A note on minimal Galois embeddings of abelian surfaces

A Note on Minimal Galois Embeddings of Abelian Surfaces

HISAOYOSHIHARA

ABSTRACT- We show that the least numberNsuch that an abelian surface has a Galois embedding inP^Nis seven and then we give examples of such surfaces.

1. Introduction.

This is a continuation of our previous paper [7]. The least numberN suchthat an abelian surface can be embedded inP^Nis four, and in that case the abelian surface has a special structure, see for example [3]. Similarly it might have some interest to study the least numberNsuchthat an abelian surfaceAcan be Galois-embedded inP^N. Moreover, in that case we want to know the structure of A. In this short note we give the answer to the problem. Before stating it, we recall from [7] some of the definitions and properties of Galois embeddings of algebraic varieties.

Let kbe the ground field of our discussions, which is assumed to be algebraic closed. Let V be a nonsingular projective algebraic variety of dimensionnandDa very ample divisor. We denote this by a pair (V;D).

Let f ˆf_D:V,!P^Nbe the embedding ofV associated withthe complete linear system jDj, wh ere N‡1ˆdimH⁰(V;O(D)). Suppose thatW is a linear subvariety of P^N suchthat dimW ˆN n 1 and W\f(V)ˆ ;.

Then consider the projectionW,p_W:P^{N >}W₀withcenterW, whereW₀ is an n-dimensional linear subvariety not meeting W. The composition pˆp_Wf is a surjective morphism fromVtoW0Pⁿ. LetK ˆk(V) and K₀ˆk(W₀) be the function fields ofV andW₀ respectively. The covering map p induces a finite extension of fields p:K₀,!K of degree dˆdegf(V)ˆDⁿ, which is the self-intersection number ofD. It is easy to see that the structure of this extension does not depend on the choice ofW0

(*) Indirizzo dell'A.: Department of Mathematics, Faculty of Science, Niigata University, Niigata 950-2181, Japan.

E-mail: yosihara@math.sc.niigata-u.ac.jp

but only onW, hence we denote byKWthe Galois closure of this extension and byGW ˆGal(KW=K0) the Galois group of KW=K0. Note that GW is isomorphic to the monodromy group of the coveringp:V !W0.

DEFINITION1. In the above situation we callG_Wthe Galois group atW.

If the extensionK=K₀is Galois, we callf andWa Galois embedding and a Galois subspace for the embedding, respectively.

DEFINITION2. A nonsingular projective algebraic varietyV is said to have a Galois embedding if there exist a very ample divisorDsuch that the embedding associated withthe complete linear system jDj has a Galois subspace. In this case the pair (V;D) is said to define a Galois embedding.

In this note we use the following notation.

Z_m : the cyclic group of orderm Dm : the dihedral group of order 2m jGj: the order of a groupG

r: exp (2p 

p 1

=6)

Aut(V) : the automorphism group of a varietyV ha₁;. . .;a_mi: the subgroup generated bya₁;. . .;a_m 1₂ : the unit matrix of degree two

We shall make use of the following criterion (cf. [7, Theorem 2.2]).

THEOREMA. Let V and D be as above. The pair(V;D)defines a Galois embedding if and onlyif the following conditions hold:

(1) There exists a subgroup G of Aut(V)such thatjGj ˆDⁿ.

(2) There exists a G-invariant linear subspace L of H⁰(V;O(D)) of dimension n‡1such that, for anys2G, the restrictionsj_Lis a multiple of the identity.

(3) The linear system L has no base points.

The original form of the study of the Galois embedding is given in [5] or [6]. We have applied the above method to abelian surfacesAoverkˆC and obtained some results.

2. Statement of Theorem.

LetAbe an abelian surface defined overkˆCandGbe a finite sub- group ofAut(A). Fix a covering morphismC² !A. An elementg2Ghas

a representation eg on the universal covering C² suchthat egzˆM(g)z‡t(g), where M(g)2GL(2;C),z2C² andt(g)2C². We call M(g) andt(g) the matrix and translation part of the representationeg, re- spectively. PutG₀ˆ fg2GjM(g)ˆ1₂gandHˆ fM(g)jg2Gg. Then, we have the following exact sequence of groups

1 !G0 !G !H !1:

ClearlyBˆA=G₀is also an abelian surface andHG=G₀is a subgroup of Aut(B).

Hereafter we assume that (A;D) defines a Galois embedding inP^Nand let G be the Galois group. Then G is a subgroup ofAut(A) and B=His isomorphic toA=GP². With the notation above, we have:

THEOREM B. [7, Theorem 3.7] If an abelian surface A has a Galois embedding, then H is isomorphic to D₃, D₄or a semidirect product Z₂^j K, where KD₄or Z_mZ_m (mˆ3; 4; 6)

Now, the answer to the question in the introduction is given as follows:

THEOREMC. Suppose that(A;D)defines a Galois embedding. Then (1) The least number N is seven, i.e., there exists an abelian surface A such that it can be embedded into P⁷ with a Galois subspace, and no abelian surface embedded inP^N(N6)has a Galois subspace.

(2) The abelian surface BˆA=G0 is isomorphic to the self-product EE of an elliptic curve, such that HˆG=G0acts on B and HD4or Z₂^j D₄.

REMARK1. The minimal Galois embedding for an elliptic curve E is given as follows. If E can be Galois-embedded in P², th en E must have an automorphism of order three with a fixed point. In fact, the elliptic curve is unique and is defined by Y²Zˆ4X³‡Z³. Th e centers of the projections are (1:0:0);(0: 

p 3

:1) and (0: 

p 3

:1). How- ever, note that every elliptic curve has a Galois embedding in P³, where the group is isomorphic to Z2Z2 (further, if the j-invariant is 1728, then it has another projection center whose Galois group is isomorphic to Z₄).

In what follows we shall give the proof of Theorem C and some examples.

In particular, there is an example whereBis the Jacobian of a curve. Indeed,

let J(C) be the Jacobian of the normalization C of the curve y²ˆ x(x⁴‡ax²‡1); a6ˆ 2, and letAbe an abelian surface which is an etale double coveringq:A !J(C). Then we shall show that (A; q(C‡C⁰)) gives a minimal Galois embedding, whereC⁰is a translated ofC.

3. Proof.

By Theorem B we havejHj ˆ2^a3^b, wh ere (a;b)ˆ(3;0), (4;0), (5;0), (1;1), (1;2) (3;2). We consider the possible values ofjGj ˆD²ˆ2m. SinceAcan be embedded inP^{m 1}by the complete linear systemjDj, we havem 14 by [3]. In view of Theorem B we havem6ˆ5, hencem6. Suppose thatmˆ6.

Then, from Theorem B again we infer thatHD3andjG0j ˆ2. Every two dimensional complex crystallographic group G with X=GP² has been classified in [4]. Referring to it, we see thatAcan be expressed asAˆC²=V suchthatVis the period matrix

Vˆ 1 r² v r²v

1 r v rv

ˆ 1 r²

1 r

1 0 v 0

0 1 0 v

; wherevis a complex number withIv>0.

Define four vectors as follows:

v1ˆ 1 1

; v2ˆ r²

r ; v3ˆ v v

ˆvv1; v4ˆ r²v rv

ˆvv2: LetL_Abe the lattice inC²generated byv₁;v₂;v₃andv₄. Letg₁be a gen- erator ofG0whose representation iseg1zˆz‡e, wherez; e2C². Sinceg12

is the identity onA, we have 2e2 L_A. LetL_Bbe the lattice generated byL_A ande. Then BˆC²=L_BˆA=hg₁iis also an abelian surface on which the groupHacts. As shown in [4],His generated byg₂ andg₃, whose matrix parts are

M₂ˆM(g₂)ˆ 0 1 1 0

andM₃ˆM(g₃)ˆ r 0 0 r²

respectively. Since 2e2 LA, the vectorecan be expressed as eˆ1

2 X⁴

iˆ1

n_iv_i;

where niˆ0 or 1 (1i4). Since G0 is a normal subgroup of G and jG0j ˆ2, g1 commutes witheachelement of G. Therefore, we infer that

Mie e2 LA(iˆ2;3). Indeed we have

M2e eˆ f (2n1‡n2)v1 (2n3‡n4)v3g=2 2 L_A: Thus we have

( n₂v₁ n₄v₃)=22 L_A: (1)

Similarly, consideringM3e e, we have

f n₂v₁‡(n₁ n₂)v₂ n₄v₃‡(n₃ n₄)v₄g=22 L_A: (2)

From (1) we getn2ˆn4ˆ0, hence from (2) we getn1ˆn3ˆ0. Since e2 L= _A, this is a contradiction. Thus we havem7. From Theorem B we inferjGj 6ˆ14, hence we havem8. We conclude that the least numberm is 8 by the examples in the next section.

REMARK 2. Referring to [4, Theorem 1], we see that in the case (i) HD4 the dimension of the moduli space is 1, but in the case (ii) HZ2^j D4the dimension is zero.

4. Examples.

When we make examples, the following lemma is useful.

LEMMA3. If M(g) 1₂is a nonsingular matrix for g2G, then we can assume t(g)ˆ0.

PROOF. We consider tGt ¹ instead of G, where t is a translation

etzˆz‡`. If M(g) 1<sub>2</sub> is nonsingular, then by putting `ˆ

(M(g) 1₂) ¹t(g), we gett(tgt ¹)ˆ0. p

In casemˆ8 we haveHD4orZ2^j D4by Theorem B. We shall give examples of both. If (i) HD4, thenjG0j ˆ2. Since G0 is a normal sub- group of G and jG₀j ˆ2, we infer that GˆG₀H. Suchexamples are given in Examples 4 and 5. On the other hand, if HZ₂^j D₄, we have GH. Suchan example is given in Example 7.

EXAMPLE4. LetAbe the abelian surface with the period matrix

1 0 v 0

0 1 0 v

suchthatIv>0:

LetLAbe the lattice generated by the column vectors of the period matrix.

Let us consider the automorphismsg1,g2 andg3ofA, whose representa- tions onC²are as follows:

ge1z ˆ z‡¹₂ n1‡n3v n2‡n4v

; ge₂z ˆ 0 1

1 0

z‡ a₁ a₂ ; ge3z ˆ 0 1

1 0

z

where (n1;n2;n3;n4)ˆ(0;0;1;1);(1;1;0;0);(1;1;1;1);

a₁‡a₂ a₁‡a₂

2 L_Aand 2a₁ 0

2 L_A:

We haveg₁²ˆg₂²ˆg₃⁴ˆid,g₂g₃g₂ˆg₃ ¹andg_ig₁ˆg₁g_i(iˆ2;3) onA.

Putting Gˆ hg1;g2;g3i, we have G0ˆ hg1i and GˆG0H where Hˆ hM(g2);M(g3)i. ClearlyHD4. The groupGis a subgroup ofAut(A) and A=GP². The very ample divisor D is given by p(L), where p:A !A=GP²andLis a line inP²(cf. [7, Lemma 3.5]). We infer from Theorem A that (A;D) defines a Galois embedding.

By [7, Corollary 3.8], if A has a Galois embedding, then the abelian surfaceBˆA=G0is isomorphic toEEfor some elliptic curveE. On th e other hand,EEcan be a Jacobian of a curve for someE(cf. [2]). So one may ask what type of genus 2 curve can give the Jacobian whose double covering has a minimal Galois embedding. Let us consider this question in the next example.

EXAMPLE 5. Let G be the curve defined by y²ˆx(x⁴‡ax²‡1), where we assumea6ˆ 2 (cf. [1, Theorem 4.8]). This curve has a singular point at1. LetCbe the normalization ofG. The genus ofCis two. Lets andtbe the birational transformations ofG defined by

s(x)ˆ x; s(y)ˆiy and t(x)ˆ1=x; t(y)ˆy=x³

respectively. Clearly we have s⁴ˆt²ˆidandtstˆs ¹. LetHbe the group generated bysandt. Then we haveH D₄. This group acts onC.

LetC(x;y) be th e function field ofC, wh erey²ˆx(x⁴‡ax²‡1). Clearly the invariant field of C(x;y) bys is C(x²). Let w1 andw2 be a basis of holomorphic 1-forms on C induced from dx=y and xdx=y respectively.

We have

s(w1)ˆiw1; s(w2)ˆ iw2 and t(w1)ˆ w2; t(w2)ˆ w1: LetJ(C) be the Jacobian ofC. Taking a base pointP2C, the Abel-Jacobi mapj_P is given as

jP :C !J(C) such th at jP(Q) Z^Q

P

w1; Z^Q

P

w2

0

@

1

A …modulo the lattice†;

whereQ2C. If Pis a fixed point of s, then we have Z^s(Q)

P

wˆ Z^Q

P

s(w);

forQ2C, wh erewˆw1orw2. Similarly ifP⁰is a fixed point oft, th en we have

Z^t(Q0)

P⁰

wˆ Z^Q0

P⁰

t(w);

whereQ⁰2C. Note that ifj_P:C !J(C) is defined witha base pointP, and jP⁰ :C !J(C) is defined witha base pointP⁰, thenjP⁰ ˆtjP, wheretis a translation inJ(C). Assume thatPis a fixed point ofs. Then, lettingesandet be the representations ofsandtonC²respectively, we obtain that they can be expressed aseszˆM(s)zandetzˆM(t)z‡v, wherez2C²,v2C²and

M(s)ˆ i 0

0 i

andM(t)ˆ 0 1

1 0

:

PutHˆ hM(s);M(t)i. Then we haveHD4. Note that the curvejP(C) is fixed bysandC=hsiis isomorphic to a smooth rational curve. We infer from the above arguments thatJ(C)=His isomorphic toP²andC=hsiis a line.

Letp:J(C) !J(C)=HP²be the quotient morphism. Thenp(L) can be expressed asC‡C⁰, wh ereLis the line andC⁰is a translation ofConJ(C).

Let A be an abelian surface suchthat q:A !J(C) is an etale double covering given as follows. ExpressJ(C)ˆC²=LandAˆC²=L0, whereL andL₀ are lattices satisfyingjL:L₀j ˆ2. Take an element`2 L n L<sub>0</sub> so that r(z)ˆz‡` is a translation of order two on A. Then we have L ˆ hL₀; `i. Since 2`2 L₀, we have M(2`)ˆ2M(`)2 L₀, where M2H.

Hence we infer thatsandtinduce automorphisms onA. We use the same

letterHto denote the group consisting of the elements which are induced fromH. LetGbe the automorphism group onAgenerated byHandr. Then putpˆpq:A !A=GP². Since deg (p)ˆ1610, we see thatp(L) is very ample (cf. [7, Lemma 3.5]). From Theorem A we infer that (A; p(L)) defines a Galois embedding inP⁷.

REMARK6. Note thatq(C)ˆCeis irreducible. Because, if not so, then q(C) can be written asC1‡C2. We have (q(C))²ˆ4 and C₁²ˆC₂²ˆ2.

SinceC₁ is ample, we have (C₁;C₂)1. This is a contradiction. Similarly q(C⁰)ˆCe⁰is also irreducible. The divisorCe‡Ce⁰gives the minimal Galois embedding ofA. IfLis the image ofC, thenCe‡Ce⁰ˆp(L).

EXAMPLE7. LetAbe the abelian surface with period matrix 1 0 i (1‡i)=2

0 1 0 (1‡i)=2

:

This abelian surface has the automorphisms g₁, g₂ and g₃, whose repre- sentations onC²are as follows:

ge₁z ˆ 1 0 0 1

z‡ e₁₁ e₁₂

; ge₂z ˆ 0 1

1 0

z‡ e₂₁ e22

; ge₃z ˆ i 0

0 i

z;

where the following vectors belong to the lattice generated by the column vectors of the period matrix:

2e012

; e21‡e22

e21‡e22

; e11 e12 2e21

e11‡e12

; (1 i)e11

(1 i)e12

; e21 ie22

e22‡ie21

:

We haveg12ˆg22ˆg34ˆid,g1g2g1ˆg2g²₃,g1g3g1ˆg3andg2g3g2ˆg3 1. Putting Gˆ hg1;g2;g3i, we see that G is isomorphic to the semidirect productZ₂^j D₄ andGis a subgroup ofAut(A) andA=GP². Th e very ample divisorDis given by p(L), wherep:A !A=GP² andL is a line in P². We infer from Theorem A that (A;D) defines a Galois em- bedding.

REMARK 8. The abelian surface A in Example 7 is isogenous to EiEi, whereEiˆC=(1; i). (In fact, we can show thatAis isomorphic to E_iE_i. ) Thanks to [2], this abelian surface cannot be a Jacobian of a curve.

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Manoscritto pervenuto in redazione il 14 aprile 2010.