Minimal Degree Liftings of Hyperbolic Curves

[摘要] Themaingoalofthispaperistoanalyzethepropertiesofliftsofhyperellipticcurves$y_0^2=f(x_0)$overperfectfieldsofcharacteristic$p>2$(tohyperellipticcurvesovertheringofWittvectors)thathaveliftsofpointswhosecoordinatefunctionshaveminimaldegrees.Itisshownthat,whentryingtominimizethedegreesofthe$wvx$-coordinate,the$(n+1)$-thentry,say$F_n$,canbetakentobeapolynomialin$x_0$suchthat$(dp^n-(d-2))/2leqdegF_nleq(dp^n+(d-2))/2$,where$d=degf(x_0)$.Besidesupperandlowerboundsforthedegrees,othertopicsdiscussedincludeanecessaryconditiontoachievethelowerboundsandliftingtheFrobenius.Computationalaspectsarealsoconsideredandthecaseofellipticcurvesisanalyzedinmoredetail.AnexplicitformulaforderivativesofcoordinatefunctionsoftheellipticTeichm"ullerliftisproved,namely$dF_n/dx_0=0$,if$p=2$,and$dF_n/dx_0=hi^{(p^n-1)/(p-1)},y_0^{p^n-1}-sum_{i=0}^{n-1}F_i^{(p^{n-i}-1)},dF_i/dx_0$,if$pgeq3$,where$hi$istheHasseinvariantofthecurve.Finally,weestablishaconnectionbetweenminimaldegreeliftingsandMochizuki'stheoryof``canonicalliftings''inthecaseofgenus2curves.

[学科分类] 数学（综合）

[关键词] elliptic curves;canonical liftings;elliptic Teichm"uller lift;Witt vectors;minimal degree liftings;Mochizuki liftings