Some Corrections from Bhupendranathtiwari

Some corrections from BhupendraNathTiwari.

EllipticCurves

p.31(4.2)whereg^{\star}(X,Y,Z)=...

Oneseeeasilythat...Oneseeseasilythat...

p.49(Isogenies)Twoellipticcurves...

Thusaregularmap\verphi:E->E^{\prime}isanasigenyif\verphi(O)=O^{\prime}

and\verphi(k^{al}):E^{al}->E^{al}issurjective.--->

Thusaregularmap\verphi:E->E^{\prime}isanasigenyif\verphi(O)=O^{\prime}

and\verphi(k^{al}):E^{al}->E^{\prime}^{al}issurjective.

p.66(5.8)Wenowbegin...

AfterProposition4.2,...--->Aftercorrolary4.2,...

p.73(6.2)...hasaninverseimagethecurve

Y^2X^2=X^3+aX,...--->Y^2X^2=X^3+\alphaX,...

p.89(ThenotionofaRiemannsurface)LetXbeaconnected...

tobeequivalentifthetheirunionisalsoacoordinatecovering...--->

tobeequivalentiftheirunionisalsoacoordinatecovering...

p.97(3.17PROOF)

Oneliminating$\alpha$frombetweenthetwoequations,wefindthat

c\tau^2+(d-a)\tau+b=0----->c\tau^2+(d-a)\tau-b=0

p.97(3.17PROOF)

...Oneliminating$\tau$frombetweenthetwoequations,wefindthat

\alpha^2-(a+d)\alpha+bc=0---->\alpha^2-(a+d)\alpha+ad-bc=0

p.120(4.2PROOF)Itremainstoshowthat

max(|ac|,|ad+bc|,|bd|)\ge½(max(|a|,|b|)(max(|c||d|)--->

max(|ac|,|ad+bc|,|bd|)\ge½(max(|a|,|b|)(max(|c|,|d|))

p.128(5.4)(b)...thatthePontryagindualofthissequenceisanexactsequence:...

Inthissequence,the\prod_{p,\infty}H^1(Q_p,E)--->\prod_{p=2,\infty} H^1(Q_p,E)

p.134Amorphism...

issimplyamapA-sets.We...--->issimplyamapofA-sets.We...

p.134(Twistofellipticcurves)

Recall(Theorem2.1,Chap.I)thatE(a,b)and...--->

Recall(Theorem2.1,Chap.II)thatE(a,b)and...

p.135(7.12)Inordertobeabletoapply...

WeapplyTheorem2.1,Chap.Iinthecase(a^{\prime},b^{\prime})=(a,b).--->

WeapplyTheorem2.1,Chap.IIinthecase(a^{\prime},b^{\prime})=(a,b).

p.141(Curvesofgenus1)Let\sigma\in\Gamma....

Now(see2.3)thereisacurveE_0.. --->Now(see2.3,Chap.II)thereisacurveE_0...

p.149(9.2PROOF)...issurjective(4.23).Inparticular,thereexistsa...--->

...issurjective(4.23,Chap.I).Inparticular,thereexistsa...

p.170(Spha\erePacking)LetEbeaconstantellipticcurve...

thenpart(a)ofTheorem11.2givesusr,...--->thenpart(a)ofProposition11.1givesusr,...

p.171(11.4PROOF)Sinceall\alpha_iand\omega_jequal-q,

iffollowsfrom(11.2a)that...--->itfollowsfrom(11.1a)that...

p.174(1.1)Forexample,..

whoseininverseC^{\times}->C/Zis...--->whoseinverseinC^{\times}->C/Zis...

p.193.(3.7PROOF)Fromthetheorem,weknowthat...

of\Delta,anditwillbefollowfromthe...--->of\Delta,anditwillfollowfromthe...

p.195.(4.1PROOF)then

(\alpha\verphi)(x:y:z)=(P(x^p,y^p,z^p),...)and

(\verphi\alpha)(x:y:z)=(P(x,y,z)^p,...)--->

(\alpha\verphi)(x:y:z)=(P(x^p,y^p,z^p):...)and

(\verphi\alpha)(x:y:z)=(P(x,y,z)^p:...)

(intherhsoftheseEqns,the,inouterbracketsshouldbe:)

p.199(4.7)ThemapsT(n)havethehavingthefollowingproperties:--->

ThemapsT(n)havethefollowingproperties:

p.212wherebotharedefined....

Onecancheckthatitalsoholomorphicatthepointatinfinity.--->

Onecancheckthatitisalsoholomorphicatthepointatinfinity.

p.215NowsupposeXisa...

overanfieldk.Weilshowed...--->overafieldk.Weilshowed...

p.219Wheni\ne0,...

...where\tilde{E}_i/\tilde{E}/S_iand...--->

...where\tilde{E}_i=\tilde{E}/S_iand...

General:Someplacesyouneedtoreplacedfbydef,oraddinthelistofnotation:def=df.