Periodic continued fractions and hyperelliptic curves

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1、PERIODICCONTINUEDFRACTIONSANDHYPERELLIPTICCURVESM-P.GROSSETANDA.P.VESELOVAbstract.Weinvestigatewhenanalgebraicfunctionoftheformφ(λ)=√−B(λ)+R(λ),whereR(λ)isapolynomialofodddegreeN=2g+1withA(λ)coefficientsinC,canbewrittenasaperiodicα-fractionoftheformλ−α1φ(λ)=[b0;b1,b2,...,bN

2、]α=b0+,λ−α2b1+b2+...b+λ−αNN−1λ−α1bN+λ−α2b1+b2+...forsomefixedsequenceαi.WeshowthatthisproblemhasanaturalanswergivenbytheclassicaltheoryofhyperellipticcurvesandtheirJacobivarieties.Wealsoconsiderpureperiodicα-fractionexpansionscorrespondingtothespecialcasewhenbN=b0.1.Introd

3、uctionConsiderthefollowingcontinuedfraction,whichwewillcallα-fractions:λ−α1(1)φ=b0+λ−α2=[b0,b1,...,]α,b1+b2+...whereα=(αi),αi∈Cisagivensequence,biarearbitrarycomplexnumbers,λisaformalparameter.InthispaperwewillconsideraspecialcaseofN-periodicα-fractions,whenthesequencesαi

4、andbiareperiodicwithperiodN:arXiv:math/0701932v1[math.GM]31Jan2007αi+N=αi,bi+N=biforalli≥1:(2)φ=[b0;b1,b2,...,bN]α.IntheparticularcasewhenbN=b0wehaveφ=[b0,b1,...,bN−1]α,whichwillbecalledapureN-periodicα-fraction.Thiskindoffractionsnaturallyappearinthetheoryofintegrablesys

5、tems,inparticularinthetheoryofperiodicdressingchain[1],buttothebestofourknowl-edgehasnotbeenstudiedsofar.WewerepartlyinspiredbyourrecentdiscussionswithVassilisPapageorgiouonthediscreteKdVequationwheresuchcontinuedfractionsappearaswell[2].12M-P.GROSSETANDA.P.VESELOVBecause

6、ofperiodicitywecanwriteformally(2)asλ−α1φ=b0+,λ−α2b1+b2+...b+λ−αNN−1bN−b0+φwhichimpliesaquadraticrelation(3)A(λ)φ2+2B(λ)φ+C(λ)=0,whereA,B,Carecertainpolynomialsinλwithcoefficientspolynomiallydependingonbi.Thustoanyperiodicα-fraction(2)correspondsanalgebraicfunctionp−B(λ)+R(

7、λ)(4)φ(λ)=,A(λ)where2(5)R(λ)=B(λ)−A(λ)C(λ)isthediscriminantof(3).Inthatcasewewillsaythat(2)isaperiodicα-fractionpexpansionofthealgebraicfunction(4)fromthehyperellipticextensionC(λ,R(λ))ofthefieldofrationalfunctionsC(λ).Weleavethequestionofconvergenceasideconcentratingonalg

8、ebraicandgeometricaspectsoftheproblem.Wewilldiscussthefollowingthreemainquestions.Question1.Whic