IUTを読むための用語集資料集スレ

[過去ﾛｸﾞ] IUTを読むための用語集資料集スレ (1002ﾚｽ)
上下前次1-新
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
抽出解除必死ﾁｪｯｶｰ(本家) (べ) 自ID ﾚｽ栞あぼーん

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

148: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/07/15(水) 22:08:22.61 ID:hRRJMwM+

>>139
追加

導手：Conductor of an elliptic curve

https://en.wikipedia.org/wiki/Conductor_of_an_elliptic_curve
Conductor of an elliptic curve
(抜粋)
Contents
1 History
2 Definition
3 Ogg's formula
4 Global conductor
5 References
6 Further reading

History
The conductor of an elliptic curve over a local field was implicitly studied (but not named) by Ogg (1967) in the form of an integer invariant ε+δ which later turned out to be the exponent of the conductor.

The conductor of an elliptic curve over the rationals was introduced and named by Weil (1967) as a constant appearing in the functional equation of its L-series, analogous to the way the conductor of a global field appears in the functional equation of its zeta function. He showed that it could be written as a product over primes with exponents given by order(Δ) ? μ + 1, which by Ogg's formula is equal to ε+δ. A similar definition works for any global field. Weil also suggested that the conductor was equal to the level of a modular form corresponding to the elliptic curve.

Serre & Tate (1968) extended the theory to conductors of abelian varieties.

Ogg's formula

Saito (1988) gave a uniform proof and generalized Ogg's formula to more general arithmetic surfaces.

References
・Saito, Takeshi (1988), "Conductor, discriminant, and the Noether formula of arithmetic surfaces", Duke Math. J., 57 (1): 151?173, doi:10.1215/S0012-7094-88-05706-7, MR 0952229

http://rio2016.5ch.net/test/read.cgi/math/1592654877/148

149: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/07/15(水) 22:22:21.33 ID:hRRJMwM+

>>148 追加

https://en.wikipedia.org/wiki/Tate%27s_algorithm
Tate's algorithm
(抜粋)
In the theory of elliptic curves, Tate's algorithm takes as input an integral model of an elliptic curve E over {\displaystyle \mathbb {Q} }\mathbb {Q} , or more generally an algebraic number field, and a prime or prime ideal p. It returns the exponent fp of p in the conductor of E, the type of reduction at p, the local index
c_p=[E(Q_p):E^0(Q_p)],
where  E^0(Q_p) is the group of Q_p-points whose reduction mod p is a non-singular point.
Also, the algorithm determines whether or not the given integral model is minimal at p, and, if not, returns an integral model with integral coefficients for which the valuation at p of the discriminant is minimal.

Tate's algorithm also gives the structure of the singular fibers given by the Kodaira symbol or Neron symbol, for which, see elliptic surfaces: in turn this determines the exponent fp of the conductor E.

Tate's algorithm can be greatly simplified if the characteristic of the residue class field is not 2 or 3; in this case the type and c and f can be read off from the valuations of j and Δ (defined below).

Tate's algorithm was introduced by John Tate (1975) as an improvement of the description of the Neron model of an elliptic curve by Neron (1964).

Contents
1 Notation
2 The algorithm
3 Implementations

Implementations
The algorithm is implemented for algebraic number fields in the PARI/GP computer algebra system, available through the function elllocalred.

http://rio2016.5ch.net/test/read.cgi/math/1592654877/149

151: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/07/15(水) 23:21:11.85 ID:hRRJMwM+

>>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

http://rio2016.5ch.net/test/read.cgi/math/1592654877/151

153: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2020/07/15(水) 23:41:30.47 ID:hRRJMwM+

>>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.

http://rio2016.5ch.net/test/read.cgi/math/1592654877/153

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ AAｻﾑﾈｲﾙ

ぬこの手ぬこTOP 0.035s