the previous considerations it is natural to ask if integrals (1.2) and (1.3) satisfy some differential equation. At this point, it is important to mention that there are deep connections between differential equations and exponential sums over finite fields [5], [7].

Integrals (1.2) and (1.3) satisfy pseudo-differential equations of Schrodinger-type. Let S([K.sup.n]) denote the C-vector space of Schwartz-Bruhat functions over [K.sup.n]. The dual space S'([K.sup.n]) is the space of distributions over [K.sup.n]. A pseudo-differential operator of Schrodinger-type A ([partial derivative]), with symbol [[absolute value of [tau] - [phi] ([xi])].sub.K], is an operator of the form

A([partial derivative]) : S([K.sup.n]) [flecha diestra] S([K.sup.n]) [PHI] [right arrow] [F.sup.-1.sub.([xi])[right arrow](x,t)] ([[absolute value of [tau] - [phi] ([xi])].sub.K] [F.sub.(y, [??])[right arrow] ([tau], [xi])([PHI], (1.6)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

is the Fourier transform. The operator A([partial derivative]) has self-adjoint extension with dense domain in [L.sup.2] ([K.sup.n]).

The initial value problem

A([partial derivative])z = 0, z(x,0) = h{x)[member of]S{[K.sup.n]) (1.8)

is the non-archimedean counterpart of (1.5). By passing to the Fourier transform in (1.8), we get

[[absolute value of [tau]-[phi]([xi])].sub.K][F.sub.(y, [??])[right arrow]([tau], [xi]] (z) = 0, (1.9)

from where it follows that any distribution of the form

z(x,t) = ([F.sup.-1]g) (x,t),

with g([xi], [tau]) a distribution with support in

{{[xi], [tau])[member of][K.sup.n+1]|[[absolute value of [tau]-[phi]([xi])].sub.K] = 0},

is a solution of A([partial derivative])z = 0. In the case when

g ([xi], [tau]) = [??]([xi])[delta]([tau] - [phi]([xi]),

where [delta] is the Dirac distribution, and [??] is the Fourier transform of h, the distribution z(x, t) takes the form (1.3). Finally, since I(x, 0) = h(x), it holds that I(x,t) is a solution for the initial value problem (1.8). In particular the exponential sums J (x,t) satisfy (1.8), when h(x, 0) is equal to the characteristic function of [R.sup.n.sub.K].

The theory of non-archimedean pseudo-differential operators is emerging motivated for its potential use in p-adic physics [12], [8].

The main result of this paper (cf. Theorem 3.1) gives an asymptotic estimation of [absolute value of I(x,t)] for min ([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K]) [right arrow] [infinity], in the case in which the singular locus of t[phi]([xi])+<x, [xi]> is a subset of {(t, x, [xi]) | x = [xi] = 0}, and [phi] is a generic polynomial with an algebraically isolated singularity at the origin. The proof of the main result depends on the description of the poles of the Igusa zeta function for non-degenerate polynomials [3], [4], [13], [14], and a theorem of Igusa that establishes a connection between the poles of local zeta functions and the asymptotic expansions of certain p-adic oscillatory integrals [6, Theorem 8.4.2 (3)].

2. Exponential sums and Newton polyhedra

We set [R.sub.+] = {x [member of] R | x [??] 0}. Let f([xi]) = [a.sub.l][[xi].sup.l] [member of] K[[xi]], [xi] = ([[xi].sub.1],..., [[xi].sub.n]) be a polynomial in n variables satisfying f(0) = 0. The Newton polyhedron [GAMMA](f) of f is defined as the convex hull in [R.sup.n.sub.+] of the set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We denote by <,> the usual inner product of [R.sup.n], and identify [R.sup.n] with its dual by means of it. We set

<[a.sub.[gamma]],x> = m([a.sub.[gamma]]),

for the equation of the supporting hyperplane of a facet [gamma] (i.e. a face of codimension 1 of [GAMMA](f)) with perpendicular vector [a.sub.[gamma]] = ([a.sub.1],..., [a.sub.n]) [member of] [N.sup.n] {0}, and [sigma] ([a.sub.[gamma]]) = [[SIGMA].sub.i] [a.sub.i].

Definition 2.1. A polynomial f([xi]) = [[SIGMA].sub.i][a.sub.i][[xi].sup.i] [member of] K[[xi]], [xi] = ([[xi].sub.1],..., [[xi].sub.n]), is called globally non-degenerate with respect to its Newton polyhedron [GAMMA](f), if it satisfies the following two properties:

(1) the origin of [K.sup.n] is a singular point of f([xi]);

(2) for every face [gamma] [subset] [GAMMA](f) (including [GAMMA](f) itself), the polynomial

[f.sub.[gamma]] ([xi]) = [summation over (i[member of][gamma])][a.sub.i][[xi].sup.i]

has the property that there is no [xi] [member of] [(K {0}).sup.n] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For a polynomial f([xi]) [member of] K [[xi]] globally non-degenerate with respect to its Newton polyhedron [GAMMA](f), we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[tau].sub.j] runs through all facets of [GAMMA](f) satisfying m([a.sub.j]) [not equal to] 0. We note that

[T.sub.0] = (-[beta][(f).sup.-1],...,-[beta][(f).sup.-1]) [member of]] [Q.sup.n]

is the intersection point of the boundary of the Newton polyhedron [GAMMA](f) with the diagonal {(t,...,t) | t [member of]T} [subset] [R.sup.n].

We put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with z [member of] K. Igusa showed that the asymptotic behavior of E(z,f,K), when [[absolute value of z].sub.K] [right arrow] [infinity], is controlled by the largest pole of the meromorphic continuation of the local zeta function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

associated to f, and a multiplicative character [chi] of [R.sup.x.sub.K]. More precisely, if [[gamma].sub.f] the maximum of the real parts of the poles of Z(s, f, [chi]), and [gamma] > -1, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

where C(K) is a constant, and [epsilon] > 0 (see e.g. [2, Corollary 1.4.5], or [6, Theorem 8.4.2 (3)]).

Theorem 2.2. Let K be a non-archimedean local field, and let f([xi]) [member of] [R.sub.K][[xi]], [xi] = ([[xi].sub.1],..., [[xi].sub.n]), be a globally non-degenerate polynomial with respect to its Newton polyhedron [GAMMA](f). If [beta](f) > -1, then

[absolute value of E(z,f)] [??]C(f,K)[[absolute value of z].sup.[beta](f)+[epsilon].sub.K] (2.3)

for [[absolute value of z].sub.K] [right arrow] [infinity], and any [epsilon] > 0, here C(f,K) is constant depending on f and K.

The theorem follows from (2.2) by showing the following two facts. First, the poles of Z(s, f, x) have the form

S = -[sigma]([a.sub.[gamma]])/ m([a.sub.[gamma]]) 2[pi]i/log q k/m([a.sub.[gamma]]), k [member of] Z,

for some facet [gamma] of [GAMMA](f) with perpendicular [a.sub.[gamma]], and m([a.sub.[gamma]]) [not equal to] 0, or

s = -1+ 2[pi]i/log q k, k [member of] Z.

Second, the maximum of the real parts of the poles of Z(s,f, [[chi].sub.triv]) is [beta](f), when [beta](f) > -1 (cf. [14, Theorems A, B]). The description of the largest pole of Z(s, f, [[chi].sub.triv]) when f is non-degenerate with respect to its Newton polyhedron [GAMMA](f) and [beta](f) > -1 follows from observations made by Varchenko in [11] and was originally noted in the p-adic case in [9] (although it is misstated there as a(f) [not equal to] -1). The case [beta](f) = -1 is treated in [4]. The case of [beta](f) < - 1 is more difficult and is established in [4] with some additional conditions on To by using a difficult result on exponential sums. The author proved the case [beta](f) [??] - 1 for polynomials with coefficients in a non-archimedean local field of arbitrary characteristic [14].

We put [parallel]x[[parallel].sub.K] = [max.sub.i] {[[absolute value of [x.sub.i]].sub.K]}, for x [member of] [K.sup.n].

Proposition 2.3. Let f([xi]) [member of] [R.sub.K] [[xi]], [xi] = ([[xi].sub.1],... [[xi].sub.n]), be a non-constant polynomial without singularities on [K.sup.n]. Then

[absolute value of E(z,f)] [less than or equal to] C(f,K)[ [absolute value of z].sup.1+[epsilon].sub.K]

for [[absolute value of z].sub.K] [right arrow] [infinity], and any [epsilon] > 0, here C(f, K) is a positive constant depending only on f and K.

Proof. The stationary phase formula implies that E(z,f) - 0 for [[absolute value of z].sub.K] [right arrow] [infinity]. The proof is a slightly variation of the proof of Lemma (2.4) in [14]. ?

Proposition 2.4. Let [phi]([xi]) [member of] [R.sub.K] [[xi]], [xi] = ([[xi].sub.1],..., [[xi].sub.n]), be a globally non-degenerate polynomial with respect [GAMMA]([phi]), with [beta] (f) > - 1. If the singular locus of the polynomial t[phi]([xi]) + <x, [xi]> is contained in {(t,x, [xi]) | x = 0}, then

[absolute value of J(x,t)] [less than or equal to] C([phi], K) [[min ([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K])].sup.[beta]([phi])+[epsilon]],

for min ([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K]) [right arrow] [infinity], and [epsilon] > 0, here C([phi], K) a positive constant depending only on [phi] and K.

Proof. For x [not equal to] 0, and t [not equal to] 0, t[phi] ([xi]) + <x, [xi]> does not have singular points on [K.sup.n]. If [parallel]x[[parallel].sub.K]/[[absolute value of t].sub.K] [less than or equal to] 1, Proposition 2.3 implies that

[absolute value of J(x,t)] [less than or equal to] [C.sub.0](K) [[absolute value of t].sup.-1+[epsilon].sub.K],

for [[absolute value of t].sub.K] [right arrow] [infinity], and [epsilon] > 0. If [[absolute value of t].sub.K]/ [parallel]x[[parallel].sub.K] < 1, Proposition 2.3 implies that

[absolute value of J(x,t)] [less than or equal to] [C.sub.1](K) [[absolute value of t].sup.-1+[epsilon].sub.K]

for [parallel]x[[parallel].sub.K] [right arrow] [infinity], and [epsilon] > 0. Therefore if x [not equal to] 0, and t [not equal to] 0,

[absolute value of J(x,t)] [less than or equal to] [C.sub.3](K) min [([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K]).sup.-1+[epsilon]], (2.4)

for min ([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K]) [right arrow] [infinity], and [epsilon] > 0.

For x = 0, and t [not equal to] 0, Theorem 2.2 implies that

[absolute value of J(x,t)] [less than or equal to] [C.sub.4]([phi],K)[[absolute value of t].sup.[beta]([phi])+[epsilon].sub.K], (2.5)

for [[absolute value of t].sub.K] [right arrow] [infinity], and [epsilon] > 0. For x [not equal to] 0, and t = 0,

J(x,t)= 0, (2.6)

for [[absolute value of t].sub.K] [greater than or equal to] 1. Since [beta] (f) > -1, estimates (2.4), (2.5), and (2.6) imply that

[absolute value of J(x,t)] [less than or equal to] C([phi],K) [[min([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K])].sup.[beta]([phi])+[epsilon]],

for min ([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K]) [right arrow] [infinity], and [epsilon] > 0.

The proof of the following Proposition is similar to the previous one.

Proposition 2.5. Let [phi]([xi]) [member of] [R.sub.K] [[xi]], [xi] = ([[xi].sub.1],..., [[xi].sub.n]), be polynomial such that t[phi] ([xi]) + <x, [xi]> has no singular points on [K.sup.n], for any t [member of] K, and x [member of] [K.sup.n], then

[absolute value of J(x,t)] [less than or equal to] C(K) [[min([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K])].sup.-1+[epsilon]],

for min ([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K]) [right arrow] [infinity], and [epsilon] > 0, here C(K) a positive constant.

3. Main Result

We shall say that the origin of [K.sup.n] is an algebraically isolated singularity of [phi]([xi]) [member of] K [[[xi].sub.1],..., [[xi].sub.n]], if the origin is the only solution of the system

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

Theorem 3.1. Let [phi]([xi]) [member of] [R.sub.K][[[xi].sub.1],..., [[xi].sub.n]] be a non constant polynomial with an algebraically isolated singularity at the origin, such that [phi]([xi]) is globally nongenerate with respect [GAMMA]([phi]), and [beta]([phi]) > -1. If the singular locus of the polynomial t[phi] ([xi]) + <x, [xi]> is contained in {(t,x, [xi]) | x = 0}, then

[absolute value of J(x,t)] [less than or equal to] C([phi], K) [[min ([parallel]x[[paralle].sub.K], [[absolute value of t].sub.K])].sup.[beta]([phi])+[epsilon]], (3.2)

for min ([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K]) [right arrow] [infinity], and any [epsilon] > 0, here C([phi], K) a positive constant depending only on [phi] and K.

Proof. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a finite covering of the support of h such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] We put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Without loss of generality we may assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] With this notation, (3.3) can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

If is the origin then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

for min ([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K]) [right arrow] [infinity], and any [epsilon] > 0, with [C.sub.1] ([phi], K) a positive constant depending only on h and K (cf. Proposition 2.4).

If [z.sub.i] is not the origin, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] does not have singularities on [K.sup.n]. Then Proposition 2.5 implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

for min([parallel]x[[parallel].sub.K], [[absolute value of t].sub.K]) [right arrow] [infinity], and any [epsilon] > 0, with [C.sub.2](K) a positive constant. The result follows from (3.5) and (3.6) by using the fact that [beta]{[phi]) > -1.

3.1. Remarks

1. The main result is valid for non-archimedean local fields of positive characteristic (cf. [14, Theorems A, B, and Corollary 6.1]).

2. Recently R. Cluckers showed that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for [parallel]y[[parallel].sub.K] [right arrow] [infinity] when f ([xi]) = ([f.sub.1] ([xi]),..., [f.sub.n] ([xi])) is a dominant polynomial map. This result does not provide any information about [alpha] [1, Chap. VI].

References

[1] Cluckers R., Cell decomposition and p-adic integration, Ph.D. Thesis 2002, available at http://front.math.ucdavis. edu/raath.LO/0301023.

[2] Denef J., Report on Igusa's local zeta function, Seminaire Bourbaki 1990/1991 (730-744) in Asterisque 201-203 (1991), 359-386.

[3] Denef J., Poles of p-adic complex powers and Newton polyhedra, Nieuw archief voor wiskunde, 13 (1995), 289-295.

[4] Denef J., Hoornaert Kathleen, Newton polyhedra and Igusa local zeta function, To appear in Journal of Number Theory.

[5] Dwork B., Bessel functions as p-adic functions of argument, Duke Math. J. 41 (1974), 711-738.

[6] Igusa Jun-Ichi, An introduction to the theory of local zeta functions, AMS /IP Studies in Advanced Mathematics, v. 14, 2000.

[7] Katz Nicholas M., Exponential sums and differential equations, Princeton University Press, 1990.

[8] Kochubei A. N., Pseudodifferential equations and stochastics over non-archimedean fields, Marcel Dekker, 2001.

[9] Lichtin B. & Meuser D., Poles of a local zeta function and Newton polygons, Compos. Math. 55 (1985), 313-332.

[10] Stein Elias M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, 1993.

[11] Varchenko A., Newton polyhedra and estimation of oscillanting integrals, Funct. Anal. Appl. 10 (1976), 175-196.

[12] Vladimirov V. S., Volovich I. V., & Zelenov E. I., p-adic Analysis and Mathematical Physics, World Scientific, Singapore, 1994.

[13] Zuniga-Galindo W. A., Igusa's local zeta functions of semiquasihomogeneous polynomials, Trans. Amer. Math. Soc. 353, (2001), 3193-3207.

[14] Zuniga-Galindo W. A., Local zeta functions and Newton polyhedra, Nagoya Math. J., 172 (2003), 31-58.

W. A. Zuniga-Galindo (1)

Dedicated to the Memory of Professor Jairo Charris

(1) Department of Mathematics and Computer Science, Barry University, 11300 N. E. Second Avenue, Miami Shores, Florida 33161, USA. E-mail: wzuniga@mail.barry.edu

AMS Subject Classification: Primary 46S10, 11S40.
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