Degree even coverings of elliptic curves by genus 2 curves
AALBANIAN JOURNALOF MATHEMATICSVolume 2, Number 3, Pages 241–248
ISSN
DEGREE EVEN COVERINGS OF ELLIPTIC CURVES BYGENUS 2 CURVES
N. Pjero, M. Ramasac¸o
Dep. of Mathematics,University of Vlora, [email protected], [email protected]
T. Shaska
Dep. of Computer Science and Electrical EngineeringUniversity of Vlora, Albania
Abstract.
In this survey we study the genus 2 curves with ( n, n )-split Jaco-bian for even n . Introduction
Let C be a genus 2 curve defined over an algebraically closed field k , of charac-teristic zero. Let ψ : C → E be a degree n maximal covering (i.e. does not factorthrough an isogeny) to an elliptic curve E defined over k . We say that C has a degree n elliptic subcover . Degree n elliptic subcovers occur in pairs. Let ( E, E (cid:48) )be such a pair. It is well known that there is an isogeny of degree n between theJacobian J C of C and the product E × E (cid:48) . The locus of such C , denoted by L n , isa 2-dimensional algebraic subvariety of the moduli space M of genus two curvesand has been the focus of many papers in the last decade; see [5, 7, 8, 9, 10, 1, 2].The space L was studied in Shaska/V¨olklein [9]. The space L was studied in[5] were an algebraic description was given as sublocus of M . Lately the space L has been studied in detail in [10]. The case of even degree has been less studiedeven though there have been some attempts lately to compute some of the cases for n = 4; see [4]. In this survey we study the genus 2 curves with ( n, n )-split Jacobianfor small n . While such curves have been studied by many authors, our approachis simply computational.2. Curves of genus 2 with split Jacobians
Most of the results of this section can be found in [11]. Let C and E be curvesof genus 2 and 1, respectively. Both are smooth, projective curves defined over k , char ( k ) = 0. Let ψ : C −→ E be a covering of degree n . From the Riemann-Hurwitz formula, (cid:80) P ∈ C ( e ψ ( P ) −
1) = 2 where e ψ ( P ) is the ramification index ofpoints P ∈ C , under ψ . Thus, we have two points of ramification index 2 or onepoint of ramification index 3. The two points of ramification index 2 can be inthe same fiber or in different fibers. Therefore, we have the following cases of thecovering ψ : c (cid:13) (Albanian J. Math.) a r X i v : . [ m a t h . AG ] M a y
42 DEGREE EVEN COVERINGS OF ELLIPTIC CURVES BY GENUS 2 CURVES
Case I:
There are P , P ∈ C , such that e ψ ( P ) = e ψ ( P ) = 2, ψ ( P ) (cid:54) = ψ ( P ),and ∀ P ∈ C \ { P , P } , e ψ ( P ) = 1. Case II:
There are P , P ∈ C , such that e ψ ( P ) = e ψ ( P ) = 2, ψ ( P ) = ψ ( P ),and ∀ P ∈ C \ { P , P } , e ψ ( P ) = 1. Case III:
There is P ∈ C such that e ψ ( P ) = 3, and ∀ P ∈ C \ { P } , e ψ ( P ) = 1.In case I (resp. II, III) the cover ψ has 2 (resp. 1) branch points in E.Denote the hyperelliptic involution of C by w . We choose O in E such that w restricted to E is the hyperelliptic involution on E . We denote the restriction of w on E by v , v ( P ) = − P . Thus, ψ ◦ w = v ◦ ψ . E[2] denotes the group of 2-torsionpoints of the elliptic curve E, which are the points fixed by v . The proof of thefollowing two lemmas is straightforward and will be omitted. Lemma 1. a) If Q ∈ E , then ∀ P ∈ ψ − ( Q ) , w ( P ) ∈ ψ − ( − Q ) .b) For all P ∈ C , e ψ ( P ) = e ψ ( w ( P )) . Let W be the set of points in C fixed by w . Every curve of genus 2 is given, upto isomorphism, by a binary sextic, so there are 6 points fixed by the hyperellipticinvolution w , namely the Weierstrass points of C . The following lemma determinesthe distribution of the Weierstrass points in fibers of 2-torsion points. Lemma 2.
The following hold: (1) ψ ( W ) ⊂ E [2](2) If n is an odd number theni) ψ ( W ) = E [2] ii) If Q ∈ E [2] then ( ψ − ( Q ) ∩ W ) = 1 mod (2)(3) If n is an even number then for all Q ∈ E [2] , ( ψ − ( Q ) ∩ W ) = 0 mod (2)Let π C : C −→ P and π E : E −→ P be the natural degree 2 projections. Thehyperelliptic involution permutes the points in the fibers of π C and π E . The ramifiedpoints of π C , π E are respectively points in W and E [2] and their ramification indexis 2. There is φ : P −→ P such that the diagram commutes.(1) C π C −→ P ψ ↓ ↓ φE π E −→ P Next, we will determine the ramification of induced coverings φ : P −→ P . Firstwe fix some notation. For a given branch point we will denote the ramificationof points in its fiber as follows. Any point P of ramification index m is denotedby ( m ). If there are k such points then we write ( m ) k . We omit writing symbolsfor unramified points, in other words (1) k will not be written. Ramification databetween two branch points will be separated by commas. We denote by π E ( E [2]) = { q , . . . , q } and π C ( W ) = { w , . . . , w } .2.0.1. The Case When n is Even. Let us assume now that deg ( ψ ) = n is an evennumber. The following theorem classifies the induced coverings in this case. Theorem 1. If n is an even number then the generic case for ψ : C −→ E inducethe following three cases for φ : P −→ P : I: (cid:16) (2) n − , (2) n − , (2) n − , (2) n , (2) (cid:17) EGREE EVEN COVERINGS OF ELLIPTIC CURVES BY GENUS 2 CURVES 243
II: (cid:16) (2) n − , (2) n − , (2) n , (2) n , (2) (cid:17) III: (cid:16) (2) n − , (2) n , (2) n , (2) n , (2) (cid:17) Each of the above cases has the following degenerations (two of the branch pointscollapse to one) I: (1) (cid:16) (2) n , (2) n − , (2) n − , (2) n (cid:17) (2) (cid:16) (2) n − , (2) n − , (4)(2) n − , (2) n (cid:17) (3) (cid:16) (2) n − , (2) n − , (2) n − , (4)(2) n − (cid:17) (4) (cid:16) (3)(2) n − , (2) n − , (2) n − , (2) n (cid:17) II: (1) (cid:16) (2) n − , (2) n − , (2) n , (2) n (cid:17) (2) (cid:16) (2) n − , (2) n , (2) n , (2) n (cid:17) (3) (cid:16) (4)(2) n − , (2) n − , (2) n , (2) n (cid:17) (4) (cid:16) (2) n − , (4)(2) n − , (2) n , (2) n (cid:17) (5) (cid:16) (2) n − , (2) n − , (2) n − , (2) n (cid:17) (6) (cid:16) (3)(2) n − , (2) n − , (4)(2) n , (2) n (cid:17) (7) (cid:16) (2) n − , (3)(2) n − , (2) n , (2) n (cid:17) III: (1) (cid:16) (2) n − , (2) n , (2) n , (4)(2) n (cid:17) (2) (cid:16) (2) n − , (4)(2) n − , (2) n , (2) n (cid:17) (3) (cid:16) (2) n , (2) n , (2) n , (4)(2) n − (cid:17) (4) (cid:16) (3)(2) n − , (2) n , (2) n , (2) n (cid:17) Proof.
We skip the details of the proof. (cid:3)
Remark 1.
The case n = 8 is the first true generic case when all the subcasesoccur. Maximal coverings ψ : C −→ E . Let ψ : C −→ E be a covering of degree n from a curve of genus 2 to an elliptic curve. The covering ψ : C −→ E is calleda maximal covering if it does not factor through a nontrivial isogeny. A map ofalgebraic curves f : X → Y induces maps between their Jacobians f ∗ : J Y → J X and f ∗ : J X → J Y . When f is maximal then f ∗ is injective and ker ( f ∗ ) is connected,see [8] for details.Let ψ : C −→ E be a covering as above which is maximal. Then ψ ∗ : E → J C is injective and the kernel of ψ , ∗ : J C → E is an elliptic curve which we denote by E ; see [2]. For a fixed Weierstrass point P ∈ C , we can embed C to its Jacobianvia i P : C −→ J C x → [( x ) − ( P )](2)
44 DEGREE EVEN COVERINGS OF ELLIPTIC CURVES BY GENUS 2 CURVES
Let g : E → J C be the natural embedding of E in J C , then there exists g ∗ : J C → E . Define ψ = g ∗ ◦ i P : C → E . So we have the following exact sequence0 → E g −→ J C ψ , ∗ −→ E → → E ψ ∗ −→ J C g ∗ −→ E → deg ( ψ ) is an odd number then the maximal covering ψ : C → E is unique upto isomorphism of elliptic curves. If the cover ψ : C −→ E is given, and therefore φ , we want to determine ψ : C −→ E and φ . The study of the relation betweenthe ramification structures of φ and φ provides information in this direction. Thefollowing lemma (see [2, pg. 160]) answers this question for the set of Weierstrasspoints W = { P , . . . , P } of C when the degree of the cover is odd. Lemma 3.
Let ψ : C −→ E , be maximal of degree n . Then, the map ψ : C → E is a maximal covering of degree n . Moreover, i) if n is odd and O i ∈ E i [2] , i = 1 , are the places such that ψ − i ( O i ) ∩ W ) = 3 , then ψ − ( O ) ∩ W and ψ − ( O ) ∩ W form a disjoint union of W . ii) if n is even and Q ∈ E [2] , then (cid:0) ψ − ( Q ) (cid:1) ∩ W = 0 or 2. The above lemma says that if ψ is maximal of even degree then the correspondinginduced covering can have only type I ramification, see Theorem 1. Example 1.
Let ψ : C → E be a degree n = 8 maximal covering of the ellipticcurve E by a genus 2 curve C . Then, we have Type I covering as in previoustheorem. Hence, the ramification is (cid:0) (2) , (2) , (2) , (2) , (2) (cid:1) This case is the first case which has all its subcases with ramifications as follows: i) (cid:0) (2) , (2) , (2) , (2) (cid:1) ii) (cid:0) (2) , (2) , (4)(2) , (2) (cid:1) iii) (cid:0) (2) , (2) , (2) , (4)(2) (cid:1) iv) (cid:0) (3)(2) , (2) , (2) , (2) (cid:1) The locus of genus 2 curves in the generic case is a 2-dimensional subvariety of themoduli space M . It would be interesting to explicitly compute such subvarietysince it is the first case which could give some clues to what happens in the generalcase for even degree.3. The locus of genus two curves with ( n, n ) split Jacobians In this section we will discuss the Hurwitz spaces of coverings with ramificationas in the previous section and the Humbert spaces of discriminant n .3.1. Hurwitz spaces of covers φ : P → P . Two covers f : X → P and f (cid:48) : X (cid:48) → P are called weakly equivalent if there is a homeomorphism h : X → X (cid:48) and an analytic automorphism g of P (i.e., a Moebius transformation) such that g ◦ f = f (cid:48) ◦ h . The covers f and f (cid:48) are called equivalent if the above holds with g = 1.Consider a cover f : X → P of degree n , with branch points p , ..., p r ∈ P .Pick p ∈ P \ { p , ..., p r } , and choose loops γ i around p i such that γ , ..., γ r is a EGREE EVEN COVERINGS OF ELLIPTIC CURVES BY GENUS 2 CURVES 245 standard generating system of the fundamental group Γ := π ( P \ { p , ..., p r } , p ),in particular, we have γ · · · γ r = 1. Such a system γ , ..., γ r is called a homotopybasis of P \ { p , ..., p r } . The group Γ acts on the fiber f − ( p ) by path lifting,inducing a transitive subgroup G of the symmetric group S n (determined by f up to conjugacy in S n ). It is called the monodromy group of f . The imagesof γ , ..., γ r in S n form a tuple of permutations σ = ( σ , ..., σ r ) called a tuple of branch cycles of f .We say a cover f : X → P of degree n is of type σ if it has σ as tuple of branchcycles relative to some homotopy basis of P minus the branch points of f . Let H σ be the set of weak equivalence classes of covers of type σ . The Hurwitz space H σ carries a natural structure of an quasiprojective variety.We have H σ = H τ if and only if the tuples σ , τ are in the same braid orbit O τ = O σ . In the case of the covers φ : P → P from above, the correspondingbraid orbit consists of all tuples in S n whose cycle type matches the ramificationstructure of φ .This and the genus of H σ in the degenerate cases (see the following table) hasbeen computed in GAP by the BRAID PACKAGE written by K. Magaard.deg Case cycle type of σ O σ ) G dim H σ genus of H σ , , , ,
2) 224 S , , , ) 4 16 1 02 (2 , , (4)(2) , ) 48 S , , , (4)(2) ) 96 S , , , ) 36 S Table 1.
The length of braid orbits, the order of the group, andthe genus of 1-dimensional subspaces for even degree maximal cov-erings.As the reader can imagine even such computations are not easy for higher n . Itis unclear what are the monodromy groups that appear in all the subcases and theformulas for the lengths of the braid orbits.3.2. Humbert surfaces.
Let A denote the moduli space of principally polarizedabelian surfaces. It is well known that A is the quotient of the Siegel upper halfspace H of symmetric complex 2 × Sp ( Z ).Let ∆ be a fixed positive integer and N ∆ be the set of matrices τ = (cid:18) z z z z (cid:19) ∈ H such that there exist nonzero integers a, b, c, d, e with the following properties: az + bz + cz + d ( z − z z ) + e = 0∆ = b − ac − de (3)The Humbert surface H ∆ of discriminant ∆ is called the image of N ∆ underthe canonical map H → A := Sp ( Z ) \ H .
46 DEGREE EVEN COVERINGS OF ELLIPTIC CURVES BY GENUS 2 CURVES
It is known that H ∆ (cid:54) = ∅ if and only if ∆ > ≡ or C defined over C , [ C ] belongs too L n if and only if theisomorphism class [ J C ] ∈ A of its (principally polarized) Jacobian J C belongs tothe Humbert surface H n , viewed as a subset of the moduli space A of principallypolarized abelian surfaces. There is a one to one correspondence between the pointsin L n and points in H n . Thus, we have the map: H σ −→ L n −→ H n ([ f ] , ( p , . . . , p r ) → [ X ] → [ J X ](4)In particular, every point in H n can be represented by an element of H of theform τ = (cid:18) z n n z (cid:19) , z , z ∈ H . There have been many attempts to explicitly describe these Humbert surfaces.For some small discriminant this has been done by several authors; see [9], [5].Geometric characterizations of such spaces for ∆ = 4 , ,
9, and 12 were given byHumbert (1900) in [3] and for ∆ = 13 , , ,
20, 21 by Birkenhake/Wilhelm (2003).4.
Computing the locus L n in M We take the most general case for maximal coverings of even degree, namely n ,Type I. The ramification structure of φ : P x → P z is (cid:16) (2) n − , (2) n − , (2) n − , (2) n , (2) (cid:17) We denote the branch points respectively q , . . . , q . Let q = 0 , q = 1 , q = ∞ .The red places in P x denote the unramified places and the black places all haveramification index 2. We pick the coordinate x such that it is x = 0 , x = 1 , x = ∞ in the unramified places of P z and respectively in the fibers of 0 , , ∞ as in thepicture.There are exactly d = n − places of index 2 in φ − (0). Let P ( x ) denote thepolynomial whose roots are exactly these places. Similarly denote by R ( x ) , Q ( x )such polynomials for fibers of 1 and ∞ . The other unramified places in the fibersof 0,1, ∞ we denote by w , w , w respectively.Then, we have z = λ · x x − w x − w · P ( x ) Q ( x )for some λ · ∈ C , λ (cid:54) = 0. Furthermore, z − λ · ( x − · x − w x − w · R ( x ) Q ( x )where P ( x ) , Q ( x ) , R ( x ) are monic polynomials of degree d = n − with no multipleroots and no common roots. EGREE EVEN COVERINGS OF ELLIPTIC CURVES BY GENUS 2 CURVES 247
Substituting for z we get a degree n equation λx ( x − w ) P ( x ) − ( x − w ) Q ( x ) − λ · ( x − x − w ) R ( x ) = 0By equaling coefficients of this polynomial with zero we get a nonlinear system of n + 1 equations. In the same way we get the corresponding equations from thefibers of the other two branch points s and t . Solving such system would determinealso w , w , w . The equation of the genus 2 curve C is given by y = x ( x − x − w )( x − w )( x − w )4.1. Degree 4 covers.
In this section we focus on the case deg( φ ) = 4 (not neces-sarily maximal). The goal is to determine all ramifications σ and explicitly compute L ( σ ). There is one generic case and one degenerate case in which the ramificationof deg( φ ) = 4 applies, as given by the above possible ramification structures.i) (2 , , , ,
2) (generic)ii) (2 , , ,
4) (degenerate)4.2.
Degenerate Case.
In this case one of the Weierstrass points has ramificationindex 3, so the cover is totally ramified at this point.Let the branch points be 0, 1, λ , and ∞ , where ∞ corresponds to the elementof index 4. Then, above the fibers of 0, 1, λ lie two Weierstrass points. The twoWeierstrass points above 0 can be written as the roots of a quadratic polynomial x + ax + b ; above 1, they are the roots of x + px + q ; and above λ , they are theroots of x + sx + t . This gives us an equation for the genus 2 curve C : C : y = ( x + ax + b )( x + px + q )( x + sx + t ) . The four branch points of the cover φ are the 2-torsion points E [2] of the ellipticcurve E , allowing us to write the elliptic subcover as E : y = x ( x − x − λ ) .
48 DEGREE EVEN COVERINGS OF ELLIPTIC CURVES BY GENUS 2 CURVES
We have the following theorem:
Theorem 2.
Let C be a genus 2 curve with a degree 4 degenerate elliptic subcover.Then C is isomorphic to the curve given by C : y = (cid:18) − b − b ) x + x (cid:19) (cid:18)
112 ( b − b + 13 ( b − x + x (cid:19)(cid:18) b −
23 ( b + 2) x + x (cid:19) E : v = u ( u − (cid:18) u − b (4 − b )16( b − (cid:19) (5) where the corresponding discriminants of the right sides must be non-zero. Hence, ∆ C : = b ( b − b − b − b ) (cid:54) = 0(6) ∆ E : = ( b − ( b − b ( b + 2) b − (cid:54) = 0 . (7) and its invariants satisfy J J − J J J + 5009676947631 J − J + 1176812184652746480 J J + 12448207102988800000 J − J J = 0186626560000 J J + 138962144767343358744576000000 J + 28242953648110 J + 619923800736 J J − J − J J + 266576269949878792320 J J − J J + 693067624145203200000 J J J + 1763516708182388736000 J J J = 0 . (8) Proof.
See [4]. (cid:3)
References [1]
G. Frey , On elliptic curves with isomorphic torsion structures and corresponding curves ofgenus 2.
Elliptic curves, modular forms, and Fermat’s last theorem (Hong Kong, 1993) ,79-98, Ser. Number Theory, I,
Internat. Press, Cambridge, MA , 1995.[2]
G. Frey and E. Kani , Curves of genus 2 covering elliptic curves and an arithmetic applica-tion.
Arithmetic algebraic geometry (Texel, 1989) , 153-176,
Progr. Math. , 89, Birkh¨auserBoston, MA, 1991.[3]
G. Humbert
Sur les fonctionnes ab´eliennes singuli`eres. I, II, III. J. Math. Pures Appl. serie5, t. V, 233–350 (1899); t. VI, 279–386 (1900); t. VII, 97–123 (1901).[4]
T. Shaska, S. Wijesiri, S. Wolf, L. Woodland , Degree four coverings of elliptic curves bygenus two curves. Albanian J. Math, vol. 2, Nr. 4, 2008.[5]
T. Shaska , Genus 2 curves with degree 3 elliptic subcovers,
Forum. Math. , vol. , 2, pg.263-280, 2004.[6] T. Shaska , Computational algebra and algebraic curves, ACM,
SIGSAM Bulletin , Comm.Comp. Alg. , Vol. , No. 4, 117-124, 2003.[7] T. Shaska , Genus 2 curves with (3,3)-split Jacobian and large automorphism group, Algo-rithmic Number Theory (Sydney, 2002), , 205-218, Lect. Not. in Comp. Sci. , 2369, Springer,Berlin, 2002.[8]
T. Shaska , Curves of genus 2 with ( n, n )-decomposable Jacobians,
J. Symbolic Comput.
T. Shaska and H. V¨olklein , Elliptic subfields and automorphisms of genus two fields,
Algebra, Arithmetic and Geometry with Applications , pg. 687 - 707, Springer (2004).[10]
K. Magaard, T. Shaska, H. V¨olklein , Genus 2 curves with degree 5 elliptic subcovers,Forum Math..[11]