A generalization of the Hasse-Witt matrix of a hypersurface

0.19 MB·

by  Alan Adolphson

A GENERALIZATION OF THE HASSE-WITT MATRIX
OF A HYPERSURFACE
7
1
ALAN ADOLPHSON AND STEVEN SPERBER
0
2
n Abstract. The Hasse-Witt matrix of a hypersurface in Pn over
a a finite field of characteristic p gives essentially complete mod p
J
information about the zeta function of the hypersurface. But if

7 the degreed of the hypersurfaceis ≤n, the zeta function is trivial
1
mod p and the Hasse-Witt matrix is zero-by-zero. We generalize

] a classical formula for the Hasse-Witt matrix to obtain a matrix
G that gives a nontrivial congruence for the zeta function for all d.
A We also describe the differential equations satisfied by this matrix
. and prove that it is generically invertible.
h
t
a
m
[ 1. Introduction

1 Let p be a prime number, let q = pa, and let F be the field of q
v q
9 elements. Let
0
N
5 a
(1.1) f (x) = λ x j ∈ F [x ,...,x ]
4 λ j q 0 n
0 j=1
. X
1 be a homogeneous polynomial of degree d. We write a = (a ,...,a )
0 j 0j nj
7 with ni=0aij = d. Let Xλ ⊆ Pn be the hypersurface defined by the
1 equation f (x) = 0 and let Z(X /F ,t) be its zeta function. We write
λ λ q
: P
v P (t)(−1)n
Xi (1.2) Z(Xλ/Fq,t) = λ
(1−t)(1−qt)···(1−qn−1t)
r
a for some rational function P (t) ∈ 1+tZ[[t]]. If d = 1 then P (t) = 1,
λ λ
so we shall always assume that d ≥ 2.
Define a nonnegative integer µ by the equation
n+1
= µ+1,
d
(cid:24) (cid:25)
where ⌈r⌉ denotes the least integer greater than or equal to the real
number r. By a result of Ax[4] (see also Katz[5, Proposition 2.4]) we
have
P (q−µt) ∈ 1+tZ[[t]].
λ
Date: January 18, 2017.
1
2 ALAN ADOLPHSONANDSTEVEN SPERBER
Our goal in this paper is to give a mod p congruence for P (q−µt). We
λ
do this by defining a generalization of the classical Hasse-Witt matrix,
which gives such a congruence for µ = 0. Presumably our matrix is
the matrix of a “higher Hasse-Witt” operation as defined by Katz[6,
Section 2.3.4], but so far we have not been able to prove this.
It will be convenient to define an augmentation of the vectors a .
j
Set
a+ = (a ,...,a ,1) ∈ Nn+2, j = 1,...,N,
j 0j nj
where N denotes the nonnegative integers. Note that the vectors a+ all
j
lie on the hyperplane n u = du in Rn+2. We shall be interested
i=0 i n+1
in the lattice points on this hyperplane that lie in (R )n+2: set
>0
P
n
U = u = (u ,...,u ) ∈ Nn+2 | u = du and u > 0 for all i .
0 n+1 i n+1 i
(cid:26) i=0 (cid:27)
X
Note that u ∈ U implies that u ≥ µ+1. Let
n+1
U = {u = (u ,...,u ) ∈ U | u = µ+1},
min 0 n+1 n+1
a nonempty set by the definition of µ. We define a matrix of polynomi-
alswithrowsandcolumnsindexedbyU : letA(Λ) = [A (Λ)] ,
min uv u,v∈Umin
where
Λν1···ΛνN
A (Λ) = (−1)µ+1 1 N ∈ Q[Λ ,...,Λ ].
uv 1 N
ν !···ν !
1 N
νX∈NN
PNj=1νja+j =pu−v
Note that since the (n + 1)-st coordinate of each a+ equals 1, the
j
condition on the summation implies that
N
ν = (p−1)(µ+1).
j
j=1
X
When µ = 0, it follows that ν ≤ p−1 for all j, hence the matrix A(Λ)
j
can be reduced modulo p. We denote by A¯(Λ) ∈ F [Λ] its reduction
p
modulo p. Using thealgorithmofKatz[6, Algorithm2.3.7.14], onethen
checks that A¯(λ) is the Hasse-Witt matrix of the hypersurface f = 0.
λ
It is somewhat surprising that even when µ > 0 we still have ν ≤ p−1
j
for all j.
Lemma 1.3. If u,v ∈ U , ν ∈ NN, and N ν a+ = pu −v, then
min j=1 j j
ν ≤ p − 1 for all j. In particular, A (Λ) ∈ (Q ∩ Z )[Λ], so A (Λ)
j uv p uv
P
can be reduced modulo p.
A GENERALIZATION OF THE HASSE-WITT MATRIX 3
The proof of Lemma 1.3 will be given in Section 2. By the results of
[1, Theorem 2.7 or Theorem 3.1], which will be recalled in Section 2,
Lemma 1.3 implies immediately that each A (Λ) is a mod p solution
uv
of an A-hypergeometric system of differential equations.
Write the rational function P (t) of (1.2) as
λ
Q (t)
λ
P (t) = ,
λ
R (t)
λ
where Q (t) and R (t) are relatively prime polynomials with
λ λ
Q (q−µt), R (q−µt) ∈ 1+tZ[t].
λ λ
If X is smooth, it is known that P (t) is a polynomial, i. e., R (t) = 1.
λ λ λ
Ourmainresultisthefollowing, whichdoesnotrequireanysmoothness
assumption.
Theorem 1.4. If n is not divisible by d, then R (q−µt) ≡ 1 (mod q)
λ
and
Q (q−µt) ≡ det I −tA¯(λpa−1)A¯(λpa−2)···A¯(λ) (mod p).
λ
Notethateven inth(cid:0)eclassical caseoftheHasse-Wi(cid:1)tt matrix(µ = 0),
this result contains something new, as we do not assume that X is a
λ
smooth hypersurface.
The proof of Theorem 1.4 will occupy Sections 3–5. To describe the
zeta function, we apply the p-adic cohomology theory of Dwork, as in
Katz[7, Sections 4–6]. Indeed, Equation (3.5) below is a refined version
of [7, Equation (4.5.33)]. We discuss the case d|n in Section 6. If d|n,
the conclusion of Theorem 1.4 need not hold, and the rational function
P (q−µt) (mod p) is instead described by Theorem 6.2. We prove the
λ
generic invertibility of the matrix A¯(Λ) in Section 7.
2. Proof of Lemma 1.3
It will be convenient for later applications to prove a more general
version of Lemma 1.3. Put S = {0,1,...,n} and let I ⊆ S. Define an
integer µ by the equation
I
|I|
= µ +1.
I
d
(cid:24) (cid:25)
Note that µ ≥ 0 if I 6= ∅, µ = −1, and, in the notation of the
I ∅
Introduction, µ = µ. Set
S
n
UI = u = (u ,...,u ) ∈ Nn+2 | u = du and u > 0 for all i ∈ I .
0 n+1 i n+1 i
(cid:26) i=0 (cid:27)
X
4 ALAN ADOLPHSONANDSTEVEN SPERBER
Note that u ∈ UI implies that u ≥ µ +1. Let
n+1 I
UI = {u = (u ,...,u ) ∈ UI | u = µ +1},
min 0 n+1 n+1 I
a nonempty set by the definition of µ . Lemma 1.3 is the special case
I
I = S of the following result.
Lemma 2.1. If u,v ∈ UI , ν ∈ NN, and N ν a+ = pu −v, then
min j=1 j j
ν ≤ p−1 for all j.
j
P
Proof. The result is trivial when I = ∅ since U∅ = {(0,...,0)}, so as-
min
sume I 6= ∅. Let u = (u ,...,u ,µ +1),v = (v ,...,v ,µ +1) ∈ UI .
0 n I 0 n I min
Fixk ∈ {1,...,N}. Weclaimthereexistsanindexi ∈ {0,...,n}such
0
that
u if i ∈ I,
(2.2) a ≥ i0 0
i0k u +1 if i 6∈ I.
( i0 0
For if (2.2) fails for all i ∈ {0,...,n}, then
0
u−a+ = (u −a ,...,u −a ,µ ) ∈ UI,
k 0 0k n nk I
contradicting the definition of µ .
I
If ν ≥ p, then
k
pu if i ∈ I,
ν a ≥ pa ≥ i0 0
k i0k i0k pu +p if i 6∈ I,
( i0 0
hence in both cases we have
ν a > pu −v .
k i0k i0 i0
But our hypothesis N ν a+ = pu−v implies that
j=1 j j
P νkai0k ≤ pui0 −vi0.
This contradiction shows that ν ≤ p−1. And since k was arbitrary,
k
(cid:3)
the lemma is established.
We recall the definition of the A-hypergeometric system of differen-
tial equations associated to the set A = {a+}N . Let L ⊆ ZN be the
j j=1
lattice of relations on A,
N
L = l = (l ,...,l ) ∈ ZN | l a+ = 0 ,
1 N j j
(cid:26) j=1 (cid:27)
X
and let β = (β ,...,β ) ∈ Cn+2. The A-hypergeometric system with
0 n+1
parameter β is the system of partial differential operators in variables
A GENERALIZATION OF THE HASSE-WITT MATRIX 5
Λ ,...,Λ consisting of the box operators
1 N
∂ lj ∂ −lj
(cid:3) = − for l ∈ L
l
∂Λ ∂Λ
lYj>0(cid:18) j(cid:19) lYj<0(cid:18) j(cid:19)
and the Euler (or homogeneity) operators
N
∂
Z = a Λ −β for i = 0,...,n
i ij j i
∂Λ
j
j=1
X
and
N
∂
Z = Λ −β .
n+1 j n+1
∂Λ
j
j=1
X
Let AI(Λ) = [AI (Λ)] , where
uv u,v∈UI
min
Λν1···ΛνN
(2.3) AI (Λ) = (−1)µI+1 1 N ∈ Q[Λ ,...,Λ ].
uv ν !···ν ! 1 N
1 N
νX∈NN
PNj=1νja+j=pu−v
NotethatinthenotationoftheIntroductionwehaveAS (Λ) = A (Λ).
uv uv
By Lemma 2.1, the polynomials AI (Λ) have p-integral coefficients.
uv
Lemma 2.1 also says that pu−v is very good in the sense of [1, Sec-
tion 2]. We may therefore apply [1, Theorem 2.7] (or [1, Theorem 3.1]
since this system is nonconfluent) to conclude that A¯I (Λ) is a mod p
uv
solution of the A-hypergeometric system with parameter β = pu − v
(or, equivalently, β = −v since we have reduced modulo p).
3. The zeta function
To make a connection between the matrix A(Λ) and the zeta func-
tion (1.2), we apply a consequence of the Dwork trace formula devel-
oped in [3] (see Equation (3.5) below). Let γ be a zero of the series
0
∞ tpi/pi having ord γ = 1/(p−1), where ord is the p-adic valua-
i=0 p 0 p
tion normalized by ord p = 1. Let L be the space of series
p 0
P
n
L = c γpun+1xu | u −du = 0, c ∈ C , and {c } is bounded .
0 u 0 i n+1 u p u
(cid:26)u∈Nn+2 i=0 (cid:27)
X X
For I ⊆ {0,...,n}, let LI be the subset of L defined by
0 0
LI = c γpun+1xu ∈ L | u > 0 for i ∈ I .
0 u 0 0 i
(cid:26)u∈Nn+2 (cid:27)
X
6 ALAN ADOLPHSONANDSTEVEN SPERBER
Let AH(t) = exp( ∞ tpi/pi) be the Artin-Hasse series, a power series
i=0
in t that has p-integral coefficients, and set
P
∞
θ(t) = AH(γ t) = θ ti.
0 i
i=0
X
We then have
i
(3.1) ord θ ≥ .
p i
p−1
We define the Frobenius operator on L . Put
0
N
ˆ ˆ a+
(3.2) θ(λ,x) = θ(λjx j ),
j=1
Y
ˆ
where λ denotes the Teichmu¨ller lifting of λ. We shall also need to
ˆ
consider the series θ (λ,x) defined by
0
a−1 N a−1
(3.3) θ0(λˆ,x) = θ (λˆjxa+j )pi = θ(λˆpi,xpi).
i=0 j=1 i=0
YY (cid:0) (cid:1) Y
Define an operator ψ on formal power series by
(3.4) ψ c xu = c xu.
u pu
(cid:18)u∈Nn+2 (cid:19) u∈Nn+2
X X
Denote by α the composition
λˆ
α := ψa ◦“multiplication by θ (λˆ,x).”
λˆ 0
The map α operates on L and is stable on each LI. The proof of
λˆ 0 0
Theorem 1.4 will be based on the following formula for the rational
function P (t) defined in (1.2). By [3, Equation 7.12] we have
λ
(3.5) P (qt) = det(I −qn+1−|I|tα | LI)(−1)n+1+|I|.
λ λˆ 0
I⊆{0,1,...,n}
Y
To exploit (3.5) we shall need p-adic estimates for the action of α
λˆ
on LI. Expand (3.3) as a series in x, say,
0
(3.6) θ (λˆ,x) = θ (λˆ)xw.
0 0,w
w∈NA
X
Note that from the definitions we have θ (λˆ) ∈ Q (ζ ,γ ). A direct
0,w p q−1 0
calculation shows that for v ∈ UI,
(3.7) α (xv) = θ (λˆ)xu,
λˆ 0,qu−v
u∈UI
X
A GENERALIZATION OF THE HASSE-WITT MATRIX 7
thus we need p-adic estimates for the θ (λˆ) with u,v ∈ UI.
0,qu−v
Expand (3.2) as a series in x:
(3.8) θ(λˆ,x) = θ (λˆ)xw,
w
w∈NA
X
where
(3.9) θ (λˆ) = θ(w)λˆν
w ν
ν∈NN
X
with
N θ if N ν a+ = w,
(3.10) θ(w) = j=1 νj j=1 j j
ν 0 if N ν a+ 6= w.
(Q Pj=1 j j
From (3.1) we have the estimate P
N
(3.11) ord θ(w) ≥ j=1νj = wn+1.
p ν p−1 p−1
P
In particular, this implies the estimate
w
ˆ n+1
(3.12) ord θ (λ) ≥ .
p w
p−1
By (3.3) and (3.8) we have
a−1
(3.13) θ (λˆ) = θ (λˆpi).
0,w u(i)
u(0),...,Xu(a−1)∈NAYi=0
Pa−1piu(i)=w
i=0
In particular, we get the formula
a−1
(3.14) θ (λˆ) = θ (λˆpi).
0,qu−v w(i)
w(0),...,Xw(a−1)∈NAYi=0
Pa−1piw(i)=qu−v
i=0
Applying (3.12) to the products on the right-hand side of (3.14) gives
a−1 a−1 (i)
w
(3.15) ord θ (λˆpi) ≥ n+1.
p w(i) p−1
(cid:18)i=1 (cid:19) i=0
Y X
ˆ
This estimate is not directly helpful for estimating θ (λ) because
0,qu−v
we lack information about the w(i). Instead we proceed as follows.
Fix w(0),...,w(a−1) ∈ NA with
a−1
(3.16) piw(i) = qu−v.
i=0
X
8 ALAN ADOLPHSONANDSTEVEN SPERBER
We construct inductively from {w(i)}a−1 a related sequence {w˜(i)}a ⊆
i=0 i=0
UI such that
(3.17) w(i) = pw˜(i+1) −w˜(i) for i = 0,...,a−1.
First of all, take w˜(0) = v. Eq. (3.16) shows that w(0) + w˜(0) = pw˜(1)
for some w˜(1) ∈ Zn+2; since w(0) ∈ NA and w˜(0) ∈ UI we conclude
that w˜(1) ∈ UI. Suppose that for some 0 < k ≤ a−1 we have defined
w˜(0),...,w˜(k) ∈ UI satisfying (3.17) for i = 0,...,k −1. Substituting
pw˜(i+1) −w˜(i) for w(i) for i = 0,...,k −1 in (3.16) gives
a−1
(3.18) −w˜(0) +pkw˜(k) + piw(i) = pau−v.
i=k
X
Since w˜(0) = v, we can divide this equation by pk to get w˜(k) +w(k) =
pw˜(k+1) for some w˜(k+1) ∈ Zn+2. Since w(k) ∈ NA and (by induction)
w˜(k) ∈ UI, we conclude that w˜(k+1) ∈ UI. This completes the inductive
construction. Note that in the special case k = a−1, this computation
gives w˜(a) = u.
Summing Eq. (3.17) over i = 0,...,a−1 and using w˜(0) = v, w˜(a) =
u, gives
a−1 a−1
(3.19) w(i) = pu−v +(p−1) w˜(i),
i=0 i=1
X X
hence
a−1 (i) a−1
w pu −v
(3.20) n+1 = n+1 n+1 + w˜(i) .
p−1 p−1 n+1
i=0 i=1
X X
For w(0),...,w(a−1) as in (3.16), we thus get from (3.15)
a−1 a−1
pu −v
(3.21) ord θ (λˆpi) ≥ n+1 n+1 + w˜(i) .
p w(i) p−1 n+1
(cid:18)i=0 (cid:19) i=1
Y X
Since w˜(i) ∈ UI, we have
(3.22) w˜(i) = µ +1 if w˜(i) ∈ UI and w˜(i) ≥ µ +2 if w˜(i) 6∈ UI .
n+1 I min n+1 I min
From (3.21) and (3.22) we get the following result.
Lemma 3.23. For u,v ∈ UI and w(0),...,w(a−1) as in (3.16), we have
a−1
pu −v
(3.24) ord θ (λˆpi) ≥ n+1 n+1 +(a−1)(µ +1).
p w(i) p−1 I
(cid:18)i=0 (cid:19)
Y
A GENERALIZATION OF THE HASSE-WITT MATRIX 9
Furthermore, if any of the terms w˜(1),...,w˜(a−1) of the associated se-
quence satisfying (3.17) is not contained in UI , then
min
a−1
pu −v
(3.25) ord θ (λˆpi) ≥ n+1 n+1 +(a−1)(µ +1)+1.
p w(i) p−1 I
(cid:18)i=0 (cid:19)
Y
ˆ
Our desired estimate for θ (λ) now follows from (3.14).
0,qu−v
Corollary 3.26. For u,v ∈ UI we have
pu −v
ˆ n+1 n+1
(3.27) ord θ (λ) ≥ +(a−1)(µ +1).
p 0,qu−v I
p−1
4. The action of α on LI
λˆ 0
In this section, we use Corollary 3.26 to study the action of α on LI.
λˆ 0
From (3.7) and the formula of Serre[8, Proposition 7] we have
∞
(4.1) det(I −tα | LI) = aI tm,
λˆ 0 m
m=0
X
where
(4.2) aI = (−1)m sgn(σ) θ (λˆ) ∈ Q (ζ ,γ ),
m 0,qu−σ(u) p q−1 0
UmX⊆UIσX∈Sm uY∈Um
the outer sum is over all subsets U ⊆ UI of cardinality m, and S is
m m
the group of permutations on m objects.
Proposition 4.3. The coefficient aI is divisible by qm(µI+1) and sat-
m
isfies the congruence
(4.4)
aI ≡ (−1)m sgn(σ) θ (λˆ) (mod pqm(µI+1)).
m 0,qu−σ(u)
UmX⊆UmIinσX∈Sm uY∈Um
In particular, aI ≡ 0 (mod pqm(µI+1)) if m > |UI |.
m min
Proof. If U ⊆ UI is a subset of cardinality m and σ is a permutation
m
of U , then by (3.27)
m
pu −σ(u)
ˆ n+1 n+1
ord θ (λ) ≥ m(a−1)(µ +1)+
p 0,qu−σ(u) I
p−1
(cid:18)uY∈Um (cid:19) uX∈Um
= m(a−1)(µ +1)+ u
I n+1
uX∈Um
≥ ma(µ +1)
I
10 ALAN ADOLPHSONANDSTEVEN SPERBER
since u ∈ U implies u ≥ µ + 1. It follows from (4.2) that aI is
m n+1 I m
divisible by qm(µI+1). Furthermore, u ≥ µ +2 if u 6∈ UI , so
n+1 I min
ord θ (λˆ) ≥ ma(µ +1)+1 if U 6⊆ UI .
p 0,qu−σ(u) I m min
(cid:18)uY∈Um (cid:19)
(cid:3)
The congruence (4.4) now follows from (4.2).
As an immediate corollary of Proposition 4.3, we have the following
result.
Corollary 4.5. The reciprocal roots of det(I−tα | LI) are all divisible
λˆ 0
by qµI+1.
Corollary 4.5 allows us to analyze the terms on the right-hand side
of (3.5).
Proposition 4.6. The reciprocal roots of det(I −qn+1−|I|tα | LI) are
λˆ 0
divisible by qµ+2 unless either |I| = n+1 or |I| = n and n is divisible
by d, in which case they are divisible by qµ+1.
Proof. Corollary 4.5 and the definition of µ imply that the reciprocal
I
roots of det(I −qn+1−|I|tα | LI) are divisible by q to the power
λˆ 0
|I|
(4.7) n+1−|I|+ .
d
(cid:24) (cid:25)
If |I| = n + 1, this reduces to µ + 1. Suppose |I| = n. From the
definition of µ we have n = µd+r with 0 ≤ r ≤ d−1. The expression
(4.7) then reduces to 1+⌈(µd+r)/d⌉, which equals µ+2 if r > 0 and
equals µ+1 if r = 0.
If |I| = n−1, expression (4.7) reduces to 2+⌈(µd+r−1)/d⌉, which
equals µ + 3 if r > 1 and equals µ + 2 if r = 0,1. Finally, note that
expression (4.7) cannot decrease when |I| decreases, so expression (4.7)
will be ≥ µ+2 for |I| < n−1. (cid:3)
From (3.5) and Proposition 4.6 we get the following result.
Proposition 4.8. If n is not divisible by d, then
(4.9) P (q−µt) ≡ det(I −q−µ−1tα | LS) (mod q).
λ λˆ 0
If n is divisible by d, then
det(I −q−µ−1tα | LS)
(4.10) P (q−µt) ≡ λˆ 0 (mod q).
λ n det(I −q−µtα | LS\{i})
i=0 λˆ 0
Q