Expresses the number of points of a variety over a finite field
In algebraic geometry, the Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism on its cohomology groups. There are several generalizations: the Frobenius endomorphism can be replaced by a more general endomorphism, in which case the points over a finite field are replaced by its fixed points, and there is also a more general version for a sheaf over the variety, where the cohomology groups are replaced by cohomology with coefficients in the sheaf.
The Grothendieck trace formula is an analogue in algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology.
One application of the Grothendieck trace formula is to express the zeta function of a variety over a finite field, or more generally the L-series of a sheaf, as a sum over traces of Frobenius on cohomology groups. This is one of the steps used in the proof of the Weil conjectures.
Behrend's trace formula generalizes the formula to algebraic stacks.
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algebraic geometry, the Grothendiecktraceformula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism...
Jacquet's relative trace formula, simple trace formula, stable traceformulaGrothendiecktraceformula, an analogue in algebraic geometry of the Lefschetz...
Alexander Grothendieck. Lidskii's theorem does not hold in general for Banach spaces. The theorem should not be confused with the Grothendiecktraceformula from...
sequence Grothendieck–Springer resolution Grothendieck–Teichmüller group Grothendieck–Teichmüller theory Grothendieck-Witt ring Grothendiecktraceformula Grothendieck...
which is itself equivalent to another norm, called the Grothendieck norm. To define the Grothendieck norm, first note that a linear operator K1 → K1 is just...
questions (approached with combinatorial means), and to use the Grothendiecktraceformula and Deligne's estimations of eigenvalues of Frobenius (as explained...
conjecture) (Grothendieck 1965). The general theorems about étale cohomology allowed Grothendieck to prove an analogue of the Lefschetz fixed-point formula for...
class of X. Behrend's traceformula Behrend's traceformula generalizes Grothendieck'straceformula; both formulas compute the trace of the Frobenius on...
supervision of Alexander Grothendieck, with a thesis titled Théorie de Hodge. Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes...
{1}{1-q^{-s}}}.} For a variety X over a finite field, it is known by Grothendieck'straceformula that ζ X ( s ) = Z ( X , q − s ) {\displaystyle \zeta _{X}(s)=Z(X...
function.) He is also known for Behrend's formula, the generalization of the Grothendieck–Lefschetz traceformula to algebraic stacks. He is the recipient...
and so the trace is not uniquely defined. However, if the order q ≤ 2/3, then there is a unique trace, as given by a theorem of Grothendieck. If L : B...
applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an...
instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are structures defined on arbitrary categories that allow...
formulae of the general theory.) It is a consequence of the Lefschetz traceformula for the Frobenius morphism that Z ( X , t ) = ∏ i = 0 2 dim X det...
over an algebraically closed field, but that restriction was removed by Grothendieck). The analogs of Cartan's theorems hold in great generality: if F {\displaystyle...
for a total of 890 pages. 1983 Selberg traceformula. Hejhal's proof of a general form of the Selberg traceformula consisted of 2 volumes with a total length...
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