The geometric sieve and the density of squarefree values of invariant polynomials

0.35 MB·

by  Manjul Bhargava

The geometric sieve and the density of squarefree values of
invariant polynomials
Manjul Bhargava
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February 4, 2014
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n Abstract
a
We develop a method for determining the density of squarefree values taken by certain
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multivariate integer polynomials that are invariants for the action of an algebraic group on
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a vector space. The method is shown to apply to the discriminant polynomials of various
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prehomogeneous and coregular representations where generic stabilizers are finite. This has

] applicationstoanumberofarithmeticdistributionquestions,e.g.,tothedensityofsmalldegree
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number fields having squarefree discriminant, and the density of certain unramified nonabelian
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extensions of quadratic fields. In separate works, the method forms an important ingredient in
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h establishing lower bounds on the average orders of Selmer groups of elliptic curves.
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a
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1 Introduction
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1 The purpose of this article is to develop a method for determining the density of squarefree values
v taken by certain multivariate integer polynomials that are invariants for an algebraic group acting
1
on a vector space. In the case of general polynomials in one or two variables having degree at most
3
0 three or six, respectively, methods of Hooley [28] or Greaves [26], respectively, may be applied;
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in other cases, if the degree of the polynomial is quite small relative to the number of variables,
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2 then the circle method may be used to extract squarefree values of the polynomial in question. In
0
contrast, our method may be applied to polynomials of high degree—even when the degree and
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1 the number of variables are comparable—so long as the polynomial has some extra structure, such
: as symmetry under the action of a “suitably large” algebraic group defined over Z (this condition
v
i will be made more precise in Section 2).
X
r
a 1.1 The density of number fields having squarefree discriminant
The most classical specific cases of arithmetic interest that our method addresses is that of deter-
mining the density of small degree number fields having squarefree discriminant. Building on the
works of Levi [32], Wright–Yukie [42], and Gan–Gross–Savin [24], it was shown in [21], [2], and [3]
that the integers that occur as the discriminants of orders in cubic, quartic, and quintic number
fields, respectively, correspondto suitableinteger values taken bycertain fixedmultivariate integral
polynomials f , f , and f , having degrees 4, 12, and 40 in 4, 12, and 40 variables, respectively.
3 4 5
This correspondence between number field discriminants and integers represented by these special
polynomialswasindeedwhatwasusedin[20],[4],and[5],inconjunctionwithgeometry-of-numbers
arguments, to determine the density of discriminants of cubic, quartic, and quintic number fields,
respectively.
To determine the density of such number fields having squarefree discriminant, we must
thus determine the density of squarefree integer values taken by these special polynomials f , f ,
3 4
1
and f . As we have noted, for general polynomials of large degree d in about d variables, this is an
5
unsolved problem. However, using the structure of these special polynomials—namely, that they
are invariants for the action of a “suitably large” algebraic group—we determine in §4 the density
of squarefree values taken by these polynomials.
Asaconsequence,weprovethatapositivedensityofallS -numberfieldsofdegreesn = 3,4,
n
and5havesquarefreediscriminant,andwedeterminethisdensityprecisely. Wesimilarlydetermine
the density of such number fields that have fundamental discriminant. Specifically, we prove:
Theorem 1.1 Let n = 3, 4, or 5, and let Nsqf(X) (resp. Nfund(X)) denote the number of isomor-
n n
phism classes of number fields of degree n having squarefree (resp. fundamental) discriminant of
absolute value less than X. Then
r (S )
(a) Nsqf(X) = 2 n ζ(2)−1·X +o(X);
n
3n!
r (S )
(b) Nfund(X) = 2 n ζ(2)−1·X +o(X),
n 2n!
where r (S ) denotes the number of 2-torsion elements in the symmetric group S .
2 n n
Note that Theorem 1.1 is true also for n = 2, provided that we count each quadratic field K with
weight 1 (i.e., with weight 1 ). We conjecture that Theorem 1.1 holds for general n.
2 #Aut(K)
In conjunction with the main results of [20], [4], and [5], which give the total density of
discriminants of cubic, quartic, and quintic fields, respectively, we conclude:
Corollary 1.2 When ordered by absolute discriminant, the proportion of S -number fields of de-
n
gree n (n ∈ {2,...,5}) having fundamental discriminant is given by
1 if n = 2;
ζ(2)−1ζ(3) if n = 3;


 ζ(2)−1 (1+p−2−p−3−p−4)−1 if n = 4;
 p
ζ(2)−1Qp(1+p−2−p−4−p−5)−1 if n = 5.



Furthermore, the proportion of S -Qnumber fields of degree n (n ∈{2,...,5}) having squarefree dis-
n
criminant is exactly 2/3 of the proportion having fundamental discriminant.
Both Theorem 1.1 and Corollary 1.2 follow from a general theorem that our methods allow
us to prove, concerning the asymptotic count of S -number fields of degree n ≤ 5 satisfying any
n
desired finite or suitable infinite set of local conditions:
Theorem 1.3 Let n = 2, 3, 4, or 5. Let Σ = (Σ ,Σ ,Σ ,...) denote an acceptable set of local
∞ 2 3
specifications for degree n extensions of Q, i.e., Σ is any subset of (isomorphism classes of) ´etale
ν
degree n extensions of Q for each place ν of Q, such that for sufficiently large primes p, the set
ν
Σ contains all ´etale extensions K of Q of degree n such that p2 ∤ Disc(K /Q ). Let N (X)
p p p p p n,Σ
denote the number of S -number fields K of degree n having absolute discriminant at most X such
n
that K ⊗Q ∈ Σ for all places ν of Q. Then
ν ν
N (X) 1 1 p−1 1 1
n,Σ
lim = · · · . (1)
X→∞ X 2 #Aut(K) p Discp(K) #Aut(K)
(cid:16)KX∈Σ∞ (cid:17)Yp (cid:16)KX∈Σp (cid:17)
2
The above theorem thus allows one to count number fields of degree at most five satisfying very
general sets of local conditions. In particular, it proves a more general version (namely, where we
allow infinitely many local conditions) of the heuristics given in [6, (4.2)].
Since having squarefree or fundamental discriminant is a local condition of the type occur-
ring in Theorem 1.3, Theorem 1.1 will follow from Theorem 1.3 once the sums in the Euler factors
in (1), i.e., the local masses, are computed (see §4 for details).
1.2 Unramified nonabelian (A - and S ×C -) extensions of quadratic fields
n n 2
The density of degree n number fields having squarefree discriminant is directly related to the
distribution of certain unramified nonabelian extensions of quadratic fields. More precisely, given a
finite group G and a quadratic field K, we may consider the set U(K;G) of all isomorphism classes
of unramified G-extensions of K, i.e., Galois extensions of K with Galois group G. An extension
L ∈ U(K;G) is not necessarily normal over Q, and its normal closure over Q has Galois group
G′ ⊂ G≀C = (G×G)⋊C . It is thus natural to partition U(K;G) into the sets U(K;G,G′),
2 2
where U(K;G,G′) denotes the set of all isomorphism classes of unramified G-extensions L of K
such that the Galois closure of L over Q has Galois group G′. If L ∈ U(K;G,G′), then we say that
L is an unramified extension of K of type (G,G′), or simply an unramified (G,G′)-extension.
Theorem 1.4 Let n = 3, 4, or 5, and let E+(G,G′) (resp. E−(G,G′)) denote the average number
of unramified (G,G′)-extensions that real (resp. imaginary) quadratic fields possess, where quadratic
fields are ordered by their absolute discriminants. Then
1
(a) E+(A ,S ) = ;
n n
n!
1
(b) E−(A ,S ) = ;
n n
2(n−2)!
(c) E+(S ,S ×C ) = ∞;
n n 2
(d) E−(S ,S ×C ) = ∞.
n n 2
In other words, the average number of unramified A -extensions (n = 3, 4, or 5) possessed by real
n
or imaginary quadratic fields is positive, and the average number of unramified S ×C -extensions
n 2
is also positive, and in fact infinite! For n = 2, note that Theorem 1 is still true, except that the
constants in (a) and (b) must each be multiplied by 2, again reflecting the fact that a quadratic
extension has two automorphisms.
The case n = 3 in Theorems 1.4(a)–(b) corresponds to abelian (A -) extensions, and is
3
due to Davenport–Heilbronn [20], who obtained these results via the use, in particular, of methods
that amount essentially to class field theory (see [18] for this nice interpretation). The cases n = 4
and n = 5 of Theorems 1.4(a)–(b) are both new, and to our knowledge are independent of and
cannotbetreated byclass field theory. Indeed,they yield information on thedistributionof certain
nonabelian unramified extensions of quadratic fields, namely, those correspondingto the groups A
4
and A ; in particular, the case n = 5 yields information about the distribution of unramified
5
extensions of a quadratic field of a nonsolvable type, namely A . Theorems 1.4(c)–(d) are also new.
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Returning to the statement of Theorem 1.4, it is an interesting question as to which groups
G,G′ lead to quantities E+(G,G′) and E−(G,G′) that exist and are finite, and what their values
are when they are finite. Both possibilities of finite and infinite already occur in Theorem 1.4. In
3
thecase of abelian G, we musthave that G′ = G⋊C (where thenontrivial element of C acts on G
2 2
by inversion). The Cohen–Lenstra heuristics [17] can then be shown to imply that, for G abelian,
1
E+(G,G⋊C ) = , (2)
2
|Aut(G)|·|G|
1
E−(G,G⋊C ) = (3)
2
|Aut(G)|
whenever |G| is odd.
Note that the cases in Theorems 1.4(a)–(b) in which G is abelian occur when n = 3, and
in these cases the values agree with those predicted by (2) and (3). It would be interesting to have
more general heuristics for E±(G,G′) that include both the abelian results and conjectures above
as well as the nonabelian results of Theorem 1.4.
In parts (c) and (d) of Theorem 1.4, it is actually possible to say something more precise;
namely, the methods of Section 4 show that
|U(K;S ,S ×C )| ∼ c+Xlog X; (4)
n n 2 n
0<Disc(K)<X
X
|U(K;S ,S ×C )| ∼ c−Xlog X, (5)
n n 2 n
−X<Disc(K)<0
X
for n = 3, 4, and 5, where c± are certain positive constants which depend on n.
n
1.3 Squarefree values taken by polynomials such as f , f , and f
3 4 5
As we have mentioned, to prove Theorem 1.1, Corollary 1.2, Theorem 1.3, and Theorem 1.4,
one must determine the densities of lattice points in Rm where the values of certain polynomials—
namely, thediscriminantpolynomialsf ,f ,orf —aresquarefree. Ingeneral, countingthenumber
3 4 5
of lattice points of bounded height where a polynomial takes squarefree values is an unsolved
problem, although conjecturally it is easy to guess what shouldhappen. Namely, if f(x ,...,x ) is
1 m
any squarefree polynomial over Z then, barring congruence obstructions, one expects that f takes
infinitely many squarefree values on Zm. More precisely, one expects
#{x ∈ Zm∩[−N,N]m : f(x) squarefree}
lim = (1−c /p2m), (6)
N→∞ (2N +1)m p
p
Y
where,foreach primep,thequantityc isthenumberofelementsx ∈(Z/p2Z)m satisfyingf(x) = 0
p
in Z/p2Z.
When m = 1, this assertion is relatively easy to prove in degrees ≤ 2, while for cubic
polynomials it was proven by Hooley [28]. For degrees ≥ 4, it appears that no single example is
known of a univariate irreducible polynomial f satisfying (6)! As for polynomials in more than one
variable, Greaves has shown that (6) holds for all binary forms of degree at most 6.
Conditionally, Granville [25] showed that (6) follows, for all univariate polynomials of any
degree, from the ABC Conjecture. More recently, Poonen [34] proved that the ABC Conjecture
implies that (a slightly weaker version of) equation (6) is true also for all multivariate polynomials.
In this article, we give three special examples of polynomials f for which we can prove
unconditionally that (6) holds; namely, these are the three polynomials that we use to prove
Theorem 1.1, Corollary 1.2, and Theorems 1.3–1.4. More precisely, let f (= f , f , or f ) denote
3 4 5
the primitive integral polynomial that generates the ring of invariants for:
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(i) the action of SL (C) on Sym (C2), the space of binary cubic forms over C;
2 3
(ii) the action of SL ×SL (C) on C2⊗Sym (C3), the space of pairs of ternary quadratic forms
2 3 2
over C; or
(iii) the action of SL ×SL (C) on C4⊗∧2(C5), the space of quadruples of 5×5 skew-symmetric
4 5
matrices over C,
respectively. Thenfor(i),(ii),or(iii),f isapolynomialofdegreeminmvariables,wherem = 4,12,
or 40, respectively (see [38], or see [2]–[3] for explicit constructions of these invariant polynomials).
We prove:
Theorem 1.5 The polynomials f in (i)–(iii) above are each irreducible over Q¯ and are of degree m
in m variables, where m = 4, 12, and 40 respectively. Moreover, for each of these polynomials f,
we have
#{x ∈ Zm∩[−N,N]m :f(x) squarefree} 2
lim = (1−c /p2m) = ζ(2)−1,
N→∞ (2N +1)m p 3
p
Y
where c is the number of elements x ∈ (Z/p2Z)m satisfying f(x)= 0 in Z/p2Z.
p
For these three discriminant polynomials f, particularly in the cases (ii) and (iii) where the
degrees are large (≥ 4) in each individual variable (and the number of variables is equal to the
degree), we do not believe that any of the previously known unconditional results and methods as
described above would apply. Thus these f give new examples of polynomials satisfying (6). It is
interesting to note that the density of squarefree values taken by each of these three discriminant
polynomials f is exactly 2ζ(2)−1, independent of f.
3
1.4 Squarefree values of discriminants of genus one models
The method, which we will describe more axiomatically in the next subsection and in Section 2,
may also be applied to various other polynomials that are invariant under the action of a suitably
large algebraic group defined over Z. Another family of classical examples on which the method
applies are the discriminant polynomials of models of genus one curves.
There are many such models of genus one curves of interest. Genus one curves with maps
to P1, P2, P3, or P4, viacomplete linear systems of degrees 2, 3, 4, or 5, arecalled genus one normal
curves of degree 2, 3, 4, or 5, respectively. They can be realized as: a double cover of P1 ramified
at four points; a cubic curve in P2; the intersection of a pair of quadrics in P3; or the intersection of
fivequadricsinP4 arisingas the4×4sub-Pfaffiansof a5×5skew-symmetric matrix of linear forms
on P4. A genus one model of degree one may be viewed simply as an elliptic curve in Weierstrass
form. (See, e.g., [23] for a beautiful exposition.)
We note that we may also consider genus one models in products of projective spaces. For
example, a genus one curve in P1×P1 is cut out by a bidegree (2,2)-form on P1×P1; and a genus
one curve in P2×P2 is similarly cut out by three bidegree (3,3)-forms on P2×P2. These cases will
be carried out in more detail in [10]. (See [9] also for other examples of such spaces of genus one
models.)
For all these genus one models over Z, we show that the discriminant polynomials of these
genus one curves all take the expected (positive) densities of squarefree values. (Recall that the
discriminantofagenusonemodelisthepolynomialwhosenonvanishingisequivalenttothesmooth-
ness of the corresponding genus one curve.) For genus one models of degree one, i.e., Weierstrass
5
elliptic curves y2 = x3 +Ax+B, the result is easy, as the discriminant polynomial −4A3 −27B2
is only of degree 2 as a polynomial in B. For higher degree genus one models, the result is much
more difficult to obtain.
Moreprecisely,letg(whichwewilldenotebyg ,g ,g ,g ,respectively)denotetheprimitive
2 3 4 5
integral discriminant polynomial of any of the following representations:
(i) the action of SL (C) on Sym4(C2), the space of binary quartic forms over C;
2
(ii) the action of SL (C) on Sym3(C3), the space of ternary cubic forms over C;
3
(iii) the action of SL × SL (C) on C2 ⊗Sym2(C4), the space of pairs of quaternary quadratic
2 4
forms over C;
(iv) the action of SL ×SL (C) on C4⊗∧2(C5), the space of quintuples of 5×5 skew-symmetric
5 5
matrices over C,
respectively. Then the discriminant polynomial g on each of these representations detects stable
orbits, i.e., g does not vanish precisely when the orbit is closed and has finite stabilizer. The
discriminant g of an element in any of these representations also corresponds to the discriminant
of the associated genus one model, i.e., g does not vanish precisely when this associated genus one
curve is smooth.
The dimensions of the representations in (i)–(iv) above are given by 5, 10, 20, and 50,
respectively, while the degrees of the corresponding discriminant polynomials are given by 6, 12,
24, and 60, respectively (see [23] for explicit constructions of these invariant polynomials). Then
we prove:
Theorem 1.6 The polynomials g in (i)–(iv) above are each irreducible. Moreover, for each of these
polynomials g, we have
#{x ∈ Zm∩[−N,N]m :g(x) squarefree}
lim = (1−c /p2m)
N→∞ (2N +1)m p
p
Y
where c denotes the number of elements x ∈ (Z/p2Z)m satisfying g(x) = 0 in Z/p2Z.
p
Thusapositivedensityof genusonemodelsover Zmappinginto P1, P2, P3,or P4, havesquarefree
discriminant. Inparticular, apositive density of binaryquarticformsover Z,and apositive density
of ternary cubic forms over Z, have squarefree discriminant.
These results, and the methods behind them, play an important role in establishing lower
bounds on the average sizes of Selmer groups of families of elliptic curves in [11, 12, 13, 14] and
in [10]. They also play a key role in proving that the local–global principle fails for a positive
proportion of plane cubic curves over Q (see [8]).
1.5 Method of proof
Let f be an integral polynomial on V(Z) ∼= Zm. As is standard in squarefree sieves (see, e.g., §3.4
for more details), the equality (6) can be proven for f whenever sufficiently good upper bounds
on sums involving w (f,H) are obtained, where w (f,H) denotes the number of points v ∈ V(Z)
p p
having height at most H (= the maximum of the absolute values of the coordinates) satisfying
p2 | f(v). It is natural to partition the set W = W (V) ⊂ V(Z) of elements v ∈ V(Z) such that
p p
p2 | f(v) into two sets: W(1), consisting of elements v ∈ V(Z) on which f vanishes modulo p2 for
p
6
“mod p reasons”, i.e., f(v′) ≡ 0 (mod p2) for any v′ ≡ v (mod p); and W(2), consisting of the
p
elements v ∈ V(Z) on which f vanishes modulo p2 for “mod p2 reasons”, i.e., there exist v′ ≡ v
(mod p) such that f(v′)6≡ 0 (mod p2).
(1) (2) (i)
As a consequence, we may write w (f,H) as a sum w + w , where w denotes the
p p p p
(i)
portion of the count of elements in W coming from W . It is well-known that good estimates
p p
(1)
on the relevant sums involving w can be obtained by “geometric sieve” or “closed-point sieve”
p
methods (the latter terminology is due to Poonen), as introduced in the work of Ekedahl [22]; see
also Poonen [33, 34] for a very clear treatment. We will prove a precise and quantitative version of
Ekedahl’s sieve estimates in §3.2, which will be useful in the applications.
The difficulty in squarefree sieves for values taken by integral polynomials thus arises in
(2)
the estimation of sums involving w . It is essentially here that Granville [25] and Poonen [34]
p
use the ABC Conjecture to obtain the desired estimates. For the polynomials arising in Theo-
rems 1.5 and 1.6, we sidestep the use of the ABC Conjecture by using instead the invariance of
these polynomials under the action of an algebraic group G defined over Z. Specifically, for the
(2)
polynomials f arising in Theorem 1.5, we show that for any element v ∈ W , there always exists
p
an element γ ∈ G(Q) such that γv ∈ V(Z) and f(γv) = f(v)/p2. Together with estimates from
the geometry-of-numbers in [19, 4, 5] giving uniform upper bounds on the number of “irreducible”
G(Z)-classes on V(Z) having bounded absolute discriminant, this is sufficient to obtain the desired
(2)
upper bounds on w .
p
With a related construction, for all but one of the polynomials g arising in Theorem 1.6
(2)
we show that for any element v ∈ W , there always exists an element γ ∈ G(Q) such that
p
(1)
γv ∈ W and f(γv) = f(v); i.e., via the action of G(Q), we turn v ∈ V(Z) on which f vanishes
p
modulo p2 for mod p2 reasons into v′ for which f vanishes modulo p2 for mod p reasons! As before,
we combine this construction with estimates from the geometry-of-numbers as in [11, 12, 13, 14],
which give uniform upper bounds on the number of “irreducible” G(Z)-classes on V(Z) having
(2)
bounded absolute discriminant, to deduce the desired upper bounds on w .
p
In Case (i) of Theorem 1.6, however, this argument does not work; we find that the
group G(Q) in this case is just too small to do the job. We get around this problem via a fur-
ther argument that we call the “embedding sieve”. Namely, we find a representation G′ on V′,
defined over Z, and an invariant polynomial f′ for this action, such that: there is a map of orbits
φ : G(Z)\V(Z) → G′(Z)\V′(Z), having preimages of absolutely bounded cardinality, for which
f′(φ(v)) = f(v). Furthermore, we choose (G′,V′) such that G′(Q) is sufficiently larger than G(Q),
while the set of irreducible orbits of G′(Z) on V′(Z) is not too large; this allows one to obtain
an estimate w(2)′ on V(Z)′, which then leads to a good estimate also for w(2). Amusingly, in the
p p
case of g in Theorem 1.6, we embed (G,V) into the representation (G′,V′) corresponding to the
2
polynomial f in Theorem 1.5!
4
Indeed, the latter argument (which will be described in more detail in §5) shows that the
method of this paper may in fact be applied to some polynomials f that do not have a very large
group of symmetries; in such cases, we simply attempt to arrange a suitable embedding where the
method does apply to give the desired estimates. Although we only apply this embedding sieve in
one case in this paper, it will serve as a starting point in a sequel to this paper where we study
squarefree values of more general polynomials that may have fewer symmetries.
Thispaperis organized as follows. InSection 2, we enumerate anaturalset of axioms on an
integral multivariate polynomialf which is sufficient to deducethat f takes theexpected density of
squarefreevalues(i.e., f satisfies(6)). InSection3,wethenprovethelatterassertion,bydeveloping
the geometric sieve method that we use to extract squarefree values of such polynomials satisfying
7
these axioms. Finally, in Sections 4 and 5, we then prove Theorems 1.5 and 1.6, by proving that
all but one of the polynomials occurring in these theorems satisfy the axioms of Section 2. For the
remaining polynomial g , we describe an extension of these axioms (the “embedding sieve”) that
2
allows us to prove (6) also for this polynomial.
2 Some general criteria for extracting squarefree values of invari-
ant polynomials
LetV bearepresentationofanalgebraicgroupGdefinedoverZ,andletf beaninteger polynomial
of degree d that is a relative invariant for the action of G on Z and whose squarefree values we
wish to extract. Let m := dim(V). We use G1 to denote the kernel of the determinant map
G → GL(V)→ G .
m
Suppose f, G, and V have the following properties:
1. There is a notion of a generic element of V(Z); the subset V(Z)gen of generic elements in
V(Z) is G(Z)-invariant, and satisfies
#{x ∈ V(Z)gen∩[−N,N]m}
µ(V(Z)gen) := lim = 1.
N→∞ (2N +1)m
2. Theorderofthestabilizer inG(Q¯)ofanyelementinV(Z)gen isfiniteandabsolutelybounded.
3. There is a continuous (but not necessarily polynomial) invariant I for the action of G1(Z) on
V(Z) that is homogeneous of degree d, i.e., I(λv) = λdI(v).
4. There is a fundamental domain F for the action of G1(Z) on V(R) such that the region
F := {v ∈ F :|I(v)| < X}is measurableandhomogeneously expanding, i.e., F =X1/dF ,
X X 1
and the volume Vol(F ) of F is finite.
X X
5. For any subset S of V(Z) defined by congruence conditions modulo finitely many prime
powers, we have
N(S;X) := #{v ∈ S ∩F generic} =Vol(F )· µ (S)+o(Xm/d), (7)
X X p
p
Y
where µ (S) denotes the density of the p-adic closure of S in V(Z ).
p p
6. Fix a prime p. If v ∈ V(Z)gen is an element such that f(v) is a multiple of p2, then there is a
nonnegative real number a = a , an absolutely bounded integer k = k ≥ 0, and an element
v v
g = g ∈G(Q) such that
v
(i) |I(gv)| = p−a|I(v)|;
(ii) the element gv lies in V(Z)gen in the reduction (mod p) of a closed G-invariant sub-
scheme Y of V (viewed as affine n-space) defined over Z, depending only on k, that has
k
codimension ≥ k;
(iii) for each fixed k, every point of Y (Z) arises as g v for some v ∈ V(Z)gen at most c times
k v
up to G(Z)-equivalence, where c is an absolute constant;
m
(iv) ·a+k−1 is bounded below by an absolute positive constant η.
d
8
Theorem 2.1 If f, G, V satisfy Conditions 1–6, then f takes the expected density of squarefree
values, i.e., f satisfies (6).
While Conditions 1–6 may seem very restrictive, we will see in Sections 4 and 5 that they
are satisfied by all but one of the polynomials in Theorems 1.5 and 1.6 (and, indeed, by many
other polynomials, e.g., by a number of the discriminant polynomials occurring in [9]). In general,
the notion of generic in Condition 1 is chosen so that the cusps of the fundamental domain F in
Condition 4 contain mostly non-generic points. Indeed, the integral points in the cusps of such
fundamental domains F tend to lie primarily on certain subvarieties; the lattice points in V(Z)
that lie outside the union of these subvarieties are then called generic. Condition 2 is of course
common, and will generally be satisfied in any representation that has stable orbits in the language
of geometric invariant theory. With Conditions 1–4 satisfied, Condition 5 can then be proven using
geometry-of-numbers methods (as developed, e.g., in the works [19, 4, 5, 11]).
Finally, Condition 6 is also true for all the representations and polynomials in Theorem 1.5
and all but one of the representations and polynomials in Theorem 1.6. However, it is not true
for the very first polynomial g in Theorem 1.6. In general, Condition 6 can be quite restrictive
2
(as opposed to Conditions 1–5), because there may not be enough symmetries in G to satisfy the
condition. In such cases, we may attempt to embed V into a larger representation that has more
symmetries for which (a suitable version of) Condition 6 is satisfied! This argument indeed works
for the remaining representation, and will be important in future applications.
Remark 2.2 In the course of proving Theorem 2.1, we will also show that the polynomials f in
this theorem—in addition to satisfying (6)—also satisfy
#{x ∈ F ∩V(Z)gen : f(x) squarefree }
lim X = (1−c /p2m) (8)
X→∞ #{x ∈ FX ∩V(Z)gen} p
p
Y
where again c denotes the number of elements x ∈ (Z/p2Z)m satisfying f(x)= 0 in Z/p2Z.
p
3 A geometric squarefree sieve
In this section, we describe the geometric sieve method that we use to extract squarefree values of
polynomials.
In §3.1–3.2 (see in particular Theorem 3.3), we lead up to a quantitative version of a certain
uniformity estimate due to Ekedahl [22] (see also Poonen [33, 34] and Poonen–Stoll [35]). Ekedahl
shows that, in appropriate situations, the usual inclusion–exclusion tail becomes “negligible” as
the cut off defining the tail gets larger and larger. For our applications here as well as in future
applications, we require precise quantitative versions of these tail estimates (i.e., how negligible
is “negligible”?), when counting lattice points in homogeneously expanding regions. Given any
variety of codimension at least 2 defined over Z, these estimates will, in particular, yield a method
for sieving out those lattice points that, for some sufficiently large p, reduce (mod p) to a point on
the reduction of that variety (mod p). The quantitative versions of the relevant tail estimates that
we prove in §3.2 enable one also to obtain second order terms or power-saving error terms in the
applications.
In this article, we are particularly concerned with sifting out those lattice points on which
a given polynomial takes non-squarefree values. The application to this scenario is described in
Subsection 3.3. In the final Subsection 3.4, we then prove Theorem 2.1 of Section 2, namely, that
anyintegralpolynomialf satisfyingtheaxiomsofSection2takestheexpectednumberofsquarefree
values.
9
3.1 The number of lattice points in a homogeneously expanding region lying
on a subvariety
We start withthe following simpleand oft-used lemmathat states that thenumberof lattice points
on a given variety in a homogeneously expanding region in Rn grows at most polynomially in the
linear scaling factor, where the degree of the polynomial is the dimension of the variety. Though
this result is well-known, we include a proof here for completeness, and as a preparatory ingredient
for the sieve estimates in §3.2.
Lemma 3.1 Let B be a compact region in Rn having finite measure. Let Y be a variety in Rn of
codimension k ≥ 1. Then we have
#{a ∈ rB∩Y ∩Zn} = O(rn−k), (9)
where the implied constant depends only on B and on Y.
Proof: We may clearly assume that Y is irreducible, for otherwise we could simply sum over the
irreducible components of Y. Since Y has codimension k in Rn, there exist polynomials f ,...,f
1 k
for which
Y ⊆ Y′′ := {a ∈ Rn|f (a) = f (a) = ··· = f (a) = 0} (10)
1 2 k
such that the irreducible component Y′ of Y′′ containing Y also has codimension k.
We prove the estimate of Lemma 3.1 for Y′ in place of Y, by induction on n, using the
polynomials f ,...,f . We always write the f as polynomials in the arguments x ,...,x . For the
1 k i 1 n
proof, we may clearly assume that each f is irreducible, for otherwise we can simply replace each
i
f by the irreducible factor of f that vanishes on Y′. If all the f do not involve some variable,
i i i
say x , then the result follows by the induction hypothesis. So we may assume that every variable
n
x ,...,x occurs in at least one f .
1 n i
We now show, via elimination theory, that we may reduce to the case where k−1 of the f ,
i
say f ,...,f , all do not involve some fixed variable, say x . Indeed, by reordering the f if
1 k−1 n i
necessary, let us assume that f is nonconstant as a polynomial in x . Let R (f ,f ) denote the
k n n i k
resultant of f and f with respect to x . Since R (f ,f ) = A f + B f for some polynomials
i k n n i k i i i k
A and B with A nonzero, the irreducible component containing Y in the variety cut out by
i i i
R (f ,f ),...,R (f ,f ),f (= the variety cut out by A f ,...,A f ,f ) is still Y′. Thus
n 1 k n 1 k−1 k 1 1 k−1 k−1 k
we may simply replace each f involving x (for i ∈ {1,...,k−1}) by R (f ,f ), and we see that
i n n i k
the irreducible component containing Y of the new Y′′ cut out by the new f is still the variety Y′
i
of codimension k, where now f ,...,f do not involve x .
1 k−1 n
Thus it suffices to prove the lemma when f ,...,f are polynomials only in x ,...,x .
1 k−1 1 n−1
Let h denote the leading coefficient of f as a polynomial in x , so h is a polynomial in
k k n k
x ,...,x . We may assume that h does not vanish on Y′, for otherwise we might as well
1 n−1 k
eliminate the leading term of f , and f ,...,f would still cut out a variety Y′′ whose irreducible
k 1 k
component containing Y is Y′. Let Z be the union of the irreducible components intersecting
Y′ ∩ {h = 0} of the variety cut out by f ,...,f ,h in Rn−1. Then Z is of codimension k
k 1 k−1 k
in Rn−1.
We now partition Zn into two sets of points: those on which h vanishes andthose on which
k
it does not. For the set of points where h vanishes, we have
k
#{a ∈ rB∩Y ∩Zn|h (a) = 0} = O(rn−1−k)·O(r) = O(rn−k), (11)
k
10