106:, 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(Î) â ÎŒ + 1, which by Ogg's formula is equal to Δ+ÎŽ. 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.
192:
of the conductor as a sum Δ + Ύ of two terms, corresponding to the tame and wild ramification. The tame ramification part Δ is defined in terms of the reduction type: Δ=0 for good reduction, Δ=1 for multiplicative reduction and Δ=2 for additive reduction. The wild ramification term Ύ is zero unless
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This is a finite product as the primes of bad reduction are contained in the set of primes divisors of the discriminant of any model for
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629:
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705:
113:
103:
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894:
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The exponent of the conductor is related to other invariants of the elliptic curve by Ogg's formula:
95:
in the form of an integer invariant Δ+Ύ which later turned out to be the exponent of the conductor.
1002:
903:
757:
Saito, Takeshi (1988), "Conductor, discriminant, and the
Noether formula of arithmetic surfaces",
79:
and the number of components of the special fiber over a local field, which can be computed using
448:
Ogg's original proof used a lot of case by case checking, especially in characteristics 2 and 3.
91:
The conductor of an elliptic curve over a local field was implicitly studied (but not named) by
442:
940:
Weil, AndrĂ© (1967), "Ăber die
Bestimmung Dirichletscher Reihen durch Funktionalgleichungen",
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57:
41:
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is the number of components (without counting multiplicities) of the singular fibre of the
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8:
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33:
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gave a uniform proof and generalized Ogg's formula to more general arithmetic surfaces.
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The conductor of an elliptic curve over the rationals was introduced and named by
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575:{\displaystyle f(E)=\prod _{\mathbf {p} }\mathbf {p} ^{f_{\mathbf {p} }}\ .}
76:
29:
644:"ModÚles minimaux des variétés abéliennes sur les corps locaux et globaux"
624:. Graduate Texts in Mathematics. Vol. 111 (2nd ed.). Springer.
505:. The global conductor is the ideal given by the product over primes of
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divides 2 or 3, and in the latter cases it is defined in terms of the
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427:{\displaystyle f_{\mathbf {p} }=\nu _{\mathbf {p} }(\Delta )+1-n\ ,}
64:. The primes involved in the conductor are precisely the primes of
445:
for E. (This is sometimes used as a definition of the conductor).
853:. Graduate Texts in Mathematics. Vol. 151. Springer-Verlag.
469:): it is 0 in the case of good reduction; otherwise it is 1 if Μ
898:
302:{\displaystyle \delta =\dim _{Z/lZ}{\text{Hom}}_{Z_{l}}(P,M).}
102:
as a constant appearing in the functional equation of its
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We can also describe Δ in terms of the valuation of the
48:, together with associated exponents, which encode the
316:
is the group of points on the elliptic curve of order
163:-integral and with the valuation of the discriminant Μ
75:
Ogg's formula expresses the conductor in terms of the
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369:
222:
851:
Advanced Topics in the
Arithmetic of Elliptic Curves
169:(Î) as small as possible. If the discriminant is a
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426:
301:
703:(1967), "Elliptic curves and wild ramification",
501:be an elliptic curve defined over a number field
994:
792:(1968), "Good reduction of abelian varieties",
56:generated by the points of finite order in the
989:listed by conductor, computed by John Cremona
867:
605:(2nd ed.). Cambridge University Press.
185:and the exponent of the conductor is zero.
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336:the Galois group of a finite extension of
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995:
602:Algorithms for Modular Elliptic Curves
756:
638:
449:
939:
649:Publications MathĂ©matiques de l'IHĂS
128:be an elliptic curve defined over a
99:
899:"The arithmetic of elliptic curves"
699:
589:with global integral coefficients.
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877:Rational Points on Elliptic Curves
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985:- tables of elliptic curves over
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706:American Journal of Mathematics
114:conductors of abelian varieties
70:NĂ©ronâOggâShafarevich criterion
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44:. It is given as a product of
18:conductor of an elliptic curve
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771:10.1215/S0012-7094-88-05706-7
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344:are defined over it (so that
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36:, which is analogous to the
7:
10:
1019:
188:We can write the exponent
86:
68:of the curve: this is the
620:Husemöller, Dale (2004).
904:Inventiones Mathematicae
340:such that the points of
159:whose coefficients are
112:extended the theory to
110:Serre & Tate (1968)
599:Cremona, John (1997).
576:
428:
303:
795:Annals of Mathematics
577:
429:
304:
201:of the extensions of
139:a prime ideal of the
42:Galois representation
24:(or more generally a
869:Silverman, Joseph H.
847:Silverman, Joseph H.
515:
367:
220:
157:Weierstrass equation
16:In mathematics, the
983:Elliptic Curve Data
879:. Springer-Verlag.
479:) < 0 and 2 if Μ
443:NĂ©ron minimal model
330:Swan representation
213:by Serre's formula
954:10.1007/BF01361551
917:10.1007/BF01389745
786:Serre, Jean-Pierre
662:10.1007/BF02684271
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20:over the field of
798:, Second Series,
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199:wild ramification
147:. We consider a
32:) is an integral
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155:: a generalised
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81:Tate's algorithm
54:field extensions
22:rational numbers
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977:External links
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77:discriminant
74:
50:ramification
46:prime ideals
30:global field
17:
15:
948:: 149â156,
713:(1): 1â21,
457:j-invariant
173:-unit then
130:local field
100:Weil (1967)
942:Math. Ann.
933:0296.14018
873:Tate, John
840:0172.46101
790:Tate, John
751:0147.39803
701:Ogg, A. P.
694:0132.41403
593:References
120:Definition
93:Ogg (1967)
970:120553723
925:120008651
895:John Tate
816:0003-486X
727:0002-9327
686:120802890
670:1618-1913
656:: 5â128,
535:∏
413:−
401:Δ
387:ν
251:
224:δ
58:group law
997:Category
897:(1974).
875:(1992).
849:(1994).
642:(1964),
348:acts on
104:L-series
962:0207658
832:0236190
824:1970722
779:0952229
743:0207694
735:2373092
678:0179172
489:) â„ 0.
328:is the
205:by the
87:History
60:of the
52:in the
968:
960:
931:
923:
883:
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838:
830:
822:
814:
777:
749:
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733:
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692:
684:
676:
668:
628:
609:
567:
437:where
419:
332:, and
966:S2CID
921:S2CID
820:JSTOR
731:JSTOR
682:S2CID
312:Here
40:of a
34:ideal
26:local
881:ISBN
855:ISBN
812:ISSN
723:ISSN
666:ISSN
626:ISBN
607:ISBN
497:Let
177:has
151:for
135:and
124:Let
950:doi
946:168
929:Zbl
913:doi
836:Zbl
804:doi
767:doi
747:Zbl
715:doi
690:Zbl
658:doi
256:Hom
231:dim
209:of
181:at
143:of
28:or
999::
964:,
958:MR
956:,
944:,
927:.
919:.
909:23
907:.
901:.
871:;
834:,
828:MR
826:,
818:,
810:,
800:88
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987:Q
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531:=
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525:E
522:(
519:f
507:K
503:K
499:E
487:j
485:(
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475:(
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465:(
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459:Μ
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398:(
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377:p
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350:M
346:G
342:M
338:K
334:G
326:P
322:l
318:l
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297:.
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291:M
288:,
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274:G
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195:p
190:f
183:p
175:E
171:p
166:p
161:p
153:E
145:K
137:p
133:K
126:E
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