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Finite Honda systems and supersingul...
~
Conrad, Brian David.
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Finite Honda systems and supersingular elliptic curves.
紀錄類型:
書目-電子資源 : Monograph/item
正題名/作者:
Finite Honda systems and supersingular elliptic curves./
作者:
Conrad, Brian David.
面頁冊數:
163 p.
附註:
Source: Dissertation Abstracts International, Volume: 57-03, Section: B, page: 1840.
Contained By:
Dissertation Abstracts International57-03B.
標題:
Mathematics. -
電子資源:
http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=9622640
Finite Honda systems and supersingular elliptic curves.
Conrad, Brian David.
Finite Honda systems and supersingular elliptic curves.
- 163 p.
Source: Dissertation Abstracts International, Volume: 57-03, Section: B, page: 1840.
Thesis (Ph.D.)--Princeton University, 1996.
The proof of the semistable Taniyama-Shimura Conjecture by Wiles and Taylor-Wiles uses as its central tool the deformation theory of Galois representations. Some restrictions in the application of Wiles' method were removed by Diamond, thus proving that an elliptic curve Subjects--Topical Terms:
515831
Mathematics.
Finite Honda systems and supersingular elliptic curves.
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Conrad, Brian David.
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Finite Honda systems and supersingular elliptic curves.
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163 p.
500
$a
Source: Dissertation Abstracts International, Volume: 57-03, Section: B, page: 1840.
500
$a
Adviser: Andrew Wiles.
502
$a
Thesis (Ph.D.)--Princeton University, 1996.
520
$a
The proof of the semistable Taniyama-Shimura Conjecture by Wiles and Taylor-Wiles uses as its central tool the deformation theory of Galois representations. Some restrictions in the application of Wiles' method were removed by Diamond, thus proving that an elliptic curve
$
is modular if it is either semistable at 3 and 5 or is just semistable at 3, provided that the representation
$\
bar\rho\sb{E,3}{:}{\rm Gal}(\bar{\bf Q}/{\bf Q}(\sqrt{-3}))\to{\rm Aut}(E\lbrack 3\rbrack(\bar{\bf Q}))\simeq{\rm GL}\sb2({\bf F}\sb3)
$i
s absolutely irreducible. However, the most critical restriction in Wiles' method comes from the fact that hitherto the local deformation theory has only been understood when the residual representation has good or potentially ordinary reduction.
520
$a
In this thesis, we will develop the local deformation theory further, so that it includes some of the remaining cases. This is only one part of the problem and more work is needed to apply this to proving that new families of elliptic curves are modular. Consider a continuous absolutely irreducible representation
$\
bar\rho{:}{\rm Gal}(\bar{\bf Q}\sb{p}/{\bf Q}\sb{p})\to{\rm GL}\sb2({\bf F}\sb{p}),
$
with det
$\
bar\rho
$
cyclotomic, and a finite extension
$
inside of
$\
bar{\bf Q}\sb{p}
$
with
$e
= e(K/{\bf Q}\sb{p})
$
satisfying
$e
\le p-1,
$
gcd(
$e
,p-1)=1,
$
and
$e
\vert (p+1).
$
These conditions have natural motivations and are always satisfied when
$e
=1.
$
Moreover, the latter two conditions are not very essential; these merely make certain calculations easier to carry out. Also, assume that
$\
bar\rho\vert\sb{{\rm Gal}(\bar{\bf Q}\sb{p}/K)}
$
is the generic fiber of a finite flat group scheme over
${
\cal O}\sb{K}
$,
required to be unipotent if
$e
=p-1.
520
$a
There is a universal deformation ring
$
classifying deformations
$\
rho
$
of
$\
bar\rho
$
to complete local noetherian
${
\bf Z}\sb{p}
$-
algebras R with residue field
${
\bf F}\sb{p}
$
such that
$\
rho\vert\sb{{\rm Gal}(\bar{\bf Q}\sb{p}/K)}
$
mod
${
\bf m}\sbsp{R}{n}
$
is the generic fiber of a finite flat group scheme over
${
\cal O}\sb{K}
$
for all
$n
\ge 1.
$
The main result of this thesis is that
$.
We also obtain similar results in more general settings.
520
$a
For the problem of proving modularity under the hypothesis that
$\
bar\rho\sb{E,5}
$
is irreducible and modular (which in fact covers all remaining cases in which there is a semistable twist at 3), we have essentially worked out the deformation-theoretic aspect of the problem at
$p
=5.
$
This is because the conditions on e and p above are satisfied when
$e
=3
$
and
$p
=5,
$
and this turns out to he the only remaining case when
$\
bar\rho\sb{E,5}
$
is irreducible and modular.
590
$a
School code: 0181.
650
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Mathematics.
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Wiles, Andrew,
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advisor
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1996
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http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=9622640
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