A strategy and a new operator to generate covariants in small characteristic

0.2 MB·

by  Florent Ulpat Rovetta

A strategy and a new operator to generate covariants in small
characteristic
Florent Ulpat Rovetta
1st February 2017
7
1
0
Abstract
2

n We present some new results about covariants in small characteristic. In section 1, we give a method to
a
construct covariants using an approach similar to Sturmfels. We apply our method to find a separating
J
systemofcovariantsforbinaryquarticincharacteristic3. Inthesecondsection,weconstructanewoperator
1
on covariants when the characteristic is small compared to thedegree of the form.
3

]
G
Introduction
A
h. We are interested in the computation of covariants of binary forms in small characteristic in a similar optical
t of [Bas15] and in association with the moduli space of hyperelliptic curves. Although, in characteristic 0 or
a
m in large characteristic, it is a classic problem (but still formidable in practice when the degree of the form is
higherthan10),insmallcharacteristic,theapproachbasedontransvections(whosearedifferentialsoperators)
[
collapse. In this context, we wanted to test the effectiveness of an alternative method following [Gey74] and

1 [Stu08]. The idea is to consider the algebra of covariants of n-points of P1 under the action of GL (k) (cf.
2
v definition 1.1 ). The main advantage of this algebra is that it admits a generator system of covariants explicit
5
andindependent ofthecharacteristic. Thecovariantsforthebinaryformsarethenobtainedasthesub-algebra
6
symmetrised by S . Although attractive in theory, our current implementation is extremely limited. Indeed
1 n
9 the group action Sn, in the modular case (i.e when the characteristic divide the order of the group), under
0 the covariants of n points fails with generics algorithms of Magma as soon as n = 6 (cf. [UR15, sec.5.2.6]).
1. However,thismethodwasusedtodetermineaseparatingset(alowerconditionthanbeingageneratorsystem,
0 see the definition 2 ) in the characteristic 3 binary quartics case. Along the way, we realized that some of the
7 invariants/covariantsappearinginsmallcharacteristiccouldbe derivedfromclassicalcovariantsbyanew easy
1
differential operation (cf. section 2.2) under certain conditions that we clarify. For the octavics, we get the
:
v new invariantofdegree1 found by [Bas15]andnew covariantsindegree4 and6 (cf. page10). This operation,
i while it enrichesthe algebraofcovariantsobtainedby reductionof those incharacteristiczero,is notsufficient
X
to get all the covariants (as we will see in an example in degree 4 at the end of this paper). The question of
r
a efficient generation in small characteristic remains wide open.
Notations. Let p be a prime number or 0 and let k be an algebraic closed field of characteristic p and C the
n
algebra of binary covariants defined over k.
1 A strategy to construct covariants in small characteristic
Exceptfor quartics (cf. [Bas15,sec 2.10.2])and Igusainvariantsfor sextic,we do not knowa generatorsystem
of invariants in every characteristic. Thanks to clever reductions and many computations, Bassonexhibited in
1
his thesis a “separating” system . He conjectures that it is generator in characteristic 3 and 7 for the octavic.
For the characteristic p≥11, generator systems are known thanks to the results of [LR12]. To get new results
forcovariants,wewillestablishatotallydifferentcomputationmethodfollowing[Gey74]and[Stu08]. Weobtain
new results for covariants of binary quartic in characteristic 3.
1i.e.: separatingtheorbits(cfdefinition2)
1
1.1 Strategy
The study of covariants of n-points of P1(k) under the action of GL (k) is a classicalframework and we recall
2
here the principal results. The main advantageof this workis that there exists an explicit generatorsystem of
covariants independent of the characteristic. Then, the covariants for binary forms come from the sub-algebra
symmetrised by S .
n
We slightly modify the results of Sturmfels [Stu08, Chap 3, sec 6] in order to be valid in every characteristic.
In the caseof invariants,this is exactly Geyer’s method [Gey74]. Let n>1 be a positive integer. Considerthe
binary form:
n
f(x,z)= a xkzn−k
k
Xk=0
=(µ x−ν z)(µ x−ν z)...(µ x−ν z).
1 1 2 2 n n
The ‘roots’ (µ ,ν ) can be seen as points (µ ,ν )∈P1.
i i i i
Definition 1. Let M be a monomial in k[µ ,ν ,µ ,ν ,...,µ ,ν , x, z] such that:
1 1 2 2 n n
M =µu1µu2···µunνv1νv2···νvnxw1zw2
1 2 n 1 2 n
and P be a polynomial in k[µ ,ν ,µ ,ν ,...,µ ,ν , x, z]. We say that:
1 1 2 2 n n
• M is regular of degree d if u +v =u +v =···=u +v =d. The integer d is called the regularity
1 1 2 2 n n
degree of M.
• P is regularof degree d if all of its monomials are regular of degree d. When P is regular for a degree d,
we say that P is regular.
• P is symmetric if, for all permutation σ ∈S :
n
P(µ ,ν ,µ ,ν ,...,µ ,ν , x, z)=P(µ ,ν ,µ ,ν ,...,µ ,ν , x, z).
1 1 2 2 n n σ(1) σ(1) σ(2) σ(2) σ(n) σ(n)
A regular monomial is reducible if it can be expressed as the product of two regular monomials of regularity
degree great than equal to 1.
We define the action of GL (k) on k[µ ,ν ,µ ,ν ,...,µ ,ν , x, z] in the following way: Let M ∈GL (k),
2 1 1 2 2 n n 2
ν ν ν
i → i =M−1· i
(cid:18) µi (cid:19) (cid:18) µi (cid:19) (cid:18) µi (cid:19)
x x x
→ i =M−1· .
(cid:18) z (cid:19) (cid:18) zi (cid:19) (cid:18) z (cid:19)
A regular polynomial P is a covariant (of n points) if there exists w ∈Z such that:
P(µ ,ν ,µ ,ν ,...,µ ,ν , x, z)=det(M)wP(µ ,ν ,µ ,ν ,...,µ ,ν , x, z).
1 1 2 2 n n 1 1 2 2 n n
P is calledaninvariant (of n points) ifitdoesnotdepend onxandz. Itis easyto define covariantsquantities
thanks to the brackets. Let 1≤i<j ≤n, we call bracket the following quantities:
[ij]:=µ ν −ν µ ,
i j i j
[iu]:=µ x−ν z.
i i
The sub-ring B(n) generated by these brackets in k[µ ,ν ,µ ,ν ,...,µ ,ν , x, z], is called the bracket ring.
1 1 2 2 n n
We also denote B (n) the sub-ring of B(n) of polynomial in the brackets which are regular of degree d for
reg
d≥0. The latter is generated by the monomials of the form:
[ij]mij,
Yi<j
where the integers m verify d= i−1m + n m . The polynomial ring of regular covariants is equal
ij j=1 ji j=i+1 ij
toBreg(n)(consequenceofthefirstPfundamentaPltheorem(cf[Wey39])). WhentheactinggroupisGL2(C),the
2
theorem3.2.1 andthe lemma 3.6.5of [Stu08] provide a demonstration. When the groupis arbitrary,the proof
is in [dCP76]. Note also that [Gey74, Satz 5] gives an elementary proof in the case of GL (k). In the section
2
1.2, we present an example of computation of generators of B (n). What remains to describe the final stage
reg
to get the of binary forms Let:
Ψ:k[a ,a ,...,a , x, z]→k[µ ,ν ,µ ,ν ,...,µ ,ν , x, z]
0 1 n 1 1 2 2 n n
ν ν
an−k →(−1)nµ1···µn·σk( 1,..., n).
µ µ
1 n
σ represents the k-th elementary symmetric polynomial function in n variables. The following theorem (from
k
[Stu08, th3.6.6]) is an elementary consequence of the previous theorem.
Let B (n) be the sub-ring of B (n) of the polynomial in the brackets which are symmetric.
reg,sym reg
Theorem1. ThemappingΨisanisomorphismbetweentheringofcovariantsofbinaryformsk[a ,...,a ,x,z]SL2(k)
0 n
andthesub-ringB (n)ofsymmetricandregularpolynomialbracketsfunctionsofk[µ ,ν ,µ ,ν ,...,µ ,ν ,x,z].
reg,sym 1 1 2 2 n n
If C(a ,...,a ) is a covariant of degree d and order r then Ψ(C) is a symmetric polynomial bracket function
0 n
such that:
1. in every monomial of Ψ(C),the index 1, 2,..., n appears d times,
2. in every monomial of Ψ(C), the letter u appears r times.
1.2 Computation of B (n)
reg
We wish to compute Cn as Breg(n)Sn. If b1,...,bt is a generatorsystem of bracketmonomial for Breg(n), then
we have a surjective morphism:
k[x ,...,x ] → B (n).
1 t reg
x → b
i i
The kernel I of this morphism is generated by the following relations.
Proposition 1. Let 1≤i<j <k <l ≤n, we have:
[ik][jl]=[ij][kl]+[il][jk],
[ik][ju]=[ij][ku]+[iu][jk].
These relations are called the syzygies.
Inordertohaveaclearerview ofageneratorsystemofB (n)the monomialswillbe representedby weighted
reg
graphs such that the vertices form a regular polygon. We represent:
• a monomial of B(n) by a graph with n vertices numbered from 1 to n and a vertex called (cid:13)u,
• the bracket [ij] by an edge connecting the vertex i to the vertex j with i<j ∈{1,...,n},
• the bracket [iu] by an edge connecting the vertex (cid:13)u to the vertex i.
For example, the bracket product [12][14]3[34][1u][2u]2∈B(4) is represented by the following weighted graph:
1
•
2
•
2
(cid:13)u 3
•
3
•
4
Thepreviousallowustoformulate5remarkswhichareveryusefultoconstructageneratorsystemforB (n).
reg
The next point follows from the definition of B (n).
reg
Point 1. Every monomial of order m and of regularity degree d is represented by a graph with m connexions
with (cid:13)u and every numbered vertex has a valence d.
3
Moreover, by Kempe’s lemma ([Stu08, th. 3.7.3 p. 132]), the covariant algebra of n points is generated by
elementsofregularitydegreeatmost2. Also,by[HMSV09,th.2.3p.7],theinvariantsofnpointsaregenerated
by the regularity degree 1, hence the following point:
Point 2. The numberedvertices have a valence at most 2. The graph corresponding to invariants has a valence
1.
By its definition, if a graphis expressedas aunionof subgraphscorrespondingto graphsofsmallerdegreeand
smaller order already computed, the associated covariant is reducible.
Point 3. The graphs having a subgraph already calculated are excluded.
When the vertices are in a regular polygon, proposition1 causes the following point (see [KR84, th.6.2 p.72]
for a proof):
Point 4. The graphs of our generator system have no edges crossing.
Thanks to point 2, the number of adjacent edges of (cid:13)u is bounded. Moreover,when n is even, point 1 imposes
another condition on the edges adjacent to (cid:13)u.
Point 5. (cid:13)u has at most 2n adjacent edges.
When n is even, (cid:13)u has an even number of adjacent edges.
Example 1. Using the five points above and considering increasing orders we get the following generators,
when n=4, called t , t , u , u , u and f.
0 1 0 1 2
1 1
• •
2 2
• •
(cid:13)u (cid:13)u
• •
3 3
t • t •
0 1
4 4
1 1 1
• • •
2 2 2
• • •
(cid:13)u (cid:13)u (cid:13)u
• • •
3 3 3
u0 • u1 • u2 •
4 4 4
1
•
2
•
(cid:13)u
•
3
f •
4
1.3 Symmetrization
The action of S on the b induces a representation G of S in GL (k). There are algorithms for computing
n i n n t
k[x1,...,xt]Gn = Rn (cf [DK02]). The latter being also valid in the modular case (i.e when p |Gn|), we use
them ‘naively’ through their Magma implementations. However this process generates a limitatio(cid:12)n when n=6.
(cid:12)
4
What remains to clarify is the link between R and C .
n n
Whenp does notdivide |S |, S is a linearlyreductive group(cf [DK02, Def 2.2.1])andthe existence ofReyn-
n n
olds operators (cf [DK02, Th 2.2.5]) preserves the surjectivity of the morphism k[x ,...,x ]→B (n) in the
1 n reg
symmetrizationprocess. Thanks to [Stu08, lem.3.7.2], the image of a generator system of R by the canonical
n
surjection k[x1,...,xn]→k[x1,...,xn]/I =Breg(n) is a generator system of Cn =Breg(n)Sn. In particular, if
p > n, we get a generalization of the result of Geyer: the covariant ring C is the reduction modulo p of the
n
covariantring in characteristic 0. In particular, the bi-graduate Poincaré series are identical.
When p |S |, S is only a reductive group (cf. [DK02, sec 2.2.2]) and the previous result is no longer valid in
n n
the gene(cid:12)ral case. To overcome this, we recall the following concept:
(cid:12)
Definition 2. Let X be an affine variety and G an automorphism group of k[X]. A subset S ⊆k[X]G is called
separating if, for every couple of points (x,y) of X, we have the following propriety: if there exists an element
f ∈k[X]G such that f(x)6=f(y), there exists an element g in S such that g(x)6=g(y).
The relation with the invariant ring is the following (cf [DK02, prop.2.3.10]):
Proposition 2. Suppose that X is irreducible and k[X]G is finitely generated. Let A ⊆ k[X]G be a separ-
ating sub-algebra finitely generated. Frac(k[X]G) is then a purely inseparable finite extension of Frac(A). In
particular, if the characteristic of k is zero then:
Frac(A)=Frac(k[X]G).
Definition2hastheadvantagetopreservethesurjectivityontransitiontoinvariants. LetGbealinearalgebraic
group. Thanks to [DK02, p. 59], if G is reductive, G regularly acts on an affine variety X and Y ⊆ X is a
sub-variety G-stable then the restriction map k[X]→k[Y] sends a separating subset of k[X]G to a separating
subsets of k[Y]G. So R is the separating algebra of covariants of binary forms of degree n but we do not
n
necessarilyhave an equality between R and C . We will see, in the following case of quartics in characteristic
n n
3, when the inclusion is strict.
Example 2. In example 1, we have seen that the covariant algebra of 4 points is generated by t , t , u , u ,
0 1 0 1
u et f. We will make the group S act and, using the function InvariantRing of Magma, we will compute a
2 4
separating system of the covariant algebra C . Knowing that S is generated by σ = (1234) and τ = (12), the
4 4
action of S on t , t , u , u , u and f is given by the following equalities:
4 0 1 0 1 2
tτ =−t et tσ =−t ,
0 0 0 1
tτ =t +t et tσ =−t ,
1 1 0 1 0
uτ =u et uσ =−(u +u +u ),
0 0 0 0 1 2
uτ =u +u et uσ =u ,
1 1 2 1 0
uτ =−u et uσ =u ,
2 2 2 1
fτ =f et fσ =f.
Using the Magma code of Appendix A, we get the following covariants:
c =−3a a +a2+12a a ,
0,2 1 3 2 4 0
c = −27/2a2a +9/2a a a −a3+36a a a −27/2a2a ,
0,3 1 4 1 2 3 2 2 4 0 3 0
c = a z4+a xz3+a x2z2+a x3z+a x4,
4,1 0 1 2 3 4
c = (a2−8/3a a )z4+(4/3a a −8a a )xz3+(4/3a2−2a a −16a a )x2z2
4,2 1 2 0 1 2 3 0 2 1 3 4 0
+(4/3a a −8a a )x3z+(a2−8/3a a )x4,
2 3 1 4 3 2 4
c = (a3−4a a a +8a a )z6+(2a2a +4a a a −8a a2+32a2a )xz5+
6,3 1 1 0 2 0 3 1 2 0 1 3 0 2 0 4
(5a2a +40a a a −20a a a )x2z4+(20a2a −20a a2)x3z3+
1 3 0 1 4 0 2 3 1 4 0 3
(20a a a −5a a2−40a a a )x4z2+(8a2a −4a a a −2a a2−32a a2)x5z+
1 2 4 1 3 0 3 4 2 4 1 3 4 2 3 0 4
(4a a a −8a a2−a3)x6.
2 3 4 1 4 3
We recover the classic covariants of characteristic zero.
5
We apply the same process in characteristic 3 (we change FF:= Rationals(); in the Magma code by FF:=
GF(3);) and we get
c =a ,
0,1 2
c =a3a3+a2a2a2+a a a2a a +a a4a +2a a3a2+2a3a3+2a2a3a +a2a2a2,
0,6 0 4 0 2 4 0 1 2 3 4 0 2 4 0 2 3 1 3 1 2 4 1 2 3
c = a z4+a xz3+a x2z2+a x3z+a x4,
4,1 0 1 2 3 4
c =a c ,
4,4 2 4,3
c =(2a2a +2a a a +a3)+(2a2a +a a a +a a2+2a2a )x+(a a a +a a a +2a2a )x2+
6,3 0 3 0 1 2 1 0 4 0 1 3 0 2 1 2 0 1 4 0 2 3 1 3
(a a2+2a2a )x3+(2a a a +2a a a +a a2)x4+(a a2+2a a a +2a2a +a a2)x5+
0 3 1 4 0 3 4 1 2 4 1 3 0 4 1 3 4 2 4 2 3
(a a2+a a a +2a3)x6,
1 4 2 3 4 3
c =c (c −a2c ),
8,4 4,1 4,3 2 4,1
c =(c −a2c )c .
8,6 4,3 2 4,1 4,3
Where
c =(a a2+2a a a +2a2a +a a2)x4+(a a a +a a a +2a a2)x3z+
4,3 0 4 1 3 4 2 4 2 3 0 3 4 1 2 4 1 3
(a a a +a a a +2a2a )xz3+(a2a +2a a a +2a a2+a2a )z4.
0 1 4 0 2 3 1 3 0 4 0 1 3 0 2 1 2
c is a covariant of degree 3 and order 4 in characteristic 3. This system is not generator of the algebra of
4,3
covariants because we cannot find c as polynomial in the c . This case provides an example where a sub-set
4,3 2i,j
of separating is not a generator system of the covariant algebra. We point out that {c ,c ,c ,c ,c } is
0,1 0,6 4,1 4,3 6,3
a separating system of C and one wonders if it is also a generator system. The theorem [DK02, Th.2.3.12]
4
would in theory lead to an algorithm to test this hypothesis but in practice the computation did not finish.
2 A new way to generate covariants in small characteristic
Here we introduce a new way to build covariantsin small characteristic. To show the validity of our approach,
our first idea was to use the differential characterizationof covariants,as in [Hil93, p.43]. It turns out however
that the result of Hilbert (theorem 2 ), originally shown in characteristic 0, admits counterexamples in small
characteristic, as discussed in section 2.1. So we approachthe proof of the theorem 3 directly. First we recall
the result of Hilbert and then we give our proof. In the following, f is a binary form defined over the field k :
n
f = a xizn−i.
i
Xi=0
2.1 Hilbert’s differential characterization of covariants
Let :
• k[a ,...,a ] be the homogeneous polynomial algebra of degree d,
0 n d
• T be the sub-group of diagonal matrices of SL (k),
2
• Γ be the sub-group of upper triangular matrices and diagonal equal to 1 of SL (k),
2
• Γ∗ be the sub-group of lower triangular matrices and diagonal equal to 1 of SL (k).
2
These three subgroups are important because they generate SL (k) and thus permit to break down the issues
2
of invariance under the action of these groups. Let M = aρ00aρ11...,aρnn be a monomial of k[a0,...,an]. We
define the weight of M by w= n iρ . We say that a non zero element I of k[a ,...,a ] is isobaric if all of
i=0 i 0 n
its monomials have the same wePight. We define two differential operators on I that preserve the degree. The
operators ∆ and D are given by:
n
∂
∆= ia
i
Xi=1 ∂ai−1
n−1
∂
D= (n−i)a .
i
∂a
Xi=0 i+1
6
Theorem 2. Suppose p=0 or p>nd+m. The polynomial C = m C xizm−i is a covariant of the binary
i=0 i
form f under the action of SL2(k) if and only if the following condPitions are satisfied:
1. C ,...,C arehomogeneousfunctionsofdegreedandisobaricofweightw,w+1,...,w+mwithnd−2w=
0 m
m,
2. DC =x∂C,
∂z
3. ∆C =z∂C.
∂x
This result is not available in every characteristic. Let f = 16 a xiz16−i be a binary form of degree
i=0 i
n=16 in characteristic 3. Note C =a11x6 a homogeneous polynoPmial of degree m=6. The polynomial C is
1 1
not a covariant of f because for M = ∈SL (k) we have
(cid:18) 0 1 (cid:19) 2
C(M.f,M.(x,z))=(x+2z)6(a +a )6=C.
11 14
Nonetheless
1. C ,...,C are homogeneous functions of degree d = 1 and isobaric of weight w = 5,6,...,11 with
0 m
nd−2w=16·1−2·5=6=m,
2. DC =6a x6 =0=x∂C,
10 ∂z
3. ∆C =12a x6 =0=z∂C.
6 ∂x
Hence, Hilbert theorem cannot be used to prove our next result. However, we will revise some elements of
the proof using the three sub-groups Γ, Γ∗ and T.
2.2 A new way to build covariants in positive characteristic
Before showing our theorem, we set some notations. Let M ∈SL (k). We have
2
n
f(M.(x,z))= a′xizn−i.
i
Xi=0
In the following, we note X =(x,z), X′ =M−1(x,z), a=(a ,...,a ) and a′ =(a′,...,a′ ). We start with a
0 n 0 n
lemma.
Lemma 1. An homogenous polynomial C ∈k[a ,...,a ] [x,z] is a covariant under the action of T if and only
0 n d
if the C are isobaric of weight w+i and nd−2w=m.
i
Proof. Write
m
C = C xizm−i.
i
Xi=0
If M = λ0−1λ0 ∈T, then a′i =λn−2iai and Cl(a′)= li=1 nj=0a′jǫi,j,l (resp. Cl(a)= li=1 nj=0ajǫi,j,l) with
l∈{0,..(cid:0).,m}(cid:1). We have P Q P Q
l n l n
Cl(a′)= λ(n−2j)ǫi,j,lajǫi,j,l = λPnj=0(n−2j)ǫi,j,l ajǫi,j,l
Xi=1jY=0 Xi=1 jY=0
l n
= λnd−2Pnj=0jǫi,j,l ajǫi,j,l.
Xi=1 jY=0
Since M also acts on (x,z) by M−1.(x,z), we get
m
M.C(a,X)=C(a′,...,a′ ,λx,λ−1z)= λ2l−mC (a′)xlzm−l.
0 n l
Xl=0
7
Suppose that C is a covariant, then M.C =C, so for all l ∈{0,...,m}
C(a)=λ2l−mC (a′).
l
This implies that for all l and for all i
n
nd−2 jǫ +2l−m=0.
i,j,l
Xj=0
n n
Inparticular, jǫ −l doesnotdependonthel ori. So,wecandefinew byputtingw = jǫ −l.
j=0 i,j,l j=0 i,j,l
n
WegetthennPd−2w=m. Moreover,theintegerw istheweightofC . TheweightofC is Pjǫ =w+l.
0 l j=0 i,j,l
P
Conversely we want to prove that C (a) = λ2l−mC(a′). Since the weight of C is n jǫ , we have
l l l j=0 i,j,l
w = n jǫ −l. Moreover,nd−2w=m, hence : P
j=0 i,j,l
P
n
nd−2 jǫ +2l−m=0.
i,j,l
Xj=0
This implies that:
C (a)=λ2l−mC (a′).
l l
So, C is a covariant under the action of T.
Since, starting from C , we get the weight of C of the covariant C, we can say that w is the weight of C.
0 i
Theorem 3. Let Q= m0 Q xizm0−i be a covariant of f of order m , degree d and weight ω . Let l be an
i=0 i 0 0 0
integer smaller than m0P/2 and p. The polynomial
1 ∂lQ
C =
zl ∂xl
is a covariant of f if and only if m −l+1 is congruent to 0 modulo p. When C is a non zero covariant, its
0
order is m −2l and its degree is d .
0 0
Remark 1. The operator was already known by Hilbert (cf [Hil93, th p.103]). But the way to use it in small
characteristic with the previous condition is new.
To show that C is a covariant under the action of SL (k), we consider the action of T, Γ and Γ∗. First we
2
analyse the action of this three sub-groups on C and then we give the proof of the theorem 3. Write again
m
C = C xizm−i.
i
Xi=0
Action of T. By definition of C, C ,...,C are homogeneous functions of degree d and isobaric of weight
0 m 0
l+ω ,l+ω +1,...,l+ω +m. We express C according to the coefficients of Q
0 0 0
m0 i!
C = Q xi−lzm0−i−l.
i
l!
Xi=l
If p|(m −l+1), then for all i∈{m −l+1,...,m },
0 0 0
i!
p| .
l!
In this case, if C is non zero, C is a homogeneous polynomial of degree m = m −2l. Moreover, Q being a
0
covariant, lemma1 ensures that m = nd −2w . The order of C can be writen m = nd −2(ω +l). So,
0 0 0 0 0
by lemma 1 , C is a covariant under the action of T. The converse is also given by lemma 1 . The condition
p|(m −l+1) is then a necessary and sufficient condition for C to be a covariantunder the action of T.
0
8
Action of Γ. We set g :(a,X)→(a′,(x+µz,z)). We aim at showing that C◦g =C. Meaning
1 ∂lQ 1 ∂lQ
( )◦g = .
zl ∂xl zl ∂xl
This is equivalent to
∂lQ ∂lQ
◦g = .
∂xl ∂xl
However,Q being a covariant under the action of Γ, we have
∂Q ∂Q◦g
= .
∂x ∂x
Moreover,
∂Q◦g ∂Q
= ◦g.
∂x ∂x
By immediate recurrence, we obtain the desired result. So, C is covariant under the action of Γ.
Action of Γ∗. We set g :(a,X)→(a′,(x,µx+z)). We want to prove that C◦g =C. Meaning
1 ∂lQ 1 ∂lQ
( )◦g = .
zl ∂xl zl ∂xl
This is equivalent to
∂lQ ∂lQ
zl( )◦g =(µx+z)l .
∂xl ∂xl
∂lQ l l ∂lQ
Using the fact that ◦g = ·(−µ)l−i· , this amounts to show
∂xl (cid:18)i(cid:19) ∂xi∂l−iz
Xi=0
l l ∂lQ l l ∂lQ
zl (−µ)l−i = µl−ixl−izi.
(cid:18)i(cid:19)∂xi∂zl−i (cid:18)i(cid:19)∂xl
Xi=0 Xi=0
This is still equivalent to
l l ∂lQ ∂lQ
µl−i xl−izi+(−1)l−i+1 zl =0
Xi=0(cid:18)i(cid:19) h∂xl ∂xi∂zl−i i
or that for all i∈{0,...,l}
∂lQ ∂lQ
xl−izi+(−1)l−i+1 zl =0.
∂xl ∂xi∂zl−i
Assume that p|(m −l+1). We develop the left hand of the expression and we get
0
m0
0= Q j(j−1)...(j−l+1)xj−izm0+i−j+
j
Xj=l
m0−l+i
(−1)l−i+1 Q j(j−1)...(j−i+1)xj−i(m −j)(m −j−1)...(m −j−l+i+1)zm0−j+i.
j 0 0 0
Xj=i
For all j ∈ {m −l,...,m}, p|j(j −1)...(j −l+1). In the same way, for all j ∈ {m −l,...,m −l+i},
0 0 0
p|j(j −1)...(j −i+1). So, the sums shall stop at m −l. For all j ∈ {i,...,l−1}, p|(m −j)(m −j −
0 0 0
1)...(m −j−l+i+1). So the two sums begin at l. Finally, since p|(m −l+1),
0 0
(m −j)(m −j−1)...(m −j−l+i+1)≡(l−1−j)(l−2−j)...(−j+i)≡(−1)l−i(j−i)...(j−l+1) (mod p)
0 0 0
This provesthe nullity of the expression. So it has been shown that if p|(m −l+1) then C is invariantunder
0
the action of Γ∗.
9
Proof of theorem 3 . According to paragraph ‘Action of T’ (p. 8), C is a covariant under the action of T if
and only if p|(m −l+1). According to paragraph‘Actionof Γ’ (p. 9), C is a covariantunder the action of Γ.
0
According to paragraph ‘Action of Γ∗’ (p. 9), if p|(m −l+1) then C is a covariant under the action of Γ∗.
0
Since SL (k) is generated by T, Γ and Γ∗, if p|(m −l+1), C is a covariant under the action of SL (k).
2 0 2
Conversely,assume that C is a covariantunder the action of SL (k). The invariance under the action of T
2
shows that p|(m −l+1).
0
Thanks to this theorem, we can construct new covariants which do not appear in characteristic zero.
• For the binary quartic in characteristic 3 (cf. example 2), we find c (Q = f and l = 2) and c
0,1 4,3
(Q=c and l=1) ;
6,3
• Forthe binary sextic incharacteristic3 (cf. [UR15, sec 5.2.6]),we find the covariantq (Q=f andl=1)
of degree 1 and order 4 ;
• For the binary sextic in characteristic 5 (cf. [UR15, sec 5.2.6] and [UR15, sec 6.6.2.3]), we find the
covariantc (Q=f and l =2) of degree 1 and order 2 ;
• For the binary octavic in characteristic 5, we find the same invariants, C = a (Q = f and l = 4) of
4
degree 1 identified by Basson and Lercier.
Itis tempting to wonderwhether itis posible,insmallcharacteristic,to geta generatorsystemofcovariantby
addingthisnewoperation. Afirstdifficultyisthefollowing. LetQ ,...,Q becovariants,l ,...,l beintegers
1 r 1 r
such that
1 ∂liQi
C =
i zli ∂xli
arethecovariantsobtainedbythisnewoperationstartingfromQ . LetQbeanelementofk[Q ,...,Q ,C ,...,C ].
i 1 r 1 r
The expression 1 ∂lQ is not necessarily in k[Q ,...,Q ,C ,...,C ]. For instance, over k =F , for r =1 con-
zl ∂xl 1 r 1 r 5
sider only the sextic binary form Q =f. The covariantof f
1
1 ∂3f2
=(a a +a a )x6+(4a a +4a a +2a2)x5z+(a a +a a +3a2)xz5+(4a a +4a a )z6
z3 ∂x3 3 6 4 5 2 6 3 5 4 0 4 1 3 2 0 3 1 2
is not in the algebra generated by f and C = 1 ∂2f. Indeed, if it was in this algebra, it would be a linear
1 z2 ∂x2
combinationoff2, fC andC2 since these arethe onlyterms ofdegree2ina . Howeverthe termsthatdo not
1 1 i
dependonxinthesethreecovariantsarea2,2a a and4a2. Wecannotgeneratethecoefficient(4a a +4a a ).
0 0 2 2 0 3 1 2
So, it is difficult to see when the new operation will saturate the algebra. Actually, we even have an example
where it does not. Consider the invariant c ∈I in characteristic 3 of the example 2. It cannot be obtained
0,6 4
using our new operator. To get it by our operation, it would have to be the derivative l times starting from
a certain covariant of order m and degree 6. The integers m and l have to verify l < m/2, m−2l = 0 and
m−l+1isamultipleof3. Sowegetthisinvariantbytakingthesecondderivativeofacertaincovariantc of
4,6
order 4 and degree 6. However, by performing the computations, we find that the algebra of covariant degree
less than 6 generated by our operator on the reduction of covariantsof characteristic zero is generatedby c ,
0,1
f =c ,c andc . Theonlytwooptionsforc arec5 c andc3 c . Thesetwooptionsdonotgivec .
4,1 4,3 6,3 4,6 0,1 4,1 0,1 4,3 0,6
Acknowledgement. It is a pleasure to thank Christelle Klein Scholz for his relecture.
References
[Bas15] Romain Basson. Arithmétique des espaces de modules des courbes hyperelliptiques de genre 3 en
caractéristique positive. Univeristé de Rennes 1, 2015. Thèse de doctorat.
[dCP76] C.deConciniandC.Procesi. Acharacteristicfreeapproachtoinvarianttheory. AdvancesinMath.,
21(3):330–354,1976.
[DK02] Harm Derksen and Gregor Kemper. Computational invariant theory. Invariant Theory and Al-
gebraic Transformation Groups, I. Springer-Verlag, Berlin, 2002. Encyclopaedia of Mathematical
Sciences, 130.
[Gey74] W. D. Geyer. Invarianten binärer Formen. In Classification of algebraic varieties and compact
complex manifolds, pages 36–69.Lecture Notes in Math., Vol. 412. Springer, Berlin, 1974.
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