The fundamental theorem of prehomogeneous vector spaces modulo <span class="mathjax-formula">$p^m$</span> (With an appendix by F. Sato)

Bulletin

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Publié avec le concours du Centre national de la recherche scientifique

de la SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Tome 135 Fascicule 4

2007

pages 475-494

FUNDAMENTAL THEOREM MODULO p ^m

Raf Cluckers

THE FUNDAMENTAL THEOREM OF

PREHOMOGENEOUS VECTOR SPACES MODULOp^m (WITH AN APPENDIX BY F. SATO)

by Raf Cluckers & Adriaan Herremans

Abstract. — For a number field K with ring of integersO_K, we prove an ana- logue over finite rings of the formO_K/P^mof the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where P is a big enough prime ideal ofO_K andm >1. In the appendix, F. Sato gives an application of the Theorems 1.1, 1.3 and the Theorems A, B, C in J. Denef and A. Gyoja [Char- acter sums associated to prehomogeneous vector spaces, Compos. Math.,113(1998), 237–346] to the functional equation of L-functions of Dirichlet type associated with prehomogeneous vector spaces.

Texte recu le 12 septembre 2006, révisé le 16 novembre 2006

Raf Cluckers, Katholieke Universiteit Leuven, Departement wiskunde, Celestijnenlaan 200B, B-3001 Leuven (Belgium). Current address: École Normale Supérieure, Département de mathématiques et applications, 45 rue d’Ulm, 75230 Pari Cedex 05 (France)

E-mail :[email protected] • Url :www.dma.ens.fr/∼cluckers/

Adriaan Herremans, Katholieke Universiteit Leuven, Departement wiskunde, Celestijnenlaan 200B, B-3001 Leuven (Belgium)

E-mail :[email protected]

2000 Mathematics Subject Classification. — 11S90, 11L07, 11M41, 11T24, 11L05, 20G40.

Key words and phrases. — Prehomogeneous vector spaces,L-functions, Bernstein-Sato poly- nomial, fundamental theorem of prehomogeneous vector spaces, exponential sums.

During the preparation of the paper, the authors were postdoctoral fellows of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.). The first author was partially supported by The European Commission - Marie Curie European Individual Fellowship with contract number HPMF CT 2005-007121.

BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 0037-9484/2007/475/$ 5.00

©Société Mathématique de France

Résumé (Théorème fondamental des espaces vectoriels préhomogènes modulo p^m. Avec un appendice par F. Sato)

SoitKun corps de nombres avec anneaux d’entiersO_K; nous prouvons un ana- logue, sur des anneaux finis de la forme O_K/P^m, du théorème fondamental sur la transformation de Fourier de l’invariante relative d’un espace vectoriel préhomogène.

Ici, Pest un idéal premier assez grand deO_K etm >1. Dans l’appendice, F. Sato donne une application des théorèmes 1.1, 1.3 et des théorèmes A, B, C de J. Denef et A. Gyoja [Character sums associated to prehomogeneous vector spaces, Compos.

Math.,113(1998), 237–346] à l’équation fonctionelle deL-fonctions de type Dirichlet associées aux espaces vectorielles préhomogènes.

1. Introduction

We prove an analogue over finite rings of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces. In general, this fundamental theorem expresses the Fourier transform of χ(f), with χ a multiplicative (quasi-)character andf a relative invariant, in terms ofχ(f^∨)⁻¹, withf^∨the dual relative invariant. Roughly speaking, M. Sato [18]

proved the fundamental theorem over archimedian local fields, J. Igusa [7] over p-adic number fields, and J. Denef and A. Gyoja [5] over finite fields of big enough characteristic. In [9], the regular finite field case is reproved. When the prehomogeneous vector space is regular and defined over a number field K we prove an analogue of the fundamental theorem over rings of the formOK/P^m, wherePis a big enough prime ideal of the ring of integersOK ofKandm >1, see Theorem1.1. This result is derived from the results of [5] by using explicit calculations of exponential sums over the rings OK/P^m.

In [16], F. Sato introducesL-functions of Dirichlet type associated to regular prehomogeneous vector spaces. In the appendix by F. Sato to this paper, our results are used to obtain functional equations for theseL-functions and, under extra conditions, their entireness.

To state the main results, we fix our notation on prehomogeneous vector spaces. Let (G, ρ, V) be a reductive prehomogeneous vector space, meaning thatGis a connected complex linear reductive algebraic group,ρ:G→GL(V) is a finite dimensional rational representation, and V has an open G-orbit which is denoted by Ω. Assume that (G, ρ, V) has a relative invariant 06=f ∈C[V] with character φ∈ Hom(G,C^×), that is, f(gv) = φ(g)f(v) for allg∈Gandv∈V. We assume thatf is aregularrelative invariant, namely, Ω =V \f⁻¹(0) is a single G-orbit. Writing ρ^∨ : G → GL(V^∨) for the dual ofρ,(G, ρ^∨, V^∨)is also a prehomogeneous vector space, with an openG-orbit which is denoted byΩ^∨, and there exists a relative invariant 06=f^∨ ∈C[V^∨] whose character is φ⁻¹. ThenΩ^∨=V^∨\f^∨−1(0). The mapF:= grad logf is

o

an isomorphism between Ω and Ω^∨ with inverse F^∨ := grad logf^∨. One has dimV = dimV^∨=:nanddegf = degf^∨=:d.

Let K be a number field with ring of integersOK. Suppose that(G, ρ, V) is defined overK. We fix a basis of theK-vector spaceV(K)and we suppose thatf is inK[V]and has coefficients inOK (with respect to the fixedK-basis ofV(K)). Similarly we suppose thatf^∨is inK[V^∨]and has coefficients inOK

(with respect to the basis ofV^∨dual to the fixed basis ofV). WriteV(OK)for the points of V(K)with coefficients inOK (with respect to the fixedK-basis of V(K)), and similarly forV^∨(OK). For I an ideal ofO, writeV(OK/I)for the reduction moduloI of the latticeV(OK).

The Bernstein-Sato polynomialb(s)off is defined by f^∨(grad_x)f(x)^s+1=b(s)f(x)^s.

Writeb0for the coefficient of the term of highest degree ofb(s); one hasb0∈K.

The following theorem is an analogue of the fundamental theorem for pre- homogeneous vector spaces.

Theorem 1.1. — Let m≥2 be an integer,P be a prime ideal ofOK above a big enough prime p∈Z,χ be a primitive multiplicative character modulo P^m (extended by zero outside the multiplicative units), and let ψ be a primitive additive character modulo P^m. Write q:= #(OK/P). For L∈V^∨(OK/P^m) write

S(L) := X

x∈V(OK/P^m)

χ f(x)

ψ L(x) . Then the following hold:

1) if f^∨(L)6≡0 modP, then S(L) =q^12mnP

y∈OK/P^mχ^d(y)ψ(y) q^12m

χb0f^∨(L)⁻¹ d^d

α(χ, m)ⁿ⁻¹κ^∨(L), whereκ^∨(L)andα(χ, m)are 1 or−1;

2) if f^∨(L)≡0 modP, thenS(L) = 0.

The essential (and typical) content of this fundamental theorem is that the discrete Fourier transform of the function χ(f)onV(OK/P^m)is equal to the functionχ(f^∨)⁻¹ onV^∨(OK/P^m)times some factors, and vice versa.

The first part of Theorem1.1is obtained by combining explicit calculations of character sums of quadratic functions (§2) and of discrete Fourier trans- forms (§4), a p-adic version of Morse’s lemma (§3), and results of [5]. The second part of Theorem1.1is established by comparing the L2-norms ofχ(f) and of its discrete Fourier transform.

We also obtain explicit formulas for the constants κ^∨(L) and α(χ, m) of Theorem1.1, by using work in [5] and elementary calculations. To state these formulas we use the notion of the discriminant of a matrix, as in [5, 9.1.0].

Definition 1.2. — For a symmetric(n, n)-matrixAwith entries in a fieldk, if ^tXAX = diag (a₁, . . . , a_m,0, . . . ,0) with X ∈GLn(k), ^tX its transposed, andai∈k^×, put

∆(A) :=

m

Y

i=1

ai∈k^×/k^×2,

withk^×2the squares ink^×, and call it thediscriminant ofA.

Write k_P for the finite field OK/P and k_P^×2 for the squares in k_P^×. For m > 1 and L in V^∨(OK/P^m) with f^∨(L) 6≡ 0 modP, denote by h^∨(L) the image in k^×_P/k^×_P² of the discriminant of the matrix (∂²logf^∨(L)/∂yi∂yj)ij, where{y1, . . . , y_n}is the previously fixedK-basis ofV^∨(K). Writeχ1

2 for the Legendre symbol mod P. We then obtain

Theorem 1.3. — The following hold in case 1) of Theorem1.1:

1) κ^∨(L) =χ1

2(−d2ⁿ⁻¹h^∨(L))^m; 2) α(χ, m) = 1formeven;

3) α(χ, m) =G(χ1 2, ψ⁰)/√

q for m odd, with ψ⁰ any additive character de- fined by y 7→ χ(1 +π^m−1_P y), π_P any element in P of P-adic order 1, χ¹

2 the Legendre symbol modP, andG(., .)the classical Gauss sum.

Remark 1.4. — It is interesting to compare the formulas of Theorems 1.1 and 1.3 to the formulas form = 1 given in [5]; it seems that for m = 1 the formulas depend more on subtle information of the Bernstein-Sato polynomial off.

Acknowledgement. — The authors would like to express gratitude to Jan Denef and Akihiko Gyoja for many helpful comments during the preparation of this paper. We are indebted to Jan Denef for giving the essential outline of the proofs of the main results, and to Fumihiro Sato for agreeing on writing an appendix to this paper.

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2. Preliminaries on character sums

Let χ be a multiplicative character mod P^m, extended by zero for a≡0 mod P. Say that χ is induced by a character χ1 mod Pⁿ for n < m if χ1(amodPⁿ) =χ(a)for all a ∈ OK/P^m. Call χ primitive mod P^m if there exists no such n < m and χ₁ modPⁿ such that χ is induced byχ₁. Anal- ogously, call an additive character ψ mod P^m on OK primitive if it is not induced by a characterψ1modPⁿforn < m. Letf be a polynomial innvari- ables over O_K. If we evaluate Pχ(f(x)) modP^m, withχ primitive mod P^m for somem >1, it is well known that only the critical points mod P contribute to the sum, i.e. the elementsc∈ OK/P^mfor whichgrad(f)_c≡0 modP.The following is an extension of this result.

Proposition 2.1. — Let P be a prime ideal of OK,m > 1 an integer, χ a primitive multiplicative character mod P^m and f ∈ OK[x] a polynomial in n variables. Define S_f as

(2.1) Sf = X

x∈(OK/P^m)ⁿ

χ f(x) .

Then

Sf = X

x∈(OK/P^m)ⁿ v_P(grad(f)x)≥¹₂(m−1)

χ f(x) ,

where v_P(grad(f)_x) is the minimum of the P-valuations of ∂f /∂x_{j x} for j= 1, . . . , n. Moreover, the same formulas hold for an additive char- acterψ instead ofχ.

Proof. — We treat the case that f is a function of one variable; the general case is completely analogous. It suffices to prove that

Sf(i, c) := X

x∈ OK/P^m vP(f⁰(x))=i x≡cmodP^m−i−1

χ f(x)

is zero for everyi < ¹₂(m−1)and everyc∈ OK/P^m−i−1. Iff(c)≡0 modP, then Sf(i, c) is trivially zero. Suppose f(c) 6≡ 0 modP. Let πP be in P of P-adic order1. Writingf(x)as a polynomial in(x−c), we get the equalities

S_f(i, c) = X

x∈ OK/P^m vP(f⁰(x))=i x≡cmodP^m−i−1

χ f(c) + (x−c)f⁰(c) +· · · (2.2)

=χ f(c) X

x∈ OK/P^m vP(f⁰(x))=i x≡cmodP^m−i−1

χ

1 + (x−c)f⁰(c) f(c) (2.3)

=qⁱχ f(c) X

z∈ OK/P

χ(1 +π_P^m−1zα) = 0, (2.4)

withαa unit inOK/P. Equality (2.3) comes from the fact that theP-valuation of the terms of degree ≥2 in x−c are at least m. Indeed2m−2i−2 ≥m ifi≤ ¹₂(m−2) or i < ¹₂(m−1). Either there are no terms in (2.3) and then it is automatically zero, or, there are terms and then equality (2.4) follows immediately. We conclude that the sum (2.4) is zero since we sum a non-trivial additive characterψ(z) :=χ(1 +π^m−1_P αz)overO_K/P; ψis indeed non-trivial sinceχis primitive.

Definition 2.2. — Let P be a prime ideal of OK not containing 2 and let m > 1 be an integer. Let χ, resp. ψ, be a primitive multiplicative, resp.

primitive additive, character modP^m.Then, put α(χ, m) :=e X

x∈ OK/P^m

χ(1 +x²), α(ψ, m) :=e X

x∈ OK/P^m

ψ(x²).

One can calculate the value ofα(χ, m), using Propositione 2.1in an elemen- tary way, to obtain the following lemma.

Lemma 2.3. — With the assumptions and notation of Definition 2.2, the fol- lowing hold

• α(χ, m) =e q^12m ifm is even;

• α(χ, m) =e q^12(m−1)G(χ1

2, ψ⁰) if m is odd. Herein, χ1

2 is the Legendre symbol mod P,ψ⁰is any additive character defined byy7→χ(1+π^m−1_P y) with πP any element in P of P-adic order1, andG(., .)is the classical Gauss sum.

Similar formulas can be obtained for α(ψ, m)e with ψ a primitive additive character modulo P^m. This lemma and an induction argument yield the fol- lowing proposition.

Proposition 2.4. — Use the assumptions and notation of Definition 2.2.

Let f be a polynomial inn variables overOK of the form f(x) =a0+a1x²₁+· · ·+anx²_n, with a₀, a₁, . . . , a_n multiplicative units mod P^m. Then

S_f =χ(a₀)χ1

2(aⁿ₀a₁· · ·a_n)^mα(χ, m)e ⁿ with χ1

2 the Legendre symbol mod P,andSf as in(2.1).

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3. Ap-adic analogue of the lemma of Morse

We prove an analogue of the Morse’s lemma (originally formulated forC^∞ functions on real manifolds), forp-adic analytic functions. This lemma is nor- mally stated as a local property, but here we can work with fixed (large) neigh- borhoods. To deal with the fact that Zp is totally disconnected, we will use a global notion of analycity forp-adic maps. Results in this section also hold for finite field extensions ofQp.

By ap-adic manifold we mean ap-adic manifold as defined in [2] or equiva- lently [20], Section 3.

Definition 3.1. — LetA⊂Zⁿp be open.

— Call a function f : A → Z^p globally analytic if there is a power series Pcixⁱ∈Zp[[x]]which converges onAsuch thatf(x) =Pcixⁱfor eachx∈A, withx= (x1, . . . , xn),i= (i1, . . . , in)andxⁱ=Qn

j=1xⁱ_j^j.

— Call f analytic if there is a finite cover of A by opens U such that the restriction of f to each U is globally analytic. Similarly, call a map f : A ⊂ Zⁿp →Z^mp analytic if it is given by analytic functions onA. An analytic bijection with analytic inverse is calledbi-analytic.

— For an analytic function f : A ⊂ Zⁿp → Zp, define the gradient of f in a∈Aas

grad(f)_a=∂f

∂x₁ a

, . . . , ∂f

∂x_{n a} and theHessian off inaas

Hs(f)_a= ∂²f

∂xi∂xj a

ij

.

— Say that ais a critical point ifgrad(f)_a= 0 and call the critical point a∈Anon-degenerate ifdet(Hs(f)_a)6= 0. Calla∈Aanon-degenerate critical point modulo pifgrad(f)_a≡0 modpanddet(Hs(f)_a)6≡0 modp.

LetM ⊂Zⁿp be a compactp-adic manifold of pure dimensiondandf :M → Z^pan analytic function. Then there exists a finite disjoint cover ofM by opens Ui and analytic isometriesπi fromUi to open balls inZ^dp such that the maps f ◦π⁻¹_i are globally analytic, see e.g. [20], Section 3.

— Call a pointa∈Ui ⊂M a critical point off ifπi(a)is acritical point of f ◦π_i⁻¹ as defined above. Similarly we speak ofnon-degenerate critical points and non-degenerate critical point modulo p. This is independent of the choice ofUi andπi.

The next lemma is ap-adic variant of the inverse function theorem.

Lemma 3.2 (see [8, Cor. 2.2.1]). — Suppose that g1, . . . , gn ∈Zp[[x1, . . . , xn]]

satisfy det(∂gi/∂xj 0)ij 6≡0 modpandgi(0)≡0 modpfori= 1, . . . , n. Then the map

g: (pZp)ⁿ−→(pZp)ⁿ, x7−→ g₁(x), . . . , g_n(x) is(globally)bi-analytic.

The following is a p-adic analogue of the Morse’s lemma. The proof goes along the same lines as in [10] and we refer to [10], Lemma 2.2 for the details.

Lemma 3.3 (Morse). — Let p 6= 2 and let f : (pZp)ⁿ → pZp be a globally analytic map(thusf is given by a single power series in Zp[[x]]) such that0 is a non-degenerate critical point modulop. Then there is a unique critical pointc of f and there is a(globally)bi-analytic map

T : (pZp)ⁿ−→(pZp)ⁿ, x7−→T(x) =u such that

f(x) =f(c) +

n

X

i=1

a_iu²_i, for all x∈(pZp)ⁿ, with ai inZ^×p. Moreover, χ¹

2(H) =χ¹

2(Q

iai)with H = 2⁻ⁿdet(Hs(f)_c).

Proof. — The uniqueness of the critical pointcis proved in Lemma3.4below.

We may suppose thatf(0) = 0andc= 0. We can writef(x) =P

i,jxixjhij(x) withh_ij(x)∈Zp[[x]]and sincep6= 2we can assume thath_ij =h_ji. Suppose by induction that we have a (globally) bi-analytic mapT_r−1 : (pZp)ⁿ →(pZp)ⁿ: x7→usuch that for eachx∈(pZp)ⁿ

f(x) =

r−1

X

i=1

aiu²_i + X

i,j≥r

uiujHij(u),

withHij(u)∈Zp[[u]],Hij =Hji andai∈Z^×p. We have det Hs(f)₀

=λdet H_ij(0)

ij

with λ a unit in Zp, sodet(Hij(0))ij 6≡0 modpand after a linear change in the last n−r+ 1 coordinates we may assume that Hrr(0) 6≡ 0 modp. Put a_r = H_rr(0). Then H_rr(u)/a_r ≡ 1 modp for each u ∈ (pZp)ⁿ. There is a well-defined square root function √ : 1 +pZp → 1 +pZp which is (globally) analytic. Put

g(u) =

»

Hrr(u)/ar

foru∈(pZ^p)ⁿ. Then,g is (globally) analytic on(pZ^p)ⁿ. We can now define a (globally) analytic mapT_r−1⁰ : (pZp)ⁿ→(pZp)ⁿ:u7→T_r−1⁰ (u) = (vi)by

v_i =u_i if i6=r and v_r=g(u)

u_r+X

i>r

u_iH_ir(u) Hrr(u)

.

o

Clearly we have thatdet(∂vi/∂uj 0) =g(0)6≡0 modp, thus, the map T_r−1⁰ is (globally) bi-analytic by Lemma 3.2. Put Tr =T_r−1⁰ ◦Tr−1. We then obtain for allx∈(pZp)ⁿ andv=T_r(x)

f(x) =

r

X

i=1

aiv²_i + X

i,j>r

vivjH_ij⁰ (v), withH_ij⁰ (v)∈Zp[[v]]. This finishes the induction argument.

The equalityχ¹

2(det(Hs(f)₀) =χ¹

2(Q

i2ai)follows by a classical argument as in [10]. To finish the proof we only have to prove the Lemma below.

Lemma 3.4. — Let p6= 2 and letf : (pZp)ⁿ→pZp be a globally analytic map (i.e., given by a power series inZp[[x]])such that0 is a non-degenerate critical point modulo p. Then f has a unique critical point c ∈(pZp)ⁿ and this is a non-degenerate critical point.

Proof. — We can write

f(x) =f(0) +X

i

aixi+g(x) withg(x) =P

ijxixjhij(x)for somehij ∈Zp[[x]]. Since0 is a non-degenerate critical point of f modulop, we see thatai ≡0 modp and det(Hs(g)₀) 6≡0 modp. By Lemma3.2the map

T : (pZp)ⁿ−→(pZp)ⁿ, x7−→grad(g)_x=∂g

∂x1

(x), . . . , ∂g

∂xn

(x) is a bi-analytic bijection. A fortiori, there is a unique pointc∈(pZp)ⁿsuch that T(c) = (−a1, . . . ,−an). Sincegrad(f)_x= grad(g)_x+ (a1, . . . , an), the condi- tionT(x) = (−a₁, . . . ,−a_n)is equivalent with the conditiongrad(f)_x= 0 for x∈(pZp)ⁿ. Thuscis the unique critical point off. Moreover,det(Hs(f)_c)6= 0 sincedet(Hs(f)_c)≡det(Hs(f)₀)6≡0 modp.

Finally, we give an application of Morse’s lemma to calculate character sums of polynomials.

Proposition 3.5. — Let p6= 2, let f ∈Zp[x1, . . . , xn] be a polynomial, and let X be a smooth subvariety of Aⁿ_Zp (smooth variety meaning a separated, reduced, irreducible scheme of finite type and smooth over Zp). Let M be the compact p-adic varietyX(Zp)\f⁻¹(pZp)and write dfor the dimension ofM. Let χ be a primitive multiplicative character mod p^m, m > 1. Let c1, . . . , c`

inM be the critical points off _M :M →Zp and suppose that the critical points c1, . . . , c` are non-degenerate modulo p. Put

S_f = X

x∈X(Zp/p^mZp)

χ f(x) .

Then the following holds Sf =

`

X

i=1

χ f(ci) χ¹

2 f(ci)^dHci

m

α(χ, m)e ^d,

with Hc_i = 2^−d∆(Hs(f)_|ci), α(χ, m)e as in Section 2, and ∆ the discriminant as defined in 1.2.

Proof. — That M is a compact p-adic variety follows from Hensel’s lemma, by using Taylor expansions for the equations defining X around points inZⁿp

with different residues modp. Similarly, there is a cover ofM by finitely many disjoint compact opens U such that for each U there is an analytic isometry π:U →(pZp)^d. For i = 1, . . . , r, let Ui be the open in this cover contain- ingc_iand we writeπ_i for the corresponding isometry. By Proposition2.1and Lemma 3.4it follows thatSf =P`

i=1Sf(ci)where S_f(c_i) = X

x∈(pZp/p^mZp)^d

χ f◦π_i⁻¹(x) .

By Morse’s lemma we can find for each critical pointc_i a bi-analytic isometric transformationTi: (pZp)^d→(pZp)^d such that

f(x) =f(ci) +

d

X

j=1

aju²_j for all x∈Ui andu=Ti(πi(x)),

witha_j inZ^×p andχ1

2(H_ci) =χ1 2(Q

ja_j). We can now calculate

Sf(ci) = X

u∈(pZ/p^mZ)^d

χ f(ci) +

d

X

j=1

aju²_j (3.1)

= X

u∈(Z/p^mZ)^d

χ f(ci) +

d

X

j=1

aju²_j (3.2)

=χ f(ci) χ1

2

f(ci)^dY

j

aj

^m

α(χ, m)e ^d (3.3)

=χ f(ci) χ1

2 f(ci)^dHc_i^m

α(χ, m)e ^d.

Equality (3.1) is clear. Also equality (3.2) is easy and follows by similar argu- ments as in the proof of Proposition 2.1. Equality (3.3) comes from Proposi- tion2.4and the last equality holds becauseχ1

2(Hc_i) =χ1 2(Q

jaj).

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4. Discrete Fourier transforms of characters of homogeneous polynomials Let P be a prime ideal of OK and let π_P ∈ P be of P-adic order 1.

Let L(x) = Pn

i=1aixi be a linear form on (OK/P^m)ⁿ, with m > 1 and a_i∈ O_K/P^m. Let f ∈ O_K[x] be a homogeneous polynomial of degree d and letχ, resp.ψ, be a primitive multiplicative, resp. primitive additive, character modP^m.

We will calculate the discrete Fourier transform of χ(f), defined by

S(L) := X

x∈(OK/P^m)ⁿ

χ f(x)

ψ L(x) .

After a linear change of variables one can assume that theP-valuation ofa1

is minimal among thev_P(a_i). Write

k:=v_P(a1) and L(x) =π^k_P(a⁰₁x1+a⁰₂x2+· · ·+a⁰_nxn) for some a⁰_i∈ OK/P^m. After applying the invertible linear transformation

(x1, . . . , xn)7−→(a⁰₁x1+· · ·+a⁰_nxn, x2, . . . , xn),

one reduces to the case thatL(x) =π^k_Px1. This reduction is used in the proof of the following result.

Proposition 4.1. — Let P be a prime ideal of OK and let π_P ∈ P be of P-adic order 1. Let L(x) = Pn

i=1aixi be a linear form on (OK/P^m)ⁿ, with m > 1 and ai ∈ OK/P^m. Let f ∈ OK[x] be a homogeneous polynomial of degree d and let χ, (resp. ψ), be a primitive multiplicative, (resp. primitive additive, character modP^m).Put k:= mini=1,...,nv_P(ai)and

S(L) = X

x∈(O_K/P^m)ⁿ

χ f(x)

ψ L(x) .

If P -d, thenS(L) = 0 ifk6= 0 and S(L) = X

y∈OK/P^m

χ^d(y)ψ(y) X

x∈(OK/P^m)ⁿ L(x)≡1 modP^m

χ f(x)

ifk= 0.

Proof. — As explained in the previous discussion, we may assume that L(x) =π_P^kx1, with π_P ∈ P ofP-adic order1. We split up the sum depending on the P-valuation ofx1.Let us denote by Ajk the subsum of S(L)over the elementsxwithv_P(x₁) =j. So,S(L) =Pm

j=0A_jk,with

Ajk:= X

y∈(OK/P^m−j)^× x1=yπ^j_P

b^x∈(OK/P

m)ⁿ⁻¹

χ f(yπ^j_P,bx)

ψ(π_P^j+ky),

where bx:= (x2, . . . , xn).Rewriteq^jAjkas X

y∈(OK/P^m)^×

bx∈(O_K/P^m)ⁿ⁻¹

χ f(yπ^j_P,bx)

ψ(π_P^j+ky) (4.1)

= X

y∈(O_K/P^m)^×

bx∈(OK/P^m)ⁿ⁻¹

χ f(yπ_P^j, yx)ˆ

ψ(π_P^j+ky) (4.2)

= Ñ

X

y∈OK/P^m

χ^d(y)ψ(π^j+k_P y) é Ñ

X b^x∈(OK/Pm)

n−1

χ f(π^j_P,bx) é

. (4.3)

Equality (4.1) holds because only the value ofymodP^m−j is relevant. Equal- ity (4.2) is just substitutingxb= (x₂, . . . , x_n)byyxb= (yx₂, . . . , yx_n); sincey is a unit the set over which we sum does not change. The last equality uses thatf is homogeneous of degreedand the fact thatχ^d(y) = 0ify6∈(OK/P^m)^×.

We want to prove that allAjk are zero except when k=j= 0. SinceP-d, we have that χ^d is still a primitive character modP^m. Therefore, there exists an a ∈ OK/P^m such that a ≡ 1 modP^m−1 and χ^d(a) 6= 1 (see Section 2).

By a classical argument, we obtain X

y∈OK/P^m

χ^d(y)ψ(π_P^j+ky) = X

y∈OK/P^m

χ^d(ay)ψ(aπ_P^j+ky)

= χ^d(a) X

y∈OK/P^m

χ^d(y)ψ(π_P^j+ky),

if j+k ≥1. Indeed, the first equation is just substituting y by ay, where a is a unit. The second uses the fact that χ^d is multiplicative and since a≡1 modP^m−1,we have ay≡y modP^m−k−j ifj+k≥1.

Sinceχ^d(a)6= 1,we conclude that X

y∈O_K/P^m

χ^d(y)ψ(π_P^j+ky) = 0.

Note that when j+k≥mthis sum is just P

y∈OK/P^mχ^d(y)which is directly seen to be zero. This proves the proposition since onlyA00is non-zero by equal- ity (4.3). This argument can also be found in [24], Chapter VII, Prop. 13.

5. Applications to prehomogeneous vector spaces We use the notation and the asumptions from the introduction.

o

Proof of Theorem 1.1. — For any v^∨ ∈V^∨ letH(v^∨)be the hyperplane in V defined by v^∨(x)−1. By [5, Lemma 9.1.2] and [5, Lemma 9.1.7], for any v^∨ with f^∨(v^∨) 6= 0, the restriction f_|H(v∨)∩Ω has the point d⁻¹F^∨(v^∨) as its unique critical point and it is a non-degenerate critical point. The fact that this holds for allv^∨ withf^∨(v^∨)6= 0is a first order statement (in the language of rings), and hence, the analogue statement is true over finite fields of big enough characteristic. That is, for all prime idealsP ofOK above big enough primes p ∈ Z with residue field kP, for all v^∨ ∈ V^∨(kP) with f^∨(v^∨) 6= 0, the point d⁻¹F^∨(v^∨) is the unique critical point of f_|H(v∨)∩Ω(kP) and it is a non-degenerate critical point.

Take a prime idealP above a big enough primep. ChooseL∈V^∨(OK/P^m) with f^∨(L)6≡0 modP. Let L₀ ∈V^∨(O_K)lie above L. Let c be d⁻¹F^∨(L₀).

LetRbe the valuation ring of theP-adic completion ofK. SinceP is supposed to lie above a big enough prime p ∈ Z, we may suppose that d is invertible in R, and hence,c lies inV(R). Make the planeH(L₀)into a vector space by choosing a zero point in H(L0)(OK). Take a basis of the lattice H(L0)(OK) in H(L0) and take coordinates on H(L0) with respect to this basis. Let Hc

be 2⁻ⁿ⁺¹ times the determinant of the Hessian in c of f_|H(L0), expressed in these coordinates. Since p ∈ Z is big enough and by the above discussion, Hc lies in R and HcmodP is nonzero in OK/P, and this is so uniformly in L∈V^∨(O_K/P^m).

Then,

S(L) = X

y∈O_K/P^m

χ^d(y)ψ(y) X

x∈V(O_K/P^m) L(x)≡1 modP^m

χ f(x) (5.1)

= X

y∈OK/P^m

χ^d(y)ψ(y) χ f(c)

χ1

2 f(c)ⁿ⁻¹Hc^m

α(χ, m)e ⁿ⁻¹ (5.2)

=q^12mnP

O_K/P^mχ^d(y)ψ(y) q^12m

χb₀f^∨(L)⁻¹ d^d

κ^∨(L)α(χ, m)e ⁿ⁻¹ q^12(n−1)m (5.3) ,

with κ^∨(L) = χ1

2(f(c)ⁿ⁻¹Hc)^m. Equality (5.1) follows from Proposi- tion 4.1. We obtain (5.2) from Proposition 3.5. If we now use the fact that f(c) = d^−df(F^∨(L)) = d^−db0f^∨(L)⁻¹ (see [5, Lemma 9.1.2] and [6, Lemma 1.8]), equation (5.3) and thus case 1) of the theorem follows with κ^∨(L) =χ1

2(f(c)ⁿ⁻¹Hc)^mandα(χ, m) =α(χ, m)e ⁿ⁻¹/q^12(n−1)m.

Now we prove case 2) of Theorem1.1with a technique of N. Kawanaka. The functionS: Hom((OK/P^m)ⁿ,OK/P^m)→C,L7→S(L)is the discrete Fourier transform of χ(f) : (O_K/P^m)ⁿ → C, x7→ χ(f(x)). By a classical result on

L2-norms of Fourier transforms on finite abelian groups, it follows that

(5.4) kSk²₂=q^mn

χ(f)

2

2=q^mnN1,

withN1:= #AandA:={x∈V(OK/P^m)|f(x)6≡0 modP}.

It follows from the formula in case 1), that|S(L)|²=q^mnforL∈A. Writing B :=V(O_K/P^m)\A, one has

(5.5) kSk²₂=X

A

q^mn+X

B

S(L)

2,

and hence,S(L) = 0forL∈B.

Proof of Theorem 1.3. — The statement about α(χ, m) follows from equa- tion (5.3) and Lemma2.3. We recall thath^∨(L)is defined in the introduction, and that the numberχ1

2(h^∨(L))is well-defined. We obtain the value ofκ^∨(L) as an immediate corollary of [5, Lemma 9.1.7].

Appendix A

L-functions of prehomogeneous vector spaces (by Fumihiro Sato)

In this note, we give an application of Theorems 1.1 and 1.3 in [3] (and Theorems A, B, C in [5]) to the functional equation ofL-functions of Dirichlet type associated with prehomogeneous vector spaces, which is a generalization of Theorem L in [16].

In the following we retain the notation in [3]. However, for simplicity, we assume that K = Q, OK =Z and P = (p) with a rational prime p. Let χ be a primitive Dirichlet character with conductor N > 1. We extend χ to Z/NZby χ(a) = 0 for a6∈ (Z/NZ)^×. Putm(p) =vp(N), thep-order of N, for any rational prime p. Since(Z/NZ)^× is isomorphic toQ

p|N(Z/p^m(p)Z)^×, χ induces a primitive characterχ^(p): (Z/p^m(p)Z)^×→C^× for each p|N.

Let (G, ρ, V) be a reductive prehomogeneous vector space defined over Q. Let P₁, . . . , P<sub>`</sub> be the fundamental relative invariants over Q, namely, the Q- irreducible relatively invariant polynomials onV. We denote byφi(1≤i≤`) the rational character of G corresponding to Pi. The fundamental relative invariants are determined uniquely up to a non-zero constant multiple in Q^× and any relative invariant inQ[V]is a monomial of them.

We fix a basis of the Q-vector space V(Q) and take a relative invariant f ∈ Q[V] with coefficients in Z (with respect to the fixed Q-basis of V(Q)).

The characterφcorresponding tof is defined over Q. In the following we assume that

o

A.1) fis aregularrelative invariant, namely,Ω =V\f⁻¹(0)is a singleG-orbit;

A.2) for every x ∈ Ω(Q), the group of Q-rational characters of the identity component ofG_x=

g∈G|ρ(g)x=x is trivial.

We denote byG⁺the identity component of the real Lie groupG(R)and put G⁺_x =G⁺∩Gx. LetΩ(R) = Ω1∪ · · · ∪Ων be the decomposition into the con- nected components (in the usual topology). By the assumption A.1), everyΩi

is a singleG⁺-orbit. Let Γ_N be an arithmetic subgroup ofG(Q)which stabi- lizes V(Z)and induces the identity mapping onV(Z)/N V(Z). Then the func- tionV(Z)3x7→χ(f(x))∈Cis ΓN-invariant and factors through V(Z/NZ).

The L-functions L_i(s;χ) (1≤i ≤ν)associated with (G, ρ, V)and χ are de- fined by

L_i(s;χ) = X

x∈Γ_N\(V(Z)∩Ω_i)

µ(x)χ f(x)

`

Y

j=1

P_j(x)

−sj

, s= (s₁, . . . , s<sub>`</sub>)∈C<sup>`</sup>,

where µ(x)is the volume of the fundamental domainG⁺_x/(ΓN ∩G⁺_x)with re- spect to the normalized Haar measure onG⁺_x (for the normalization of the Haar measure on G⁺_x, see [15], §4). By the assumptions A.1), A.2) and [13, Theo- rem 1.1], theL-functions converge absolutely when the real parts of s1, . . . , s`

are sufficiently large.

We take a relative invariant f^∨ of the dual prehomogeneous vector space (G, ρ^∨, V^∨) with coefficients in Z (with respect to the basis of V^∨ dual to the fixed basis of V) that corresponds to the character φ⁻¹. Then f^∨ and (G, ρ^∨, V^∨)satisfy the assumptions A.1) and A.2). Put Ω^∨ =V^∨\f^∨−1(0).

We may order the fundamental relative invariants P₁^∨, . . . , P_`^∨ of (G, ρ^∨, V^∨) such that the character corresponding to P_i^∨ isφ⁻¹_i .

Our final assumption is the following:

A.3) for every prime factor p of N with m(p) > 1 (resp. m(p) = 1), The- orems 1.1 and 1.3 in [3] (resp. Theorems A, B, C in [5]) hold for χ^(p) andf.

Let ψ : Z/NZ → C^× be an additive primitive character. By the Chi- nese remainder theorem, ψ determines an additive primitive character ψ^(p):Z/p^m(p)Z→C^× for each p | N. For L ∈ V^∨(Z), let us consider the character sum

S(χ, f;L) = X

x∈V(Z/NZ)

χ f(x)

ψ L(x)

=Y

p|N

X

x∈V(Z/p^m(p)Z)

χ^(p) f(x)

ψ^(p) L(x) .

Then, by assumptions A.1) and A.3), we have (A.4) S(χ, f;L) =χ⁻¹(f^∨(L))Y

p|N

κ^∨_p(L)g(χ^(p), f),

with LmodN ∈ Ω^∨(Z/NZ) and where κ^∨_p(L) is the constant κ^∨(L) = ±1 given for each (sufficiently large)pby Theorem1.3in [3] or Theorems B and C in [5], and g(χ^(p), f) is a constant independent ofL whose explicit value can be easily seen from Theorems1.1and1.3in [3] or Theorem A in [5] according as m(p)>1or m(p) = 1. Put

κ^∨(L) =Y

p|N

κ^∨_p(L) and g(χ, f) =Y

p|N

g(χ^(p), f).

Now we define theL-functions associated with(G, ρ^∨, V^∨)by L^∨_i(s;χ⁻¹) = X

L∈ΓN\(V^∨(Z)∩Ω^∨_i)

µ^∨(L)κ^∨(L)χ⁻¹ f^∨(L)

`

Y

j=1

P_j^∨(L)

−sj

.

Assumptions A.1), A.2) and [13, Thm. 1.1] again imply that theseL-functions converge absolutely when the real parts ofs₁, . . . , s_` are sufficiently large. The abscissa of absolute convergence is independent ofχ andN.

To describe analytic properties of theL-functions, we need some more nota- tional preliminaries. Letb(s) =b(s1, . . . , s`)be the Bernstein-Sato polynomial defined by

(A.5) Y^`

i=1

P_i^∨(grad_x)Y^`

i=1

Pi(x)^si =b(s)

`

Y

i=1

Pi(x)^si−1.

It is known that the functionb(s)is a product of inhomogeneous linear forms s1, . . . , s` of integral coefficients (see [18]). We also need the Bernstein-Sato polynomial b_f(s)off, which is defined by

f^∨(grad_x)f(x)^s+1=b_f(s)f(x)^s.

It is known that the roots ofb_f(s)are negative rational numbers.

Each of the assumptions A.1) and A.2) implies that there exists a δ = (δ1, . . . , δ`)∈(<sup>1</sup><sub>2</sub>Z)<sup>`</sup> such that the relative invariantQ`

i=1Pi(x)^2δi corresponds to the character detρ(g)² (see [19, Prop. 8] or [18, Prop. 11] for A.1) and [15, Lemma 4.1] for A.2)).

Finally we recall the fundamental theorem of the theory of prehomogeneous vector spaces over the real number field R. For i = 1, . . . , ν and s with Re(s₁), . . . ,Re(s_`)>0 , we define a continuous function|P(x)|^s_Ω

i onV(R)by P(x)

s

Ω_i=n Q`

j=1|P_j(x)|^sj (x∈Ω_i),

0 (x6∈Ωi).

The function|P(x)|^s_Ωi depends holomorphically onsand is extended to a tem- pered distribution onV(R)depending meromorphically ons∈C^`. We denote

o

the tempered distribution by the same symbol. We can also define the tem- pered distributions|P^∨(L)|^s_Ω∨

i onV^∨(R)quite similarly. Then the fundamental theorem reads

(A.6) Z

V(R)

P(x)

s−δ

Ω_i exp 2πiL(x) dx=

ν

X

j=1

γ_ij(s) P^∨(L)

−s Ω^∨_j,

where γ_ij(s) (i, j= 1, . . . , ν)have elementary (but not explicit) expressions in terms of the gamma function and the exponential function (see [17], [15]).

Theorem A.1. — 1) The L-functions Li(s;χ) and L^∨_i(s;χ⁻¹) multiplied by b(s−δ) have analytic continuations to holomorphic functions of s inC^` and satisfy the following functional equation

g(χ, f)·L^∨_j(δ−s;χ⁻¹) =N^{d1s1+···+d`s`}

ν

X

i=1

γ_ij(s)L_i(s;χ),

where d_i (1 ≤ i ≤ `) is the degree of the fundamental relative invariant P_i andγij(s) is the same as above.

2) Assume thatχ satisfies at least one of the following conditions:

• m(p)≥2 for somepdividing the conductorN of χ;

• the order of χ^(p) for some p | N with m(p) = 1 is different from the reduced denominators of the roots of the Bernstein-Sato polynomialbf(s).

Then the L-functions Li(s;χ)andL^∨_i(s;χ⁻¹)are entire functions ofsin C^`. Since the proof of the theorem is almost the same as the one of Theorem 2 and Corollary 1 of [15], we shall give only an outline of the proof.

Denote by A the ring of adeles of Q and by A0 = Q0

p<∞Qp the ring of finite adeles ofQ. Denote byΦ_p(x_p)the characteristic function of V(Zp) and put Φ0(x0) = Q

p<∞Φp(xp)for x0 = (xp)∈ A0. The function Φ0(x0) is the characteristic function of Q

p<∞V(Zp). LetΦ∞(x∞)be a rapidly decreasing C^∞-function onV(R)and define a function Φ_χ onV(A)by

Φχ(x) = Φ_∞(x_∞)Y

p|N

χ^(p) f(xp)

Φ0(x0) x= (x_∞, x0)∈V(A) .

The functionΦχis a Schwartz-Bruhat function onV(A)and the Poisson sum- mation formula implies the identity

(A.7) X

x∈V(Q)

Φχ ρ(g)x

=

detρ(g)

−1 A

X

L∈V^∨(Q)

Φbχ ρ^∨(g)L

g∈G(A) ,

whereΦbχ is the Fourier transform ofΦχ, which is defined by an additive char- acter of A/Qof conductor 1, more specifically, the additive character whose

p-component is of conductor 1 and coincides withψ^(p)(N x)onp^−m(p)Zp ifp divides N. It is easy to see that

(A.8) Φbχ(L) =N⁻ⁿS(χ, f;N L0)Φ0(N L0)bΦ_∞(L_∞),

with L= (L_∞, L0)∈V^∨(A) andn= dimV. We note here that the function S(χ, f;L)originally defined onV^∨(Z)can naturally be extended to a function onQ

p<∞V^∨(Zp). By the usual technique of unfolding, we have Z

G⁺/Γ_N

`

Y

j=1

φ_j(g_∞)

sj X

x∈Ω(Q)

Φ_χ ρ(g_∞,1)x dg_∞ (A.9)

=

ν

X

i=1

Li(s;χ)

|P(x)|^s−δ_Ω

i ,Φ∞ .

By the identities (A.4) and (A.8), we also have Z

G⁺/ΓN

`

Y

j=1

φj(g∞)

−sj X

L∈Ω^∨(Q)

Φbχ ρ^∨(g∞,1)L dg∞

=N^{Σ`j=1djsj−n}g(χ, f)

ν

X

i=1

L^∨_i(s;χ⁻¹)

|P^∨(L)|^s−δ_Ω∨ i

,Φb∞ . Now Theorem A.1 can be proved in the same manner as in [15, §6] by using these integral representations of the L-functions, the Poisson summation for- mula (A.7), and the fundamental theorem (A.6) overR. We note here that the Poisson summation formula (A.7) is used in the form

X

x∈Ω(Q)

Φ_χ ρ(g)x

=

detρ(g)

−1 A

X

L∈Ω^∨(Q)

Φb_χ ρ^∨(g)L

+I(Φ_χ, g),

where

I(Φχ, g) =

detρ(g)

−1 A

X

L6∈Ω^∨(Q)

Φbχ ρ^∨(g)L

−X

x6∈Ω(Q)

Φχ ρ(g)x .

The contribution ofI(Φχ, g)to the integral representation contains the infor- mation on the poles of theL-functions and is in general very hard to calculate explicitly. However, if we take a test function of the form

Φ_∞(x_∞) =Y^`

`=1

P_`^∨(grad_x)

Φ⁰_∞(x_∞)

for aC^∞-functionΦ⁰_∞ with compact support inΩ(R), thenI(Φχ, g)vanishes and, by using (A.5), we can prove the first assertion of Theorem A.1. If the assumption of the second assertion of Theorem A.1 is fulfilled, then Theo- rem1.1, 2) in [3] and Remark 5.2.3.3 in [5] imply the vanishing ofI(Φχ,(g_∞,1)) for g∞ ∈ G⁺ and arbitrary Φ∞ with compact support inΩ(R). The second assertion of TheoremA.1follows immediately from this observation.

o

Remark A.2. — The simplest example of the L-functions considered in this note is the Dirichlet L-function, which is the L-function associated with (GL₁, ρ, V(1)), whereρ is the standard 1-dimensional representation of GL₁. Further examples of the L-functions associated with prehomogeneous vector spaces have been studied in [21], [4], [11], [12], [22] and [23]. It is noteworthy that, unlike the case of the DirichletL-functions, theL-functionsL_i(s;χ)may have poles even for non-trivialχ, ifχdoes not satisfy any one of the conditions in Theorem A.1, 2) (for concrete examples, see [4, Thm. 6.2], [11, Thm. 4.2]

and [12, Thm. 5]). It is an interesting problem to determine the conditions onχunder which Li(s;χ)has actually poles.

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