arithmetic elliptic curves in general position (comments)

《arithmetic elliptic curves in general position (comments)》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、COMMENTSON“ARITHMETICELLIPTICCURVESINGENERALPOSITION”ShinichiMochizukiDecember2015(1.)Thenotation“ordv(−):Fv→Z”inthefinalsentenceofthefirstparagraphfollowing[GenEll],Definition1.1,shouldread“ord(−):F×→Z”.vv(2.)In[GenEll],Definition1.2,(ii),thenon-resp’dandfirstresp’ditemsinthedisplayshouldberevers

2、ed!Thatistosay,thenotation“αFβ”correspondsto“α(x)−β(x)≤C”;thenotation“αFβ”correspondsto“β(x)−α(x)≤C”.(3.)Thefirstportionofthefirstsentenceofthestatementof[GenEll],Corollary4.4,shouldread:“LetQbeanalgebraicclosureofQ;...”.(4.)The“log-diffM([EL]))”inthesecondinequalityofthefinaldisplayoftheellsta

3、tementof[GenEll],Corollary4.4,shouldread“log-diffM([EL])”.ell(5.)TheequalityhtE≈(deg(E)/deg(ωX))·htωXimplicitinthefinal“≈”ofthefinaldisplayoftheproofof[GenEll],Theorem2.1,shouldbereplacedbyaninequalityhtE2·(deg(E)/deg(ωX))·htωX[whichfollowsimmediatelyfrom[GenEll],Proposition1.4,(ii)],andtheexpr

4、ession“deg(E)/deg(ω)”intheinequalityimposedonthechoiceofshouldbereplacedXbytheexpression“2·(deg(E)/deg(ωX))”.(6.)Thephrase“Corollary2.1”inthefirstparagraphof[GenEll],§2,shouldread“Theorem2.1”.(7.)Theinequality“xS◦≤xS◦”attheendofthesecondtolastsentenceoftheproofof[GenEll],Corollary4.3,shouldr

5、ead“xS◦≤xS•”.TypesetbyAMS-TEX12SHINICHIMOCHIZUKI(8.)Supposethatweareinthesituationof[GenEll],Example1.3,(ii).LetU⊆Xbeanopensubscheme.Thena“compactlyboundedsubset”KV⊆U(Q)(⊆X(Q))ofU(Q)istobeunderstoodasasubsetwhichformsacompactlyboundedsubsetofX(Q)[i.e.,inthesensediscussedin[GenEll],Example1.3,

6、(ii)]and,moreover,satisfiesthepropertythatforeachv∈Varcdef=VV(Q)arc(respectively,v∈Vnondef=VV(Q)non),thecompactdomainK⊆Xarc(respectively,K⊆X(Q))vvvis,infact,containedinU(C)⊆X(C)=Xarc(respectively,U(Q)⊆X(Q)).vvInparticular,thisconventionshouldbeappliedtotheuseoftheterm“compactlyboundedsubset”

7、inthestatementsof[GenEll],Theorem2.1;[GenEll],Lemma3.7;[GenEll],Theorem3.8;[GenEll],Corollary4.4.Althoughthisconventionwasnotdiscussedexplicitlyin[GenEll],Example1.3,(ii),itis,ineffect,discussedexplicitlyinthediscussionof“compactlyboundedsubse