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{{distinguish|Néron–Ogg–Shafarevich criterion}}
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In mathematics, the '''Grothendieck–Ogg–Shafarevich formula''' describes the [[Euler characteristic]] of complete curve with coefficients in an  [[abelian variety]] or [[constructible sheaf]], in terms of local data involving the [[Swan conductor]]. {{harvs|txt|last=Ogg|author-link=Andrew Ogg|year=1962}} and {{harvs|txt|last=Shafarevich|authorlink=Igor Shafarevich|year=1961}} proved the formula for abelian varieties with tame ramification over curves, and {{harvs|txt|last=Grothendieck|authorlink=Alexander Grothendieck|year=1977|loc=Exp. X formula 7.2}} extended the formula to constructible sheaves over a curve {{harv|Raynaud|1965}}.
 
==Statement==
 
Suppose that ''F'' is a constructible sheaf over a genus ''g'' smooth projective curve ''C'', of rank ''n'' outside a finite set ''X'' of points where it has stalk 0. Then
:<math>\chi(C,F) = n(2-2g) -\sum_{x\in X}(n+Sw_x(F))</math>
where ''Sw'' is the Swan conductor at a point.
 
==References==
 
*{{citation
| last = Grothendieck
| first = Alexandre
| authorlink = Alexandre Grothendieck
| title = Séminaire de Géométrie Algébrique du Bois Marie - 1965-66 - Cohomologie l-adique et Fonctions L - (SGA 5)    |series=Lecture notes in mathematics |volume = 589
| year = 1977
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| location = Berlin; New York
| language = French
| pages = xii+484
| isbn = 3540082484
| nopp = true
|doi = 10.1007/BFb0096802
}}
*{{Citation | last1=Ogg | first1=A. P. | title=Cohomology of abelian varieties over function fields | jstor=1970272 | mr=0155824  | year=1962 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=76 | pages=185–212}}
*{{Citation | last1=Raynaud | first1=Michel | author1-link=Michel Raynaud | title=Séminaire Bourbaki, Vol. 9 | url=http://www.numdam.org/item?id=SB_1964-1966__9__129_0 | publisher=[[Société Mathématique de France]] | location=Paris | series=Exp. No. 286 | mr=1608794  | year=1965 | chapter=Caractéristique d'Euler-Poincaré d'un faisceau et cohomologie des variétés abéliennes | pages= 129–147}}
*{{Citation | last1=Shafarevich | first1=Igor R. | title=Principal homogeneous spaces defined over a function field | mr=0162806  | year=1961 | journal=Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova | issn=0371-9685 | volume=64 | pages=316–346}}
 
{{DEFAULTSORT:Grothendieck-Ogg-Shafarevich formula}}
[[Category:Elliptic curves]]
[[Category:Abelian varieties]]

Latest revision as of 17:36, 2 December 2014

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