mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Grothendieck–Ogg–Shafarevich formula describes the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...

of a complete curve with coefficients in an

abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...

constructible sheaf In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space ''X'', such that ''X'' is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origi ...

, in terms of local data involving the Swan conductor. and proved the formula for abelian varieties with tame ramification over curves, and extended the formula to constructible sheaves over a curve .

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 :

\chi(C,F) = n(2-2g) -\sum_(n+Sw_x(F))

where ''Sw'' is the Swan conductor at a point.

References

* * * * {{DEFAULTSORT:Grothendieck-Ogg-Shafarevich formula Elliptic curves Abelian varieties