On the congruence class modulo prime numbers of the number of rational points of a variety

0.29 MB·

by  Lucile Devin

ON THE CONGRUENCE CLASS MODULO PRIME NUMBERS
OF THE NUMBER OF RATIONAL POINTS OF A VARIETY
LUCILEDEVIN
Abstract. Let X be a scheme of finite type over Z. For p ∈ P the set of prime numbers, let NX(p)
be the number of Fp-points of X/Fp. For fixed n ≥ 1 and a1,...,an ∈ Z, we study the set Tni=1{p ∈
P −ΣX,NX(p) 6= ai [mod p]} where ΣX is the finite set of primes of bad reduction for X. In case
6
dimX ≤3, weshowthe setiseither emptyorhas positivelower-density. Wealsoaddress thequestion of
1
thesizeofthesmallestprimeinthatset. Usingsievemethods,weobtainforexampleanupperboundfor
0
thesizeoftheleastprimeof{p∈P,p∤NX(p)}onaverageinparticularfamiliesofhyperellipticcurves.
2

t
c
O
Introduction
2
] “reLdeutctXionbemaodsuchloemp”eooffXfin.iTtehteyqpueaonvteitryZN. F(opr)e:=verXy p(rFim)e nisutmhbeenrupmobneer ocfanFd-epfioninetXsopf=XX. F×orZfiFxpedthpe,
T X p p p
| |
theWeilConjecturegivesapreciseestimateforN (p). HoweverthearithmeticpropertiesofN (p)remain
N X X
mysterious to a large extent. The aim of this article is to study properties of N (p)[mod p].
X
.
h The main focus of this work is the set p ,p∤N (p) . The principal motivation comes from work of
X
t Fouvry and Katz ([6]). In loc. cit. the au{th∈orPs relate the }possibility to obtain sharp estimates for certain
a
m exponential sums over the rational points of X and the size of the set p ,p ∤ N (p) . More precisely
X
{ ∈ P }
the authors state ([6, Th.8.1]) that if X/C is smooth and if the set p ,p ∤ N (p) is infinite, then a
[ X
{ ∈ P }
deep geometric invariant (called the A-number) associated to X is non-zero.

3
Let us give a bit more detail on what the A-number is and on how Fouvry and Katz use that invariant.
v
Given an affine scheme X AN of finite type over Z such that X/C is smooth, a function f on X (i.e. a
3 ⊂ Z
7 morphism f :X A1Z), a finite field k and a non-trivialadditive characterψ of k, Fouvry and Katz define
→
3 A(X,f,k,ψ)astherankofacertainlissesheafdefinedusingtheℓ-adicFouriertransform([6,Part4]). (See
2 the introductionand the first partof[13] for the precisedefinition of A(X,0,F ,exp(2iπ•)).) A remarkable
p p
0
point about A-numbers is made explicit in [6, Lem. 4.3]: A(X,f,k,ψ) = 0 is equivalent to the fact that
.
1 there exists a dense open subset U in AN such that for any finite extension E of k and any h U(E) the
0 k ∈
exponential sum
6
1
ψ tr f(x)+ h x
: E/k i i
v !!
i x∈XX(E) Xi
X vanishes.
r Assuming the A-number does not vanish, Fouvry and Katz ([6, Cor. 4.5]) prove a very precise estimate
a
for that type of exponentialsums. The resultimproves substantially a previous resultof Katz and Laumon
[14] about the dimension of the set of parameters h for which the exponential sum has a given size.
The philosophy underlying the present work is that “most” schemes X of finite type over Z should have
a non-zero A-number. More generally, we can study sets of the form p ,p∤(N (p) a) for arbitrary
X
{ ∈P − }
a Z, or finite intersections of such sets. In fact we do not even need the scheme X to be affine, nor is it
∈
required that the generic fibre be smooth. We show that the sets of primes we are interested in are either
2010 Mathematics Subject Classification. Primary11R45,11G25; Secondary11N36.
Key words and phrases. Chebotarev DensityTheorem,algebraicvarietiesoverfinitefields,largeandlargersieve.
1
2 LUCILEDEVIN
empty or that they have positive lower density. In the latter case this proves a strong form of Fouvry and
Katz’s criterion ([6, Th. 8.1]).
To state our main result, let us first recall the definition of the densities. Let E be a subset of the set of
primes . Define the upper-density and lower-density of E as
P
p E,p x
dens (E)=limsup|{ ∈ ≤ }|
sup
p ,p x
x→∞
|{ ∈P ≤ }|
and
p E,p x
dens (E)=liminf |{ ∈ ≤ }|.
inf
x→∞ p ,p x
|{ ∈P ≤ }|
Ifthesequantitiescoincide,wesaythatthe setE hasa(natural)density. We denotethis valuebydens(E).
It is clear that if dens(E)>0 or if dens (E)>0 then E is infinite.
inf
We can now state the main result of this paper.
Theorem 0.1. Let X be a scheme of finite type over Z. Suppose
either dim(X/Q) 2
• ≤
or dim(X/Q)=3 and there is a projective resolution of singularities Y of X such that b (Y)=0.
3
•
Let Σ′ be the finite set of primes of bad reduction for X. Then if there exists a prime p / Σ′ satisfying
X 0 ∈ X
p ∤N (p ), one has
0 X 0
dens p ,p∤N (p) >0.
inf X
{ ∈P }
More generally, for every a ,...,a Z, if there exists a prime p / Σ′ satisfying p ∤ n (N (p )
1 n ∈ 0 ∈ X 0 i=1 X 0 −
a ), one has
i
Q
n
dens p ,N (p) a [mod p] >0.
inf X i
{ ∈P 6≡ }
i=1
\
Inparticularinthecasedim(X/Q) 2,noassumptionaboutthegeometryofaresolutionofsingularities
≤
of X is needed. Here b (Y) is the third Betti number of Y (definitions will be recalled later). In the case
3
dim(X)=3, there is no reason to believe that the assumption b (Y)=0 is generic. For example a smooth
3
hypersurface Y in P4 has often b (Y) = 0. Still this condition is not empty and we present a way to
3
6
construct schemes satisfying this condition in section 2.3.
Theorem0.1is provedin section2.1, asa consequenceofTheorem1.1andTheorem2.1. The readerwill
find there more precisions about the set of bad reduction Σ′ . Theorem 1.1 is essentially Serre’s theorem
X
[23, Th. 6.3]about the distribution of N (p) [mod m] as p varies and m is fixed. The idea of Theorem2.1
X
is to get rid of the higher degree cohomology to reduce to Galois representations whose traces of Frobenius
are bounded by a multiple of p. It is quite easy in the case dim(X) = 1. When dim(X) = 2 we combine
the arguments for curves with Poincar´e Duality. However the method does not seem to apply in higher
dimension without strong hypotheses. In section 3, we present a variant of Theorem 0.1 where we do not
even require the existence of a suitable prime p (see Theorem 3.1).
0
Combining [6, Th. 8.1], [6, Cor. 4.5] and Theorem 0.1 we deduce the following strong low-dimensional
version of [6, Cor. 4.5].
Corollary 0.2. With notations and assumptions as in Theorem 0.1 assume there exists D Z such that
∈
X[1/D] An is a smooth closed subscheme of relative dimension d with geometrically connectedfibres.
→ Z[1/D]
Assume there exists a prime p / Σ′ such that p ∤N (p ) then:
0 ∈ X 0 X 0
(i) for every function f on X, for all primes p outside of a finite set Σ′′ containing Σ′ , for all α 1,
X X ≥
and for all additive characters ψ of F , the A-number A(X,f,F ,ψ) is non-zero,
pα pα
(ii) for f fixedthereexists aconstantC depending on X andf, andaclosedsubschemeX An of
2 ⊂ Z[1/D]
relative dimension at most n 2 such that for every p / Σ′′, for every α 1, for every non-trivial
− ∈ X ≥
CONGRUENCE CLASS OF THE NUMBER OF RATIONAL POINTS OF A VARIETY 3
additive character ψ of F and for every h (An X )(F ) one has
pα ∈ Z[1/D]− 2 pα
αd
(cid:12) ψ(f(x)+ hixi)(cid:12)≤Cp 2 .
(cid:12)(cid:12)x∈XX(Fpα) Xi (cid:12)(cid:12)
(cid:12) (cid:12)
We note in passing that a simil(cid:12)ar phenomenon (“only one p(cid:12)rime is needed instead of infinitely many”)
(cid:12) (cid:12)
appears in the theory of arithmetic groups, see Lubotzky’s paper “one for almost all” [18].
Lower bounds (and even sometimes the determination of the exact value) for some A-numbers already
appearinarticlesbyKatz(see[12],[13]),butmostofthemonlyholdforparticularvarietiesgivenbycertain
forms of equation. The arguments given by Katz are of geometric nature. In Example 2 (Section 2.3), we
give a new example of a variety with non-zero A-number. The argument is computational and comes as a
consequence of Theorem 0.1.
In the case of an affine smooth hypersurface of A3, Katz [12, Rem. (ii) p. 150] gives an explicit formula
for the A-number involving the degree of its equation. Using this formula we show in section 5.2 that the
converse of Corollary 0.2(i) is false. Precisely, we consider in Proposition 5.1 a family of affine surfaces S
satisfying p N (p) for every prime p, while the A-numbers of these surfaces are non-zero.
S
|
Besides the close link with A-number of varieties the study of the density of p ,N (p) S(p)
X
{ ∈ P ∈ }
where S(p) is a set that may (but does not have to) depend on p lies at the heart of many other important
problems in arithmetic geometry. For instance the Sato–Tate conjecture solves completely (and in a very
precise way) the case where X is an elliptic curve and S(p) is an interval (p+1+a√p,p+1+b√p) (with
a and b independent of p).
In Serre’s book [23] the Chebotarev Density Theorem is used to prove results about the density of sets
of the type p ,N (p) a [mod m] . Some of the ideas underlying [23] can already be found in his
X
{ ∈ P ≡ }
article [22] especially in Section 8 about elliptic curves. Serre’s result is also used in the recent preprint of
Sawin [21] where the author gives explicit values for the density of the set of ordinary primes for abelian
surfaces (over Q).
The second main point that this present paper addresses is the question of effectiveness in Theorem 0.1.
How far does one have to go to find a suitable prime?
We solve the question by using a double sieve as in [5]. It is based on a result of Kowalski [16, Th.
8.15] and Gallagher’s larger sieve. We obtain an upper bound for the least prime in p ,p∤N (p) on
X
{ ∈P }
average over a 1-parameter family of hyperelliptic curves X.
Theorem 0.3. Let g 2 be an integer and let f Z[T] be a separable polynomial of degree 2g. For each
≥ ∈
u Z we consider the curve C with affine model
u
∈
C :y2 =f(t)(t u).
u
−
Let T 1. There exists a constant K depending only on g such that for every α ,...,α Z, for “most”
g 1 n
u Z≥[ T,T], the least prime p of good reduction for C and satisfying p∤ n (N (p)∈ α ) is at most
∈ ∩ − u i=1 Cu − i
of size
(2Kglog(T))γ/2(log(2Kglog(T)))γ2(1−γ+22n−2),Q
where one can take γ =4g2+2g+4.
This theorem is proven in section 4 under a more general and precise form. In particular we give more
quantitative precisions on what is meant by “for most”. Theorem 0.3 is a consequence of Theorem 4.1 and
a theoremofYuaboutbig monodromy(see e.g. [8]). The idea underlyingthis resultis thatthe leastprime
p not dividing N (p) should be small compared to the coefficients of an equation defining C.
C
Finallyinsection5.1wepresentsomeexamplesofcurvesforwhichtheleastprimep∤N (p)canbecome
C
arbitrarily large.
4 LUCILEDEVIN
Notations. For X a scheme of finite type over Z, we denote X0 := X Z Q the generic fiber. Given p
×
a prime, Xp := X Z Fp is the “reduction modulo p” of X. In this paper the words curve, surface and
×
threefold mean scheme of finite type over Z of relative dimension 1, 2, or 3 respectively. By f(x) g(x)
≪
or f(x)=O(g(x)) we mean that there exists a constant C 0 such that f(x) Cg(x) for all x such that
≥ | |≤
f(x) is defined. The “implicit constant” C may depend on some parameters.
Acknowledgements. Thispapercontainssomeofthe resultsofmydoctoraldissertation. Ithankmyadvisor
Florent Jouve for all his advice, help and time spent correcting the first drafts of this paper. I am grateful
to Antoine Chambert-Loir and E´tienne Fouvry for encouraging me to get more general statements. Many
mathematicianshelpedmetounderstandthegeometryofthreefolds;IthankOlivierBenoist,DavidHarari,
Olivier Wittenberg, Franc¸ois Charles and Alena Pirutka for their explanations. I would like to thank
Davide Lombardo for his comments, and particularly for providing Example 7. I have also benefited from
conversations with Jean-Louis Colliot-Th´el`ene, G´erard Laumon, Yang Cao, Tiago Jardim da Fonseca and
Cong Xue. Most of the computations presented here were performed with Sage [2].
1. Serre’s result
The proof of Theorem 0.1 uses a generalized version of Serre’s Theorem [23, Th. 6.3]. This is Theorem
1.1 below. We give a proof of this statement but we do not get into the details when it is not necessary
since the main ideas are essentially contained in the first six chapters of [23].
First we need a way to compute N (p). Combining the Grothendieck–Lefschetz trace formula with a
X
comparisontheoremforcohomologieswithcompactsupportSerregetsforad-dimensionalseparatedscheme
X over Z, the existence of a finite set Σ such that, for p not in Σ and for any prime ℓ=p,
X X
⊂P 6
2d
N (p)= ( 1)itr(Frob Hi(X Q,Q )),
X − p | c × ℓ
i=0
X
where Frob is a representative of the image of the geometric Frobenius at p (we assume a choice of
p
isomorphismhas been made). For details of the proof,see [23, Part4.8.2-4.8.4]and [3, p. 49-50](note that
theargumentusesthefactthatthesheafQ islocallyconstant). In[23,3]thesetΣ isnotgivenexplicitly:
ℓ X
it comesfromadeepstratificationtheorem[14,Th. 3.2.1]. Inthe caseX/Cisproperandsmooththen one
can take Σ to be the locus of bad reduction of X [19, p. 230 Cor. 4.2].
X
Fix a prime ℓ. For simplicity we will write Hi(X,ℓ) for Hi(X Q,Q ). We are interested in functions
c × ℓ
defined over primes of the following type:
(1) f : (Σ ℓ ) Z
X,i X
P − ∪{ } →
p tr(Frob Hi(X,ℓ)).
p
7→ |
This kind of functions can be decomposed:
Σ ℓ Frob Γ :=Gal(Q /Q) ρℓ GL(Hi(X,ℓ)) tr Q
P − X ∪{ } ΣX,ℓ ΣX,ℓ ℓ
where Q /Q is the maximal Galois extension unramified outside Σ := Σ ℓ . For every prime
ΣX,ℓ X,ℓ X ∪{ }
p / Σ , let Frob denote the corresponding geometric Frobenius element of Γ , it is well defined up to
∈ X,ℓ p ΣX,ℓ
conjugation. The second arrow above is given by the action of Γ on Hi(X,ℓ) which globally fixes the
ΣX,ℓ
imageofHi(X Q,Z )inHi(X,ℓ). Thuswecanseetheimageρ (Γ )asasubgroupofGL (Z )where
c × ℓ ℓ ΣX,ℓ bi ℓ
b =dimHi(X,ℓ) is the i-th Betti number of X. By the Weil conjectures, the image of f is in fact in Z
i X,i
and independent of ℓ.
There is a natural way to extend f at 1: it is the value of the function tr ρ at identity in Γ . We
X,i ◦ ℓ ΣX,ℓ
set
(2) f (1):=b (X)=dimHi(X,ℓ).
X,i i
CONGRUENCE CLASS OF THE NUMBER OF RATIONAL POINTS OF A VARIETY 5
Now we can state a generalized version of Serre’s Theorem.
Theorem 1.1. Let (X ) be a finite set of schemes of finite type over Z. For all j let Σ be the finite set
j j Xj
defined as above. Let f :( Σ ) Z be a Z-linear combination of functions f .
P −∪j Xj → Xj,i
Then for all a,m Z, the set
∈
(3) E (f)= p Σ ,p∤m,f(p) a [mod m]
a,m { ∈P −∪j Xj ≡ }
satisfies one of the two following properties:
either E (f)= ,
a,m
• ∅
or dens(E (f)) exists and is a positive rational number.
a,m
•
Moreover, if f(1) a [mod m] then E (f)= .
a,m
≡ 6 ∅
The same result holds for finite unions or intersections of sets E (f ).
ai,mi i
The sets E (f) are examples of Frobenian sets as introduced by Serre in [23, Sec. 3.3]. In particular
a,m
for the function N = ( 1)if , the value N (1) is the Euler-Poincar´e characteristic χ (X) of X/C.
X i − X,i X c
For ease of exposition we state the following particular case of Theorem 1.1. This is very close to [23, Th.
P
1.4].
Corollary 1.2. Let X be a scheme of finite type over Z. Let Σ be the finite set defined as above. Then
X
for all a,m Z, the set
∈
E (N )= p Σ ,p∤m,N (p) a [mod m]
a,m X X X
{ ∈P − ≡ }
satisfies one of the two following properties:
either E (N )= ,
a,m X
• ∅
or dens(E (N )) is a positive rational number.
a,m X
•
Moreover, if χ (X) a [mod m] then E (N )= .
c a,m X
≡ 6 ∅
Proof of Theorem 1.1 (Serre). We can assume f = f . Indeed the set E (f) is a finite union of finite
X,i a,m
intersections ofsets E (f ). By the Chinese Remainder Theoremit is enoughto provethe theoremfor
b,m Xj,i
m=ℓk with ℓ prime. Reducing f modulo ℓk, we get
X,i
Frob
Σ Γ GL (Z )
P − X,ℓ ΣX,ℓ bi ℓ
φ
ℓk
tr
G GL (Z/ℓkZ) Z/ℓkZ
ℓk bi
where G is the quotient of Γ by the kernel of φ . The group G can be seen as a subgroup of the
ℓk ΣX,ℓ ℓk ℓk
finite group GL (Z/ℓkZ), hence it is finite. Therefore G = Gal(E/Q) is a finite Galois group. This is a
bi ℓk
situation where the Chebotarev Density Theorem applies. Let C = g G ,tr(g) = a [mod ℓk] , it is a
a { ∈ ℓk }
unionofconjugacyclassesinG . HencethesetofprimesE (f )= p Σ ,φ (Frob ) C
ℓk a,ℓk X,i { ∈P− X,ℓ ℓk p,Gℓk ∈ a}
has a density given by
C
a
dens(E (f ))= | |
a,ℓk X,i G
| ℓk|
which is rational and positive if and only if C is non-empty, i.e. E (f ) is non-empty.
a a,ℓk X,i
Moreover,if f (1) a [mod ℓk] then the identity element of G is in C .
X,i ≡ ℓk a
In the case one has a finite intersection of E (f ), the proof follows using the Chebotarev Density
a,ℓk Xj,i
Theorem for a product of Galois groups. (cid:3)
6 LUCILEDEVIN
2. Proof of the main result
Corollary 1.2 deals with the congruence classes of N (p) modulo a fixed m as p varies among primes.
X
The point is now to use this idea to get information about N (p) modulo p as p varies. In this section, we
X
define a function, close to N , to which we can apply Theorem 1.1 and get results about N (p) modulo p.
X X
The following result is our main technical tool.
Theorem 2.1. Let X be as in Theorem 0.1. There exists a finite set Σ′ containing Σ (defined in section
X X
1), such that for every a Z, there exists a function M defined over Σ′ satisfying:
∈ X,a P − X
(i) M is a Z-linear combination of functions f of type (1) for some schemes U,
X,a U,i
(ii) for all p Σ′ , one has M (p) N (p) a [mod p],
∈P − X X,a ≡ X −
(iii) there exists explicit constants b (X),b (X) Z and A=A(X,a)>0 such that for all p Σ′
− + ∈ ∈P− X
and p A one has b (X)p<M (p)<b (X)p.
− X,a +
≥
We give more precision about the set Σ′ (which we use also in Theorem 0.1) in section 2.2. In the case
X
dim(X) 2,itcanbelargerthanΣ ,itdependsonachoiceofaprojectiveresolutionofX (seeProposition
X
≥
2.7). We postpone the proof of Theorem 2.1 to section 2.2.
2.1. Proof of Theorem 0.1. We now show how to deduce Theorem 0.1 by combining Theorem 1.1 and
Theorem 2.1.
Proof of Theorem 0.1. Letk Z. UsingTheorem2.1fori 1,...,n yieldsafunctionM andbounds
∈ ∈{ } X,ai
b (i) Z, A >0 such that for every p>A ,
± i i
∈
(b−(i)+k)p<MX,ai(p)+kfA1(p)<(b+(i)+k)p.
Suppose as in Theorem 0.1 that there exists p / Σ′ such that for every i, one has N (p )=a [mod p ].
0 ∈ X X 0 6 i 0
Let m =M (p ). By Theorem 2.1(ii), for every k Z one has, using the notation (3),
i X,ai 0 ∈
n
p0 ∈ Emi+kp0,0(MX,ai +kfA1).
i=1
\
Hence the intersection is not empty. By Theorem 1.1 (using Theorem 2.1(i)), we deduce that this set has
positive density. Let A=max (A ), b =min (b (i)) and b =max (b (i)). Then for k large enough, one
i i − i − + i +
has
n n
Emi+kp0,0(MX,ai +kfA1)∩[A,∞)⊂ {p∈P,NX(p)6≡ai [mod p]}.
i=1 i=1
\ \
Indeed, since p 2, one can choose k b such that for every i one has m +kp b +k. Then let
0 − i 0 +
≥ ≥ − ≥
p∈Emi+kp0,0(MX,ai +kfA1)∩[A,∞). One has
0<MX,ai(p)+kfA1(p)<(b++k)p
and
MX,ai(p)+kfA1(p)≡0 [mod mi+kp0].
Hence p does not divide MX,ai(p)+kfA1(p). By Theorem 2.1(ii), this concludes the proof. (cid:3)
2.2. Proof of Theorem 2.1. We are now reduced to proving Theorem 2.1. It suffices to deal with the
case a = 0, and note M = M af if a = 0, where is the point. The argument used depends on
X,a X,0 •,0
− 6 •
the dimension of the scheme and uses the result for the lower dimensions. Therefore we give a proof for
each dimension d=1,2,3, starting with d=1.
The 1-dimensional case is a corollary of Lang–Weil’s Theorem.
CONGRUENCE CLASS OF THE NUMBER OF RATIONAL POINTS OF A VARIETY 7
Lemma 2.2. Let X be a 1-dimensional scheme of finite type over Z. Let b (X) be the number of 1-
2
dimensional irreducible components of X/C. There exists a constant C(X) > 0 such that for every prime
p / Σ , one has
X
∈
(4) 0 NX(p) b2(X)p+C(X)p12.
≤ ≤
Remark 1. By definition b (X) is the secondBetti number ofX. As we wantinequality (4) to be true for
2
every prime, we cannot hope for a better estimate without assuming anything about the field of definition
of the irreducible components of X/C.
Proposition 2.3. In case X is of dimension 1 over Z, the function M =N satisfies Theorem 2.1. One
X X
can take Σ′ =Σ , b (X)= 1 and b (X)=b (X)+1.
X X − − + 2
Proof. This follows from Lemma 2.2. (cid:3)
Remark 2. In the case of curves, the argument is very close to the one given in [20, Prop. 2.7.1].
IncasethedimensionoverZis2or3,webeginwiththe easiercaseofasmoothprojectivevariety. Then
by Hironaka’s resolution of singularities we will deduce the general case.
The following lemma is an easy corollary of Poincar´eDuality.
Lemma 2.4. Let p be a prime number and let Y be a smooth projective variety of dimension d over F .
p p
For all primes ℓ=p and for all i d+1,...2d , one has
6 ∈{ }
tr(Frob Hi(Y F ,Q ))=pi−dtr(Frob H2d−i(Y F ,Q )).
p | c p× p ℓ p | c p× p ℓ
Proof. Using Deligne’s Theorem (Weil’s Conjectures), one can write
bi
tr(Frob Hi(Y F ,Q ))= α
p | c p× p ℓ i,j
j=1
X
with |αi,j|=p2i. By Poincar´eDuality bi =b2d−i and (up to reordering) αi,j = α2dp−di,j for all j. Hence
bi bi pd i bi pd−2i
αi,j = =p2
α α
2d−i,j 2d−i,j
j=1 j=1 j=1
X X X
bi α
i 2d−i,j
= p2 σ
j=1 (cid:18) pd−2i (cid:19)
X
= pi−dtr(Frob H2d−i(Y F ,Q ))
p | c p× p ℓ
where z σ(z) denotes complex conjugation. (cid:3)
7→
We deduce Theorem 2.1 for smooth projective schemes.
Proposition 2.5. Let X be a scheme of finite type over Z, suppose X/C is either a smooth projective
surface, or a smooth projective threefold satisfying b (X) = 0. Then the function M = 2 ( 1)if
3 X i=0 − X,i
satisfies Theorem 2.1. One can take Σ′ =Σ , b (X)= b (X) 1 and b (X)=b (X)+1.
X X − − 2 − + 2 P
Proof. Use the fact that tr(Frob Hi(Y F ,Q )) Z for every i, and Lemma 2.4. (cid:3)
p | c p× p ℓ ∈
For the proof of the general case, we need a corollary to Hironaka’s resolution of singularities (e.g. [15,
Th.3.36]).
Lemma 2.6. Let X be a scheme of finite type over Z then there exists a smooth open dense subscheme U
0
of X which is Q-isomorphic to an open dense subscheme V of a smooth projective variety Y defined over
0 0 0
Q.
8 LUCILEDEVIN
In particular there exist D Z≥1 and schemes U,V,Y SpecZ[1/D] whose generic fibers are U0,V0,Y0
∈ →
respectively such that for every prime p ∤ D, the subscheme U is F -isomorphic to V and Y is smooth
p p p p
projective over F .
p
Remark 3. In Hironaka’s Theorem, the subscheme U is the smooth locus of X . In particular if X/C is
0 0
smooth, one can take U =X.
As U is an open dense subscheme of X, one has dim(X U) dim(X) 1. Similarly, dim(Y V)
− ≤ − − ≤
dim(X) 1. Using these observations we conclude the proof of Theorem 2.1 via an induction argument.
−
Proposition2.7. LetX beaschemeoffinitetypeover Z, andU,V,Y andD given byLemma2.6. Suppose
Y/C is either a smooth projective surface, or a smooth projective threefold satisfying b (Y) = 0. Then the
3
function M =M M +M satisfies Theorem 2.1, for thesetΣ′ =Σ Σ′ Σ′ p∤D .
X Y− Y−V X−U X X∪ Y−V∪ X−U∪{ }
Onecantakeb (X)= b (Y) b (Y V)+b (X U)+1andb (X)=b (Y) b (Y V)+b (X U) 1.
− 2 + − + 2 − +
− − − − − − − −
Proof. By Lemma 2.6, the scheme U is Z[1/D]-isomorphic to V. Define (by induction on the dimension)
Σ′ =Σ Σ′ Σ′ p D . For every prime p / Σ′ , one has
X X ∪ Y−V ∪ X−U ∪{ | } ∈ X
N (p)=N (p),
U V
i.e.
N (p) N (p)=N (p) N (p).
X Y X−U Y−V
− −
We invoke Proposition 2.5 for the scheme Y. In case dim(X)=2, we invoke Proposition2.3 for the curves
X U and Y V. In case dim(X)= 3, we use an induction argument: we invoke Proposition 2.7 for the
sur−faces X U−and Y V. Hence the function M :=M M +M satisfies Theorem 2.1. (cid:3)
X Y Y−V X−U
− − −
2.3. The condition on the third Betti number of threefolds is not empty. As we see in the proof
of Theorem 2.1, the condition b (Y) = 0 is needed in the case of threefolds. We present now examples of
3
threefolds to which we would like to apply Theorem 0.1.
Example 1 (Hypersurfaces). Thefirstexampleonecouldthinkofisthecaseofahypersurface. LetY P4
⊆
be a smooth projective hypersurface defined over Z by an equation of degree d. From [4, Chap. 5 3] we
§
get b (Y) = (d−1)5+1 1 which is zero if d = 1 or 2, and is positive as soon as d > 2. Hence we cannot
3 d −
apply Theorem 0.1 to non-rationalhypersurfaces.
We present now an example of a family of schemes better suited for the application of Theorem 0.1. In
turn we obtain a new example of a variety with non-zero A-number.
LetS be a projectivesurfacedefinedoverZ, smoothoverC with b (S)=0(e.g. S is a K3-surface). We
1
build a smooth scheme Y with a morphism g :Y S such that for all s S, the fiber Y is isomorphic to
s
P1. Then the Leray spectral sequence [19, App. B→] for g : Y S is Ei,j∈:= Hi(S,Rjg C) Hi+j(Y,C).
→ 2 ∗ ⇒
As Ei,3−i =0 for each i 0,1,2,3 we get H3(Y,C)=0.
2 ∈{ }
Remark 4. A first example of a scheme equipped with such a fibration is S P1 but probably counting
×
the number of F -points on S P1 is not that interesting.
p
×
These schemes are exactly the Severi-Brauer schemes over S of relative order 2. In our situation (S a
projective smooth surface over C), [7, Part 8] ensures that the Severi-Brauer schemes over S of relative
order 2 are classified by the 2-torsion subgroup of the Brauer group of S, noted Br(S)[2]. This group can
beseenasasubsetofthesetofquaternionalgebrasoverthefunctionfieldofS. IfBr(S)[2]isnon-trivial(it
is the case when S is a K3-surface), a non trivial element of Br(S)[2] yields equations for a Severi-Brauer
scheme not of type S P1.
×
Precisely, let S be a K3-surface in the weighted projective space P(1,1,1,3) given by an equation
f(x,y,z) = w2 where f is a homogeneous polynomial of degree 6. A non-trivial element of Br(S)[2]
CONGRUENCE CLASS OF THE NUMBER OF RATIONAL POINTS OF A VARIETY 9
can be givenon anopen affine subscheme O of S as a pair (a,b) where a andb are rationalfunctions in the
variable s=(x,y,z,w) O. Define the scheme U(a,b) in O P2 by the equations:
∈ ×
U(a,b):a(s)u2+b(s)v2 =t2.
Then U(a,b) is birational to a Severi-Brauer scheme over S of relative order 2. In particular it admits a
smooth projective completion Y(a,b) satisfying b (Y(a,b)) = 0. Writing a = α/d, b = β/d with α,β,d
3
polynomials, we can define a larger scheme X(a,b) in P(1,1,1,3) P2 by the equations:
×
f(x,y,z) = w2
X(a,b):
α(x,y,z)u2+β(x,y,z)v2=d(x,y,z)t2
(cid:26)
in the variables [x : y : z : w] P(1,1,1,3), [t : u : v] P2. Then U(a,b) is an open dense subscheme of
∈ ∈
X(a,b)henceX(a,b)isalsobirationaltoY(a,b). Aswewantanaffinescheme,weconsidertheintersection
with some hyperplane: the affine scheme in A5 given by the equations
f(x,y,1) = w2
X(a,b):
α(x,y,1)u2+β(x,y,1)v2=d(x,y,1)
(cid:26)
admits Y(a,b) as a smooth projective model.
It is not alwayseasy to describe a non-trivialelement ofthe groupBr(S)[2], but there are some surfaces
well studied in the literature.
Example 2. In [1] the authors study a K3-surface S that we can define by an equation w2 = f(x,y,z)
where
f(x,y,z) =x6+6x5y+12x5z+x4y2+22x4yz+28x3y3 38x3y2z+46x3yz2+4x3z3+24x2y4
−
4x2y3z 37x2y2z2 36x2yz3 4x2z4+48xy4z 24xy3z2+34xy2z3+4xyz4
− − − − −
+20y5z+20y4z2 8y3z3 11y2z4 4yz5.
− − −
Then [1, Prop. 11] gives a non-trivial element of Br(S)[2] as a quaternion algebra of parameter (a,b)
with
a=x2+14xy 23y2 8yz
− −
and
b=b b =(x 4y z)(3x3+2x2y 4x2z+8xyz+3xz2 16y3 11y2z 8yz2 z3).
1 2
− − − − − − −
Let X be the 3-dimensional scheme in A5 given by the equations
1
f(x,y,1) =w2
X : .
1 a(x,y,1)u2+b(x,y,1)v2= 1
(cid:26)
We can apply Theorem 0.1 to the affine scheme X . The primes of bad reduction for the surface S are
1
given in [1, Rk. 12]. Building X does not give rise to more primes of bad reduction. Hence X has good
1 1
reduction at the prime 7. We compute
N (7)=584=0 [mod 7],
X1 6
and we deduce dens p ,p∤N (p) >0.
inf X
{ ∈P }
The threefold X is smooth over C. Hence Corollary 0.2 holds for X . In particular X has non-zero
1 1 1
A-number.
Let D be the product of the elements of Σ . Let f be a function on X , there exists a constantC, and
X1 1
a closed subscheme X A5 of relative dimension at most 3 such that for every p / Σ , for every
2 ⊂ Z[1/D] ∈ X1
α 1, for every non-trivial additive character ψ of F and for every h (A5 X )(F ) one has
≥ pα ∈ Z[1/D]− 2 pα
(cid:12) ψ(f(x)+ hixi)(cid:12)≤Cp32α.
(cid:12)(cid:12)x∈XX1(Fpα) Xi (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
10 LUCILEDEVIN
3. Value at 1 of Frobenian functions
Partof Theorem 1.1 also deals with the value at 1 of Z-linear combinations of functions of type (1). We
canuse the value at1 (see (2)) ofthe function M to ensurethat the setE (M ) (defined by (3)) is not
X a,m X
empty. We deduce the following variation of Theorem 0.1.
Theorem 3.1. Let X be as in Theorem 0.1, and let a ,...,a Z. Denote by M and b (X,a ) the
1 n ∈ X,ai ± i
functions and bounds obtained by applying Theorem 2.1, respectively. If for every i one has
(5) max(M (1) b (X,a ), M (1) b (X,a )) b (X,a ) b (X,a )
| X,ai − − i | | X,ai − + i | ≥ + i − − i
then
n
dens p ,N (p) a [mod p] >0.
inf X i
{ ∈P 6≡ }
i=1
\
Proof. For ease of notation, we present the proof in the case n = 1. For the general case one can use the
fact that the finite intersection of Frobenian sets is still a Frobenian set. As in the proof of Theorem 0.1,
all we need is to prove that there exist k ∈Z, and mk large enough such that the set Emk,0(MX,a+kfA1)
is not empty and up to discarding a finite set of primes it yields a subset of p ,N (p) a [mod p] .
X
{ ∈P 6≡ }
Suppose that
max(M (1) b , M (1) b )= M (1) b ,
X,a − X,a + X,a −
| − | | − | | − |
and choose k = b . One has for every large enough p
−
−
(6) 0<MX(p) b−fA1(p)<(b+ b−)p.
− −
Also evaluating the function at 1 we obtain M (1) b . Set m = M (1) b b b . Then by
X − − −b− | X − −| ≥ +− −
Theorem 1.1 the set Em−b−,0(MX,a −b−fA1) is not empty, hence it has positive density. By (6), (up to
discarding a finite set of primes) it is a subset of p ,N (p) a [mod p] .
X
Setting k = b , we see that the proof is simil{ar∈ifPM (1) 6≡b < M } (1) b . (cid:3)
+ X,a − X,a +
− | − | | − |
Let us now consider cases where the values of M (1), b (X),b (X) are given and satisfy (5). Then
X − +
contrary to Theorem 0.1 finding a prime p is not required. The case of irreducible curves is particularly
0
easy.
Corollary 3.2. Let C be an absolutely irreducible curve over Z. For every a Z χ (C) 1 , one has
c
∈ −{ − }
dens p Σ ,p∤(N (p) a) >0.
inf C C
{ ∈P − − }
Remark 5. More precise results are already known if the smooth projective model of C has genus g < 2.
If g =0, then C is a rational curve and N (p) is easy to compute for all p. If g = 1, and C has a rational
C
point, then its smooth projective model is an elliptic curve, and dens p ,p ∤ N (p) a is very well
C
{ ∈ P − }
understood thanks to the Sato–Tate Conjecture (which is now a theorem).
Proof of Corollary 3.2. The curve C is irreducible and defined over Z, hence the Lang–Weilbound ensures
that for large enough p we always have
0<N (p)<2p.
C
We take M =N a, b =0 and b =2. The condition (5) in Theorem 3.1 is now
C C − +
−
max(χ (C) a, χ (C) a 2) 2.
c c
| − | | − − | ≥
(cid:3)
The easiest case besides irreducible curves is the case of irreducible smooth surfaces.