On the Poincar´e series for diagonal forms over algebraic number ﬁelds

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LXIII.2 (1993)

On the Poincar´ e series for diagonal forms over algebraic number fields

by

Jun Wang (Dalian)

1. Introduction. Let p be a fixed prime and f (x 1 , . . . , x s ) a polynomial with coefficients in Z p , the p-adic integers. Let c n denote the number of solutions of f = 0 over the ring Z/p ⁿ Z, with c 0 = 1. Then the Poincar´ e series P f (t) is the generating function

P f (t) =

∞

X

n=0

c n t ⁿ .

This series was introduced by Borevich and Shafarevich [1, p. 47], who conjectured that P f (t) is a rational function of t for all polynomials. This was proved by Igusa in 1975 in a more general setting, by using a mixture of analytic and algebraic methods [5, 6]. Since the proof is nonconstructive, deriving explicit formulas for P f (t) is an interesting problem. In this di- rection Goldman [2, 3] treated strongly nondegenerate forms and algebraic curves all of whose singularities are “locally” of the form αx ^a = βy ^b , while polynomials of form P x ^d _i

ⁱ

with p - d i were investigated earlier by E. Steven- son [7], using Jacobi sums. In [8] explicit formulas for P f (t) were derived for diagonal forms. This paper generalizes the results of [8] to algebraic number fields.

Let F be a finite extension of the rational field, and P a prime ideal of F with norm N (P ) = q which is a rational prime power. Using the previous notations, we let c n denote the number of solutions of the congruence (1) a 1 x ^d ₁

¹

+ . . . + a s x ^d _s

^s

≡ 0 (mod P ⁿ ) ,

where d 1 , . . . , d s are positive integers, a 1 , . . . , a s are integers of F prime to P , and write P (t) = P ∞

n=0 c n t ⁿ .

It is clear that c n = q ^n(s−1) if d i = 1, for some i, 1 ≤ i ≤ s. Therefore we assume that d 1 , . . . , d s are all integers greater than 1.

Research supported by the National Natural Science Foundation of China.

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Throughout this paper, we set d = lcm{d 1 , . . . , d s }, f _i = d/d i , r = f 1 + . . . + f s and c n = q ^−n(s−1) c n .

2. Exponential sums. For the prime ideal P of F , choose an ideal C of F such that (P, C) = 1 and P C = (θ) is principal. Then we may assume that any integer u in F is of the form

u = θ ^j ξ (j ≥ 0, (ξ, P ) = 1) .

In this case we write ord P u = j. Let D represent the different of F (see [4, Ch. 36]), and choose B, (B, P ) = 1 such that (ζ) = B/P ⁿ D is principal.

We set ζ m = ζθ ^n−m , 0 ≤ m ≤ n, such that ζ = ζ n , and define further e m (u) = e ^{2πi tr(uζ}

^m

⁾ ,

where the symbol tr(γ) denotes the trace in F . The function e m (u) defines an additive character (mod P ^m ) and has the following simple properties:

e 0 (u) = 1, e m (u) = e m (u ⁰ ) if u ≡ u ⁰ (mod P ^m ) , (2)

e m (uθ ^j ) = e m−j (u) (0 ≤ j ≤ m) , (3)

X

z (mod P

^m

)

e m (uz) = n q ^m if u ≡ 0 (mod P ^m ), 0 otherwise.

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For k ≥ 1, we define

S m (u, k) = X

z (mod P

^m

)

e m (uz ^k ) , S 0 (u, k) = 1 . It is clear that if m ≥ j ≥ 0, then

(5) S m (uθ ^j , k) = q ^j S m−j (u, k) .

The following lemmas are useful in the proof of the main theorem.

Lemma 1. For any positive integer k , there is an integer a ≥ k such that whenever m ≥ a, then

(6) S m (u, k) = q ^k−1 S m−k (u, k), (u, P ) = 1 .

P r o o f. Suppose ord P k = l. Then take a to be a positive integer which is greater than k and all of i(l + 1)/(i − 1), i = 2, . . . , k. Thus, when m ≥ a we have

(7) i(m − l − 1) ≥ m , i = 2, . . . , k . From this it follows that m ≥ l + 1 and

{z (mod P ^m )} = {y + xθ ^m−l−1 | y (mod P ^m−l−1 ), x (mod P ^l+1 )} . Using the binomial theorem and (7) we have

(y + xθ ^m−l−1 ) ^k ≡ y ^k + ky ^k−1 xθ ^m−l−1 (mod P ^m ) ,

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and

S m (u, k) = X

y (mod P

^m−l−1

)

e m (uy ^k ) X

x (mod P

^l+1

)

e l+1 (uky ^k−1 x) .

Since ord P k = l, by (4), the inner sum is 0 unless y ≡ 0 (mod P ), in which case it has the value q ^l+1 . Hence we have, by setting y = y 1 θ, y 1 (mod P ^m−l−2 ),

S m (u, k) = q ^l+1 X

y

1

(mod P

^m−l−2

)

e m−k (uy ^k ₁ ) = q ^k−1 S m−k (u, k) .

Let a(k) be the least positive integer such that (6) holds when m ≥ a(k), and write

(8) % = max{a(d 1 ), . . . , a(d s )} . Lemma 2. Put T m = q ^−ms P

(v,P

^m

)=1 S m (va 1 , d 1 ) . . . S m (va s , d s ). Then T d+j = q ^d−r T j for j ≥ % − 1.

P r o o f. Since j ≥ % − 1 and d i ≥ 2, we have d i + j ≥ a(d i ). By Lemma 1 one gets

S d+j (u, d i ) = q ^f

ⁱ

^(d

ⁱ

⁻¹⁾ S j (u, d i ) , i = 1, 2, . . . , s . Therefore,

T d+j = q ^−(d+j)s X

(v,P

^d+j

)=1

S d+j (va 1 , d 1 ) . . . S d+j (va s , d s )

= q ^−(d+j)s X

(v,P

^d+j

)=1 s

Y

i=1

q ^f

ⁱ

^(d

ⁱ

⁻¹⁾ S j (va i , d i ) = q ^d−r T j .

3. Main results

Theorem. Let % be as in (8). We have (i) recursion: for n ≥ %,

c n+d = c + q ^d−r c n , (ii) the Poincar´ e series is given by

P (t) = (1 − q ^s−1 t)( P %+d−1

i=0 c i t ⁱ − q ^ds−r P %−1

i=0 c ⁱ t ^d+i ) + cq ^(%+d)(s−1) t ^%+d

(1 − q ^s−1 t)(1 − q ^ds−r t ^d ) ,

where c = c %+d−1 − q ^d−r c %−1 is a constant depending only upon the diagonal

form as in (1).

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P r o o f. (i) From (4) we have c n = q ⁻ⁿ X

x

1

,...,x

s

(mod P

ⁿ

)

X

u (mod P

ⁿ

)

e n (u(a 1 x ^d ₁

¹

+ . . . + a s x ^d _s

^s

))

= q ⁻ⁿ X

u (mod P

ⁿ

)

S n (ua 1 , d 1 ) . . . S n (ua s , d s ) .

In the summation over u (mod P ⁿ ), we may set u = vθ ^n−m , 0 ≤ m ≤ n, v (mod P ^m ) and (v, P ^m ) = 1. From (5) one has

c n = q ^n(s−1)

n

X

m=0

q ^−ms X

(v,P

^m

)=1

S m (va 1 , d 1 ) . . . S m (va s , d s )

= q ^n(s−1)

n

X

m=0

T m .

Set n = % + l, l ≥ 0. By Lemma 2, we have c n+d =

n+d

X

m=0

T m =

%+d−1

X

m=0

T m +

l

X

m=0

T %+d+m = c %+d−1 +

l

X

m=0

q ^d−r T %+m

= c %+d−1 + q ^d−r (c n − c %−1 ) = c + q ^d−r c n . (ii) Put q ^s−1 t = t 1 . Then

P (t) =

∞

X

n=0

c n t ⁿ =

%+d−1

X

i=0

c i t ⁱ +

∞

X

n=%

c n+d t ^n+d

=

%+d−1

X

i=0

c i t ⁱ +

∞

X

n=%

c n+d t ^n+d ₁ =

%+d−1

X

i=0

c i t ⁱ +

∞

X

n=%

(c + q ^d−r c n )t ^n+d ₁

=

%+d−1

X

i=0

c i t ⁱ + ct ^%+d ₁ (1 − t 1 ) ⁻¹ + q ^d−r t ^d ₁ P (t) −

%−1

X

i=0

c i t ⁱ . This gives the result of the theorem.

References

[1] Z. I. B o r e v i c h and I. R. S h a f a r e v i c h, Number Theory , Academic Press, New York 1966.

[2] J. R. G o l d m a n, Numbers of solutions of congruences: Poincar´ e series for strongly nondegenerate forms, Proc. Amer. Math. Soc. 87 (1983), 586–590.

[3] —, Numbers of solutions of congruences: Poincar´ e series for algebraic curves, Adv.

in Math. 62 (1986), 68–83.

[4] E. H e c k e, Lectures on the Theory of Algebraic Numbers, Springer, New York 1981.

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[5] J. I g u s a, Complex powers and asymptotic expansions, II , J. Reine Angew. Math.

278/279 (1979), 307–321.

[6] —, Some observations on higher degree characters, Amer. J. Math. 99 (1977), 393- 417.

[7] E. S t e v e n s o n, The rationality of the Poincar´ e series of a diagonal form, Thesis, Princeton University, 1978.

[8] J. W a n g, On the Poincar´ e series for diagonal forms, Proc. Amer. Math. Soc., to appear.

INSTITUTE OF MATHEMATICAL SCIENCES DALIAN UNIVERSITY OF TECHNOLOGY DALIAN 116024, PEOPLE’S REPUBLIC OF CHINA

Received on 4.7.1991 (2157)

On the Poincar´e series for diagonal forms over algebraic number ﬁelds

LXIII.2 (1993)

On the Poincar´ e series for diagonal forms over algebraic number fields

by

Jun Wang (Dalian)

1. Introduction. Let p be a fixed prime and f (x 1 , . . . , x s ) a polynomial with coefficients in Z p , the p-adic integers. Let c n denote the number of solutions of f = 0 over the ring Z/p n Z, with c 0 = 1. Then the Poincar´ e series P f (t) is the generating function

P f (t) =

∞

X

n=0

c n t n .

with p - d i were investigated earlier by E. Steven- son [7], using Jacobi sums. In [8] explicit formulas for P f (t) were derived for diagonal forms. This paper generalizes the results of [8] to algebraic number fields.

Let F be a finite extension of the rational field, and P a prime ideal of F with norm N (P ) = q which is a rational prime power. Using the previous notations, we let c n denote the number of solutions of the congruence (1) a 1 x d 1

+ . . . + a s x d s

≡ 0 (mod P n ) ,

where d 1 , . . . , d s are positive integers, a 1 , . . . , a s are integers of F prime to P , and write P (t) = P ∞

n=0 c n t n .

It is clear that c n = q n(s−1) if d i = 1, for some i, 1 ≤ i ≤ s. Therefore we assume that d 1 , . . . , d s are all integers greater than 1.

Research supported by the National Natural Science Foundation of China.

Throughout this paper, we set d = lcm{d 1 , . . . , d s }, f i = d/d i , r = f 1 + . . . + f s and c n = q −n(s−1) c n .

2. Exponential sums. For the prime ideal P of F , choose an ideal C of F such that (P, C) = 1 and P C = (θ) is principal. Then we may assume that any integer u in F is of the form

u = θ j ξ (j ≥ 0, (ξ, P ) = 1) .

In this case we write ord P u = j. Let D represent the different of F (see [4, Ch. 36]), and choose B, (B, P ) = 1 such that (ζ) = B/P n D is principal.

We set ζ m = ζθ n−m , 0 ≤ m ≤ n, such that ζ = ζ n , and define further e m (u) = e 2πi tr(uζ

) ,

where the symbol tr(γ) denotes the trace in F . The function e m (u) defines an additive character (mod P m ) and has the following simple properties:

e 0 (u) = 1, e m (u) = e m (u 0 ) if u ≡ u 0 (mod P m ) , (2)

e m (uθ j ) = e m−j (u) (0 ≤ j ≤ m) , (3)

X

z (mod P

)

e m (uz) = n q m if u ≡ 0 (mod P m ), 0 otherwise.

(4)

For k ≥ 1, we define

S m (u, k) = X

z (mod P

)

e m (uz k ) , S 0 (u, k) = 1 . It is clear that if m ≥ j ≥ 0, then

(5) S m (uθ j , k) = q j S m−j (u, k) .

The following lemmas are useful in the proof of the main theorem.

Lemma 1. For any positive integer k , there is an integer a ≥ k such that whenever m ≥ a, then

(6) S m (u, k) = q k−1 S m−k (u, k), (u, P ) = 1 .

P r o o f. Suppose ord P k = l. Then take a to be a positive integer which is greater than k and all of i(l + 1)/(i − 1), i = 2, . . . , k. Thus, when m ≥ a we have

(7) i(m − l − 1) ≥ m , i = 2, . . . , k . From this it follows that m ≥ l + 1 and

{z (mod P m )} = {y + xθ m−l−1 | y (mod P m−l−1 ), x (mod P l+1 )} . Using the binomial theorem and (7) we have

(y + xθ m−l−1 ) k ≡ y k + ky k−1 xθ m−l−1 (mod P m ) ,

and

S m (u, k) = X

y (mod P

)

e m (uy k ) X

x (mod P

)

e l+1 (uky k−1 x) .

Since ord P k = l, by (4), the inner sum is 0 unless y ≡ 0 (mod P ), in which case it has the value q l+1 . Hence we have, by setting y = y 1 θ, y 1 (mod P m−l−2 ),

S m (u, k) = q l+1 X

y

(mod P

)

e m−k (uy k 1 ) = q k−1 S m−k (u, k) .

Let a(k) be the least positive integer such that (6) holds when m ≥ a(k), and write

(8) % = max{a(d 1 ), . . . , a(d s )} . Lemma 2. Put T m = q −ms P

(v,P

)=1 S m (va 1 , d 1 ) . . . S m (va s , d s ). Then T d+j = q d−r T j for j ≥ % − 1.

P r o o f. Since j ≥ % − 1 and d i ≥ 2, we have d i + j ≥ a(d i ). By Lemma 1 one gets

S d+j (u, d i ) = q f

(d

−1) S j (u, d i ) , i = 1, 2, . . . , s . Therefore,

T d+j = q −(d+j)s X

(v,P

)=1

S d+j (va 1 , d 1 ) . . . S d+j (va s , d s )

= q −(d+j)s X

(v,P

)=1 s

Y

i=1

q f

(d

−1) S j (va i , d i ) = q d−r T j .

1. Introduction. Let p be a fixed prime and f (x 1 , . . . , x s ) a polynomial with coefficients in Z p , the p-adic integers. Let c n denote the number of solutions of f = 0 over the ring Z/p ⁿ Z, with c 0 = 1. Then the Poincar´ e series P f (t) is the generating function

c n t ⁿ .

Let F be a finite extension of the rational field, and P a prime ideal of F with norm N (P ) = q which is a rational prime power. Using the previous notations, we let c n denote the number of solutions of the congruence (1) a 1 x ^d ₁

+ . . . + a s x ^d _s

≡ 0 (mod P ⁿ ) ,

n=0 c n t ⁿ .

It is clear that c n = q ^n(s−1) if d i = 1, for some i, 1 ≤ i ≤ s. Therefore we assume that d 1 , . . . , d s are all integers greater than 1.

Throughout this paper, we set d = lcm{d 1 , . . . , d s }, f _i = d/d i , r = f 1 + . . . + f s and c n = q ^−n(s−1) c n .

u = θ ^j ξ (j ≥ 0, (ξ, P ) = 1) .

In this case we write ord P u = j. Let D represent the different of F (see [4, Ch. 36]), and choose B, (B, P ) = 1 such that (ζ) = B/P ⁿ D is principal.

We set ζ m = ζθ ^n−m , 0 ≤ m ≤ n, such that ζ = ζ n , and define further e m (u) = e ^{2πi tr(uζ}

⁾ ,

where the symbol tr(γ) denotes the trace in F . The function e m (u) defines an additive character (mod P ^m ) and has the following simple properties:

e 0 (u) = 1, e m (u) = e m (u ⁰ ) if u ≡ u ⁰ (mod P ^m ) , (2)

e m (uθ ^j ) = e m−j (u) (0 ≤ j ≤ m) , (3)

e m (uz) = n q ^m if u ≡ 0 (mod P ^m ), 0 otherwise.

e m (uz ^k ) , S 0 (u, k) = 1 . It is clear that if m ≥ j ≥ 0, then

(5) S m (uθ ^j , k) = q ^j S m−j (u, k) .

(6) S m (u, k) = q ^k−1 S m−k (u, k), (u, P ) = 1 .

{z (mod P ^m )} = {y + xθ ^m−l−1 | y (mod P ^m−l−1 ), x (mod P ^l+1 )} . Using the binomial theorem and (7) we have

(y + xθ ^m−l−1 ) ^k ≡ y ^k + ky ^k−1 xθ ^m−l−1 (mod P ^m ) ,

e m (uy ^k ) X

e l+1 (uky ^k−1 x) .

Since ord P k = l, by (4), the inner sum is 0 unless y ≡ 0 (mod P ), in which case it has the value q ^l+1 . Hence we have, by setting y = y 1 θ, y 1 (mod P ^m−l−2 ),

S m (u, k) = q ^l+1 X

e m−k (uy ^k ₁ ) = q ^k−1 S m−k (u, k) .

(8) % = max{a(d 1 ), . . . , a(d s )} . Lemma 2. Put T m = q ^−ms P

)=1 S m (va 1 , d 1 ) . . . S m (va s , d s ). Then T d+j = q ^d−r T j for j ≥ % − 1.

S d+j (u, d i ) = q ^f

^(d

⁻¹⁾ S j (u, d i ) , i = 1, 2, . . . , s . Therefore,

T d+j = q ^−(d+j)s X

= q ^−(d+j)s X

q ^f

^(d

⁻¹⁾ S j (va i , d i ) = q ^d−r T j .

c n+d = c + q ^d−r c n , (ii) the Poincar´ e series is given by

P (t) = (1 − q ^s−1 t)( P %+d−1

i=0 c i t ⁱ − q ^ds−r P %−1

i=0 c ⁱ t ^d+i ) + cq ^(%+d)(s−1) t ^%+d

(1 − q ^s−1 t)(1 − q ^ds−r t ^d ) ,

where c = c %+d−1 − q ^d−r c %−1 is a constant depending only upon the diagonal

P r o o f. (i) From (4) we have c n = q ⁻ⁿ X

e n (u(a 1 x ^d ₁

+ . . . + a s x ^d _s

= q ⁻ⁿ X

In the summation over u (mod P ⁿ ), we may set u = vθ ^n−m , 0 ≤ m ≤ n, v (mod P ^m ) and (v, P ^m ) = 1. From (5) one has

c n = q ^n(s−1)

q ^−ms X

= q ^n(s−1)

q ^d−r T %+m

= c %+d−1 + q ^d−r (c n − c %−1 ) = c + q ^d−r c n . (ii) Put q ^s−1 t = t 1 . Then

c n t ⁿ =

c i t ⁱ +

c n+d t ^n+d