Simultaneous zeros of a Cubic and Quadratic form

0.18 MB·English

by  Jahan Zahid

Simultaneous zeros of a Cubic
and Quadratic form
0
1
0
Jahan Zahid
2

n
a
J
1 Introduction

7

Consider a system of forms
]
T
N F(x)=(F1(x),...,Fr(x))
.
h of degrees d ,...,d respectively in the variables x=(x ,...,x ) over a p-adic
1 r 1 n
at field K. It had been conjectured by Artin [1, Preface] that F necessarily has a
m non-trivial zero in K provided n > d2. It should be noted that there exist
i
[ systems with n = d2i which havePonly the trivial p-adic zero, so this is the
best we can hope foPr.
1
Artin’s conjecture has been verified in only a handful of cases. For example
v
it is a classical result due to Hasse [8] that every quadratic form with n > 4
5
5 variables has a non-trivial p-adic zero. The case of cubic forms was settled
0 independently by Dem’yanov [6], Lewis [10] and Springer [14]. Dem’yanov [7]
1 and later Birch, Lewis & Murphy [3] proved the conjecture for a system of two
.
1 quadratic forms. The purpose of this paper is to prove the next case of the
0 conjecture for two forms, provided we have a large enough residue class field.
0 More precisely we shall prove
1
: Theorem 1. Any system of a cubic and quadratic form in at least 14 variables
v
i definedoverK,hasanon-trivialzeroinK providedthecardinalityoftheresidue
X
class field exceeds 293.
r
a
It should be noted that Artin’s conjecture was shown to be false in general
byTerjanian[15],whofoundacounterexampleofaquarticformin18variables
with no zero in Q . If however K : Q = e and d = (d ,...,d ), then by a
2 p 1 r
| |
remarkabletheoremofAx&Kochen[2],thereexistsanintegerp(d,e)suchthat
any system F with n> d2 has a non-trivial zero provided the characteristic
i
of the residue class fieldPexceeds p(d,e).
We remark that Theorem 1 is stronger than anything we can deduce from
the Ax–Kochen theorem for a number of reasons. Firstly we have an explicit
boundonthecardinalityoftheresidueclassfieldforwhichArtin’sconjectureis
true. Secondly we have a condition depending on the cardinality of the residue
class field, rather than the characteristic. Consequently we are now able to say
thatArtin’sconjectureholdsforacubicandquadraticformoveranyunramified
extension of Q of degree at least 9. Where it was not possible to make this
p
deduction before.
As an outline to prove Theorem 1 we shall generalise a p-adic minimization
proceduredue toSchmidt[13]toholdforsystemsofformsofarbitrarydegrees.
1
WeshallthenderivesomeGeometricinformationofthesystemovertheresidue
classfield, forthosesystems whichterminateinthe minimizationprocess. This
will allow us to find a non-singular zero in the residue class field to which we
can apply Hensel’s Lemma.
Acknowledgments: This work forms part of the authors doctoral thesis
atthe UniversityofOxford. Iverygratefullyacknowledgethe financialsupport
I received from EPSRC.
I would also like to thank my supervisor Prof. Roger Heath-Brown, for
suggesting this problem and his excellent guidance over the last few years. I
have also benefited from numerous conversations with Dr. Damiano Testa and
Prof. Trevor Wooley, to whom I am glad to express my gratitude.
2 Some preliminaries
Let denote the ring of integers of K, and denote the residue class field by
K
O
F . Letπ denote a uniformizer for . If α K 0 ,we may write α=πsu,
q K
O ∈ −{ }
where u is a unit in . We define the π-adic order v() by setting v(α) = s.
K
O ·
We also define the π-adic valuation by setting α = p−s, where p denotes
|·| | |
the characteristic of the residue class field F . Recall that
q
F (x),...,F (x) K[x]
1 r
∈
denotes an arbitrary system of forms of degrees d ,...,d in the variables
1 r
x = (x ,...,x ) over K. We shall assume that d ... d unless we state
1 n 1 r
≥ ≥
otherwise and for brevity write the above system of forms as F. We are inter-
estedindetermining the existence ofapointx Kn 0 suchthatF(x)=0.
∈ −{ }
Clearly we may assume that the coefficients of the forms F and the variables x
are in , since this does not affect existence of a zero.
K
O
ByaslightabuseofnotationwewriteGL(n, )todenotethesetof(n n)-
K
O ×
matrices over with non-zero (rather than unitary) determinant. So let
K
O
τ GL(n, ) and write F to denote F(τx) = (F (τx),...,F (τx))t. We
K τ 1 r
∈ O
also write T =(t ) to denote the (r r) upper triangular matrix with entries
ij
×
t (x)=π−ciG (x), (1)
ij ij
where c 0 and G [x] to denote an arbitrary form of degree
i ij K
≥ ∈O
degF degF 0,
i j
− ≥
for 1 i<j r and define the diagonal terms G :=1.
ii
≤ ≤
Let Ω=(ω ,...,ω ) where ω >0 for each i. Then we write
1 r i
F F′,
≻Ω
if F and F′ are both defined over and
K
O
F′ =TF
τ
with
c ω s>0 (2)
i i
−
X
2
where s = v(detτ) and the c are as in (1). If F′ = TF , it is clear that if F′
i τ
has a zero if and only if F has a zero.
We say that F is Ω-bottomless if there is an infinite chain
F F(1) F(2)
≻Ω ≻Ω ≻Ω···
otherwise, F will be called Ω-bottomed. We also say that F is Ω-reduced if
there does not exist any F′ such that
F F′.
≻Ω
WesaythattwosystemsFandF′ areequivalent,ifbothsystemsaredefined
over and
K
O F′ =TF
τ
where c =0 for all 1 i r and v(detτ)=0 in T and τ as in (1). The order
i
≤ ≤
o(F), of a system F is the least positive integer m such that F is equivalent to
asystemthatcontainsm variablesexplicitly. We alsodefine the h-invariantfor
a system F, denoted h(F) as the least integer h such that we can write
F (x)=x H (x)+...+x H (x) (mod π), (3)
i 1 i1 h ih
for all 1 i r and all systems equivalent to F. Note that since the F are
i
≤ ≤
defined over , considering them modulo π is well defined.
K
O
Given any set S = e ,...,e of positive integers we define v as the least
1 s S
{ }
integerv suchthat everysystemconsisting ofs forms ofdegrees e ,...,e have
1 s
a non-trivialzero providedthe number of variables in the system is at least v .
S
If φ denotes the empty set we define v :=1. We remark here that it is due to
φ
a classical theorem of Brauer [4], the number v is always finite.
S
Although the next theorem is likely to have further applications, it will for
the purpose of this paper play a crucial part in the minimisation procedure for
a system of a cubic and quadratic form.
Theorem 2. Let S d ,...,d denote any subset of cardinality r 1 with
1 r
⊂ { } −
indexing set I such that v is maximal. Let j I then provided
S
6∈
n v +d2 (4)
≥ S j
there exists some Ω = (ω ,...,ω ) such that ω > d for each 1 i r and
1 r i i
≤ ≤
such that every Ω-bottomless system F defined over has a non-trivial p-adic
K
O
zero.
3 Proof of Theorem 2
Since the field K has characteristic 0, given any form F of degree d there is
a unique form M (x ,...,x ) which is linear in each vector x and which is
F 1 d j
symmetric in x ,...,x , such that
1 d
F(x)=M (x,...,x).
F
Lete =(1,0,...,0),e ,...,e beunitvectors. WesaythatF=(F ,...,F )
1 2 n 1 r
is Ω-special if there are non-negative integers a ,...,a and b ,...,b with
1 n 1 r
a + +a <ω b + +ω b (5)
1 n 1 1 r r
··· ···
3
such that
M (e ,...,e )=0 (6)
Fi j1 jdi
for each 1 i r and d -tuple (j ,...,j ) for which,
≤ ≤ i 1 di
a + +a <b . (7)
j1 ··· jdi i
NotethefollowingimportantcorrespondencebetweenΩ-bottomlesssystems
and Ω-special systems.
Theorem 3. Every Ω-bottomless system is equivalent to a Ω-special system.
We prove this in due course. Now we make use of this to prove Theorem 2.
Proof of Theorem . By Theorem 3 we may suppose that F is Ω-special. For
ease of notation we may assume that
b b
a ... a and 1 ... r, (8)
1 ≤ ≤ n d ≤ ≤ d
1 r
dropping any previous ordering we had on d ,...,d . If there is a subset S
1 r
⊂
d ,...,d with indexing set I such that
1 r
{ }
d a <b for all i I
i vS i 6∈
then F has a non-trivial zero. For if such a subset S exists then by (7) one has
M (e ,...,e )=0 for all i I
Fi j1 jdi 6∈
and every 1 j ,...,j v . Therefore the system (F ) vanishes on the
≤ 1 di ≤ S i i6∈I
v -dimensionalsubspacespannedby e ,...,e andonthis subspacewe can
S { 1 vS}
find a zero of (F ) and therefore a zero of F. Consequently we may assume
i i∈I
thatfor eachS d ,...,d thereexists some i,whichby the ordering(8)we
1 r
⊂{ }
may assume to be min 1 i d:i I such that
{ ≤ ≤ 6∈ }
d a b .
i vS ≥ i
We define S := φ and S := d ,...,d for 1 i r 1 and for ease of
0 i 1 i
{ } ≤ ≤ −
notation write w =v for 1 i r. Then it follows that
i Si−1 ≤ ≤
d a b (9)
i wi ≥ i
for every 1 i r. Note that by assumption (4) we have that
≤ ≤
n w +d2 d2+ +d2+1.
≥ r r ≥ 1 ··· r
Moreover for each 0 i r 1 we claim that
≤ ≤ −
w w d2. (10)
i+1− i ≥ i
ForifweletT=(F ,...,F )denoteasystemwithw 1variableswithonly
1 i−1 i
−
the trivial zero and F denote a form in d2 variables with only the trivial zero
i i
withitsvariablesdistinctfromthevariablesinTthenitisclearthatthesystem
4
T (F ) has only the trivial zero. Therefore it follows that w w +d2 as
∪ i i+1 ≥ i i
claimed. By (8), (9) and (10) it follows that
a + +a d2a +d2a + +d2a +a
1 ··· n ≥ 1 w1 2 w2 ··· r wr n
1 1 1
d2+ a + d2+ a + + d2+ a
≥ (cid:16) 1 r(cid:17) w1 (cid:16) 2 r(cid:17) w2 ··· (cid:16) r r(cid:17) wr
(d +ǫ)b +(d +ǫ)b + +(d +ǫ)b
1 1 2 2 r r
≥ ···
taking ǫ = (rd )−1, where d := max d ,...,d . Hence if we let ω =
max max 1 r i
{ }
d +ǫfor1 i r,itfollowsthateveryΩ-specialsystemmusthaveanon-trivial
i
≤ ≤
zero completing the proof of the theorem.
WeshallnowproceedbyprovingTheorem3,generalisingwhereappropriate
the method of Schmidt [13]. Given two systems F and F′ defined over , we
K
O
write
k
F F′ (11)
≻Ω
if k 1 and
≥ F F′
≻Ω
with the condition (2) strengthened to
c ω (s+k) 0.
i i
− ≥
X
The system F shall be called Ω-high if for every k there is a F′ such that (11)
holds. Note the following lemma.
Lemma 1. Suppose F is a Ω-bottomless system, then it is Ω-high.
Proof. Fix a k 1, then if F is Ω-bottomless there exists an infinite chain
≥
F=F(1) F(2) ... F(k) .
≻Ω ≻Ω ≻Ω ≻Ω···
For each m 1 we may write
≥
F(m+1) =T F(m)
m τm
where τ GL(n, ) with v(detτ ) = s and T = (t ) is an (r r)
m K m m m ij,m
∈ O ×
upper triangular matrix with entries
t (x)=π−ci,mG (x),
ij,m ij,m
for 1 i j r where c 0 and G [x] is a form of degree1
i,m ij,m K
≤ ≤ ≤ ≥ ∈O
degF degF 0
i j
− ≥
with G =1. Let Q be any positive integer such that ω 1 for every i, then
ii i ≥ Q
by condition (2) we have that
c ω (s + 1) 0
i,m i− m Q ≥
Xi
1weassumeonceagainthatd1≥d2≥...≥dr
5
for each 1 m r. If T denotes the product of T and τ denotes the product
m
≤ ≤
of τ for 1 m kQ, then we have that F(kQ+1) = TF . Crucially we also
m τ
≤ ≤
have that
c ω s +k 0.
i,m i m
Xi (cid:16)Xm (cid:17) −(cid:16)Xm (cid:17)≥
Therefore
k
F F(kQ+1)
≻Ω
as required.
We shall now note two Lemmata, the proofs of which can be found in
Schmidt’s paper [13, Lemmata 8 and 10].
Lemma 2. Let A ,...,A and B ,...,B be linear forms with integer coeffi-
1 l 1 m
cients in the vector x = (x ,...,x ). Let x ,x ,... be a sequence of vectors
1 n 1 2
with
A (x ) 0 (i=1,...,l; k =1,2,...).
i k
≥
Then there exists a subsequence, y ,y ,... say, a constant B and an integer
1 2
vector a with
A (a) 0 (i=1,...,l)
i
≥
such that
lim B (y )=+ for j withB (a)>0
j k j
k→∞ ∞
and
B (y ) B for j withB (a) 0.
j k j
≤ ≤
Before stating the next lemma we need to introduce some terminology. If
a ,...,a are linearly independent vectorsin Kn, we call the set of linear com-
1 s
binations
c a + +c a
1 1 s s
···
a lattice, where c . We call a ,...,a , a basis of the lattice.
i K 1 s
∈O
Lemma 3. Suppose M is a sublattice of Λ. Then there exists a basis l ,...,l
1 s
of Λ and a basis of m ,...,m of M such that
1 s
m =πa1l ,...,m =πasl
1 1 s s
for some non-negative integers a ,...,a .
1 s
Proof of Theorem 3. SupposeFisΩ-bottomless,thenbyLemma1itisΩ-high.
Hence for every k 1 there are maps T and τ GL(n, ) such that T F
≥ k k ∈ OK k τk
is defined over and
K
O
c ω (s +k) 0 (12)
i,k i k
− ≥
Xi
where v(detτ )=s andT =(t ) is an(r r) upper triangularmatrix with
k k k ij,k
×
entries
tij,k(x)=π−ci,kGij,k(x),
6
for 1 i j r where c 0 and G [x] is a form of degree
i,k ij,k K
≤ ≤ ≤ ≥ ∈O
degF degF 0
i j
− ≥
with G =1. For ease of notation we write the ith row of the vector
ii
diag(πc1,k,...,πcr,k)TkF(x)
as
r
R (x):= G (x)F (x)
i,k ij,k j
Xj=i
for 1 i r. If Λ denotes the lattice τ n then by assumption
≤ ≤ k kOK
π−ci,kRi,k(x) K
∈O
for all 1 i r and every x Λ . By Lemma 3, Λ has a basis
k k
≤ ≤ ∈
Λ :πu1u ,...,πunu (13)
k 1 n
where u ,...,u is a basis of n. Next we let M denote the multilinear
1 n OK Ri,k
forms associated with R for 1 i r, then there exists some fixed non-
i,k
≤ ≤
negative integer γ such that
M (x ,...,x ) π−γ
Ri,k 1 di ∈ OK
for all 1 i r and any x ,...,x Λ . Consequently taking the basis
≤ ≤ 1 di ∈ k
vectors of Λ (13) we get
k
π−ci,kMRi,k(πuj1uj1,...,πujdiujdi)∈π−γOK
or
|MRi,k(uj1,...,ujdi)|≤pγ−(ci,k−uj1−...−ujdi)
for all 1 j ,...,j n.
≤ 1 di ≤
Note that the u ,u ,c and R all depend on k. Also note that since
i i i,k i,k
u ,...,u is a basis of n we must have that det(u ,...,u ) = 1. By
1 n OK | 1 n |
the compactness of n there must exist a subsequence of the sequence of
OK
integers k = 1,2,..., such that on this subsequence u ,...,u tend respec-
1 n
tively to a ,...,a and the forms R ,...,R tend respectively to the forms
1 n 1,k r,k
R ,...,R (x). Tobeclearthesystem(R ,...,R )hasazeroifandonly
1 r K 1 r
∈O
if the system F = (F ,...,F ) has a zero, since G = 1 for each 1 i r in
1 r ii
≤ ≤
R . Moreover we have that det(a ,...,a ) = 1, hence a ,...,a is a basis
i,k 1 n 1 n
| |
of n. There exists a map σ defined over such that
OK OK
σe =a , for 1 i n.
i i
≤ ≤
With each k in our subsequence we define the vector
(A ,...,A )=(u ,...,u ,c ,...,c )
1 n+r 1 n 1,k r,k
We also define
r n
B = c ω u
i,k i j
−
Xi=1 Xj=1
7
and
B =c (u + +u )
i i,k− j1 ··· jdi
for 1 j ,...,j n and 1 i r. We apply Lemma 2 to the forms
≤ 1 di ≤ ≤ ≤
A ,B,B , but before we do this note that the form B tends to + . This is
i j
∞
because s =u + +u and so by equation (12) one has
k 1 n
···
c ω u k.
i,k i j
− ≥
Xi Xj
Now by Lemma 2 there is a vector
a=(a ,...,a ,b ,...,b )
1 n 1 r
with non-negative integer components, such that
ω b + +ω b (a + +a )>0.
1 1 r r 1 n
··· − ···
Moreoverwe havethatB tends to + for allvalues i andj ,...,j forwhich
i ∞ 1 di
b (a + +a )>0. (14)
i− j1 ··· jdi
Taking the limit we obtain
M (a ,...,a )=0
Ri j1 jdi
for all j ,...,j satisfying (14). If we set G = R where R = (R ,...,R ),
1 di σ 1 r
then G satisfies precisely the conditions required to be Ω-special. Finally since
R is equivalent to F, we deduce that F is equivalent to a Ω-special system as
required.
4 Preliminaries for a Cubic and Quadratic form
Throughout this section and subsequent sections F= (F,G) will denote a sys-
temofacubicandquadraticforminnvariablesdefinedovertheringofintegers
of some p-adic field K. Let π denote a uniformizer for and let F be
K K q
O O
the residue class field of K, where q denotes its cardinality.
Note the following corollary of Theorem 2.
Corollary 1. Let (F,G) denote an arbitrary system of a cubic and quadratic
form in n 14 variables. Then there exists some ω >3 and ω >2 such that
1 2
≥
every (ω ,ω )-bottomless system (F,G) has a non-trivial p-adic zero.
1 2
Therefore it follows that to prove Theorem 1 it is sufficient to consider
(ω ,ω )-reduced systems for some ω >3 and ω >2.
1 2 1 2
RecallthatwesaythattwosystemsFandF′ areequivalent,ifbothsystems
are defined over and
K
O F′ =TF
τ
where
1 L
T = ,
(cid:18) 0 1 (cid:19)
for some linear form L [x] and τ GL(n, ) with v(detτ) = 0. The
K K
∈ O ∈ O
8
ordero(F),ofasystemFistheleastpositiveintegermsuchthatFisequivalent
to a system that contains m variables explicitly. Also recall the h-invariant of
F, denoted h(F) is the least integer h such that we can write
F(x)=x H (x)+...+x H (x) (mod π),
1 11 h 1h
and
G(x)=x H (x)+...+x H (x) (mod π),
1 21 h 2h
for all systems equivalent to F. We similarly define the h-invariant of a single
formintheobviousway. NoteherethatsinceFisdefinedover ,considering
K
O
it modulo π is well defined. Note the following lemma which will play a crucial
part in our proof.
Lemma 4. Suppose F = (F,G) is (α,β)-reduced, for some α > 3 and β > 2,
then
h(G)>2, h(F LG)>3 and h(F)>5
−
for every linear form L(x) [x].
K
∈O
Proof. Letτ be the diagonaln nmatrixwhichhas π asits firstr entries and
r
×
1 otherwise. If h(G) 2 then we may write
≤
G=x L +x L (mod π),
1 1 2 2
for some linear forms L [x]. If we let
i K
∈O
1 0
T = ,
(cid:18) 0 π−1 (cid:19)
then F′ = TF is defined over . However β 2 > 0, contradicting our
τ2 OK −
assumption that F is (α,β)-reduced (cf. condition (2), p.2). If h(F LG) 3
− ≤
for some linear form L(x) [x], then we may write
K
∈O
F LG=x Q +x Q +x Q (mod π),
1 1 2 2 3 3
−
for some quadratic forms Q [x]. If we let
i K
∈O
π−1 π−1L
T = − ,
(cid:18) 0 1 (cid:19)
then F′ = TF is defined over . However α 3 > 0, contradicting that F
τ3 OK −
is (α,β)-reduced. Finally if h(F) 5 then we may write
≤
F′ =(F LG,G)=(x Q +...+x Q ,x L +...+x L ) (mod π),
1 1 5 5 1 1 5 5
−
for some quadratic and linear forms Q ,L ,L [x]. This time we let
i i K
∈O
π−1 0
T = ,
(cid:18) 0 π−1 (cid:19)
thenF′′ =TF′ isdefinedover . Howeverα+β 5>0,contradictingthat
τ5 OK −
F is (α,β)-reduced.
9
5 Reduced systems
In this section we will work with our system F=(F,G) modulo π, which from
now on we shall denote as f = (f,g). We assume that F is (α,β)-reduced, for
some α>3 and β >2. Hence f will satisfy the conclusion of Lemma 4 viz.
h(g)>2 h(f lg)>3 and h(f)>5 (15)
−
for every linear form l(x) F [x]. We also denote m=o(f).
q
∈
The aim of this section is to show that we can find a non singular zero of
f which by Hensel’s Lemma will lift to give us a zero of our original system F.
For clarity we outline the steps we will take in order to prove this:
Step 1: We prove that we can find a zero e say, of f such that g =0.
1
∇ 6
Therefore we are able to write our system f in the shape
f(x) = x2f +x f +f
1 1 1 2 3
g(x) = x g +g
1 1 2
where g 0. If e is a non singular zero then we’re done. Otherwise we can
1 1
6≡
find some λ F such that f =λg . We now consider the equivalent system:
q 1 1
∈
(f λx g)(x) = x (f λg )+f
1 1 2 2 3
− −
g(x) = x g +g .
1 1 2
We may therefore assume that f is equivalent to one of two situations:
(i) deg f =0,
x1
(ii) deg f =1.
x1
Step 2: In case (i) we show that we may write f as
f(x) = f(x )
2
g(x) = x x +g (x )
1 2 2 3
where we define x :=(x ,x ,...,x ). Next we will find a non singular zero
i i i+1 m
x of f such that x = 0. Hence by setting x = x−1g (x ), we get a non
2 2 6 1 2 3 3
singular zero of the system f as required.
Step 3: In case (ii) we show that we can write f as
f(x) = x f (x )+f (x )
1 2 3 3 2
g(x) = x x +g (x ).
1 2 2 3
Next we define the quartic form
H(x ):=x f (x ) (f g )(x ).
2 2 3 2 2 2 3
−
It will follow that if we can find a non singularzero of H such that x =0 then
2
6
we can find a non singular zero of the system f. Finding a non singular zero of
H suchthatx =0requiresablendofideaswhichutilizes the information(15)
2
6
we have at our disposal about the h-invariant of the system.
Having described the outline of the proof we proceed with Step 1.
10