[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
抽出解除 ﾚｽ栞
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
ﾘﾛｰﾄﾞ規制です｡10分ほどで解除するので､他のﾌﾞﾗｳｻﾞへ避難してください｡
148(3): 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/07/15(水) 22:08:22.61 ID:hRRJMwM+(1/6) AAS
>>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
149(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/07/15(水) 22:22:21.33 ID:hRRJMwM+(2/6) AAS
>>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.
150: 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2020/07/15(水) 22:26:46.33 ID:hRRJMwM+(3/6) AAS
>>148
>Serre & Tate (1968) extended the theory to conductors of abelian varieties.
これだな
https://en.wikipedia.org/wiki/Conductor_of_an_abelian_variety
Conductor of an abelian variety
(抜粋)
In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is.
It is connected to the ramification in the field generated by the torsion points.
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
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 0.039s