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151(2): 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/07/15(水) 23:21:11.85 ID:hRRJMwM+(4/6) AAS
>>148 追加
https://mathoverflow.net/questions/2022/definition-and-meaning-of-the-conductor-of-an-elliptic-curve
<mathoverflow>
Definition and meaning of the conductor of an elliptic curve
(抜粋)
I never really understood the definition of the conductor of an elliptic curve.
asked Oct 23 '09 at 3:15
Sam Derbyshire
5 Answers
＜39＞
Saito proved that
Art(X/R)=ν(Δ)
where Δ∈R is the ''discriminant'' of X which mesures the defect of a functorial isomorphism which involves powers of the relative dualizing sheaf of X/R.
When C is an elliptic curve, one can prove that Δ is actually the discriminant of a minimal Weierstrass equation over R, and le tour est joue !
This paper of Saito was apparently not very known by the number theorists. Some more details are given in a text (in French).
http://www.ufr-mi.u-bordeaux.fr/~liu/Notes/ogg.ps
So Ogg's formula should be called Ogg-Saito's formula. That some people do.
answered Jan 26 '10 at 22:50
Qing Liu
152: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/07/15(水) 23:28:35.83 ID:hRRJMwM+(5/6) AAS
>>151 追加
https://www.lmfdb.org/knowledge/show/ec.conductor
LMFDB
Conductor of an elliptic curve (reviewed)
(抜粋)
The conductor of an elliptic curve E defined over a number field K is an ideal of the ring of integers of K that is divisible by the prime ideals of bad reduction and no others.
153: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/07/15(水) 23:41:30.47 ID:hRRJMwM+(6/6) AAS
>>151 追加
これは、米高校生の数学ソフトによる 計算レポートだが
なかなかレベル高いね
https://scholarcommons.sc.edu/cgi/viewcontent.cgi?article=1194&context=jscas
The Relationship between Conductor and Discriminant of an Elliptic Curve over Q
Nico Adamo
Heathwood Hall Episcopal School, 9th Grade, Columbia SC
(抜粋)
Saito (1988) establishes a relationship between two invariants associated with a smooth projective curve,
the conductor and discriminant. Saito defined the conductor of an arbitrary scheme
of finite type using p-adic etale cohomology. He used a definition of Deligne for the discriminant
as measuring defects in a canonical isomorphism between powers of relative dualizing sheaf of
smooth projective curves. The researcher in this paper uses the fact that this relationship is
analogous to that of conductor to discriminant in the case of elliptic curves, Saito’s result, as
well as analysis of data on conductors and discriminants to determine whether patterns exist
between discriminant and conductor of elliptic curves. The researcher finds such patterns do
in fact exist and discusses two main patterns: that of the conductor dividing the discriminant
and that of the conductor ”branching” in a predictable way. These patterns also allow for
easier algorithms for computing conductors.
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