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Grothendieck–Ogg–Shafarevich formula

In mathematics, the Grothendieck–Ogg–Shafarevich formula describes the Euler characteristic of a complete curve with coefficients in an abelian variety or constructible sheaf, 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_{x\in X}(n+Sw_x(F)) where Sw is the Swan conductor at a point.

References

{{citation | author-link = Alexander Grothendieck | no-pp = true

Wikipedia Source

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elliptic-curves abelian-varieties

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