Calculation And Application Of Lefschetz Trace Formula Of Stacks

Stack theory is a generalization of scheme theory.Stack theory is closely related to the fields of number theory and geometric Langlands,and provides new methods for solving some problems in these fields.Higgs bundles are extremely important geometric objects.In recent years,Higgs bundles have been used in the resolution of many major problems.Especially in geometric Langlands,the stack of Higgs bundles and the related Hitchin map are extremely important research objects.First,this article reviews some knowledge of algebraic geometry and category theory,and then explains the shortcomings of scheme theory,that is,some common functors may not be representable by schems,so the Higgs bundle needs to be considered by stack theory.Next,we study the stack of Higgs twisted bundles,and first verify that the parameter space of Higgs twisted bundles is a stack.Then study the deformation of the Higgs twisted bundles stack,the method used is as follows: because the cohomological dimension of the category formed by Higgs twisted bundle is not equal to 1,it is necessary to select a suitable Euler form and then find two special line bundles to construct a Quot scheme,and then consider the cotangent bundle of the Quot scheme,get a moment map of the cotangent bundle to a Lie algebra,and construct a symplectic quotient,and finally prove that the symplectic quotient at 0 is the Higgs twisted bundles stack.Finally,consider the symplectic quotients at non-zero positions,which are isomorphic under the action of G_m,so on one hard,the calculation of number of rational points of the Higgs twisted bundles stack reduces to the calculation of the number of rational points of deformed fibers.On the other hand,one can use Lefschetz trace formula to Higgs twisted bundles stack,and hope to obtain an identity about the number of rational points of the stack.