Families of Hyperelliptic Curves with Real Multiplication

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Families of Hyperelliptic Curves with Real Multiplication Familles de courbes hyperelliptiques à multiplications réelles Arithmetic algebraic geometry (Texel, 1989), Progr. Math. 89 (Birkhäuser Boston, 1991) J.-F. Mestre Translated from the French by Benjamin Smith This version was compiled on February 9, 2012 For all integers n, we let Gn denote the polynomial Gn(T )= bn/2c∏ k=1 ( T ?2cos (2kpi n )) , where bxc denotes the integer part of x. We say that a curve C of genus bn/2c, defined over a field k, has real multiplication by Gn if there exists a correspondence C on C such that Gn is the characteristic polynomial of the endomorphism induced by C on the regular differentials on C . The endomorphism ring of the Jacobian JC of such a curve C contains a subring isomorphic to Z[X ]/(Gn(X )) whose elements are invariant under the Rosati involution. In particular, if n is an odd prime, then JC has real multiplication by Z[2cos 2pin ] in the usual terminology (see [9], for example). In this article we construct, for all integers n ≥ 4, a 2-dimensional family of hyperelliptic curves of genus bn/2c defined over C with real multiplication by Gn .

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Families of Hyperelliptic Curves with Real Multiplication
Familles de courbes hyperelliptiques À multiplications rÉelles
Arithmetic algebraic geometry (Texel, 1989), Progr. Math.89(Birkhuser Boston, 1991)
J.-F. Mestre Translated from the French by Benjamin Smith
This version was compiled on February 9, 2012
For all integersn, we letGndenote the polynomial µ ¶ bn/2c³ ´ Y 2kπ Gn(T)=T2 cos , n k=1
wherebxcdenotes the integer part ofx. We say that a curveCof genusbn/2c, defined over a fieldk, hasreal multiplication by Gnif there exists a correspondenceConCsuch thatGnis the characteristic polynomial of the endomorphism induced byCon the regular differentials onC. The endomorphism ring of the JacobianJCof such a curveCcontains a subring isomorphic to Z[X]/(Gn(X)) whose elements are invariant under the Rosati involution. In particular, ifnis an odd 2π prime, thenJChas real multiplication byZ] in the usual terminology (see [9], for example).[2 cos n In this article we construct, for all integersn4, a 2-dimensional family of hyperelliptic curves of genusbn/2cdefined overCwith real multiplication byGn. More precisely, for every elliptic curveE defined over a fieldkof characteristic zero together with ak-rational cyclic subgroupGof ordernwe define a one-parameter family of hyperelliptic curves of genusbn/2cdefined overkwith real multipli-cation byGn. IfGis generated by ak-rational point, then the associated correspondence isk-rational. In the casen=5 we recover a known construction, due to Humbert (cf. for example [5, p. 374], [10, p. 20], and also [2]), which we recall here: letXbe a curve of genus 2 whose Jacobian has real p multiplication byZ[(1+5)/2], and letwbe the hyperelliptic involution ofX. LetCbe a plane conic, andf:X/w〉 →Can isomorphism. IfPis the image onCof a Weierstrass point ofX, then there exists a numberingP1, . . . ,P5of the images onCof the Weierstrass points ofXnot equal toPsuch that there exists a conic passing throughPand inscribed in the pentagon formed byP1, . . . ,P5(that is, tangent to the linesP1P2,P2P3, . . . ,P5P1this statement with the elliptic curve-theoretic). Comparing interpretation of Poncelet’s theorem, we see that the data ofXis equivalent to the data of an elliptic curveEwith a point of order 5, a double coveringφfromEto a curve of genus 0, and a point of this curve distinct from the 4 ramification points ofφ. We construct the family of hyperelliptic curves mentioned above in §1. More generally, for each isogenyf:E1E2of elliptic curves defined over a fieldkwe define a hyperelliptic curveCfoverk(T), whereTis a free parameter; for each elementRof the kernel offthere is an associated correspon-denceCRonCf, such that the characteristic polynomial of the endomorphism induced byCRon the regular differentials onCfis a product of polynomialsGm. This construction allows us, for example, to obtain a 2-parameter family, defined overQ, of hyper-elliptic curves of genus 19 whose Jacobians are isogenous to a product of 19 elliptic curves. We give some examples based on some isogenies with cyclic kernels in §2. Forn=the curve5, 7, 9, X1(n) classifying elliptic curves equipped with a point of ordernisQ-isomorphic to the projective line.
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