Moduli stack of principal bundles

In algebraic geometry, given a smooth projective curve X over a finite field and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by , is an algebraic stack given by:[1] for any -algebra R,

the category of principal G-bundles over the relative curve .

In particular, the category of -points of , that is, , is the category of G-bundles over X.

Similarly, can also be defined when the curve X is over the field of complex numbers. Roughly, in the complex case, one can define as the quotient stack of the space of holomorphic connections on X by the gauge group. Replacing the quotient stack (which is not a topological space) by a homotopy quotient (which is a topological space) gives the homotopy type of .

In the finite field case, it is not common to define the homotopy type of . But one can still define a (smooth) cohomology and homology of .

Basic properties

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It is known that   is a smooth stack of dimension   where   is the genus of X. It is not of finite type but locally of finite type; one thus usually uses a stratification by open substacks of finite type (cf. the Harder–Narasimhan stratification), also for parahoric G over curve X see [2] and for G only a flat group scheme of finite type over X see.[3]

If G is a split reductive group, then the set of connected components   is in a natural bijection with the fundamental group  .[4]

The Atiyah–Bott formula

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Behrend's trace formula

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This is a (conjectural) version of the Lefschetz trace formula for   when X is over a finite field, introduced by Behrend in 1993.[5] It states:[6] if G is a smooth affine group scheme with semisimple connected generic fiber, then

 

where (see also Behrend's trace formula for the details)

  • l is a prime number that is not p and the ring   of l-adic integers is viewed as a subring of  .
  •   is the geometric Frobenius.
  •  , the sum running over all isomorphism classes of G-bundles on X and convergent.
  •   for a graded vector space  , provided the series on the right absolutely converges.

A priori, neither left nor right side in the formula converges. Thus, the formula states that the two sides converge to finite numbers and that those numbers coincide.

Notes

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  1. ^ Lurie, Jacob (April 3, 2013), Tamagawa Numbers in the Function Field Case (Lecture 2) (PDF), archived from the original (PDF) on 2013-04-11, retrieved 2014-01-30
  2. ^ Heinloth 2010, Proposition 2.1.2
  3. ^ Arasteh Rad, E.; Hartl, Urs (2021), "Uniformizing the moduli stacks of global G-shtukas", International Mathematics Research Notices (21): 16121–16192, arXiv:1302.6351, doi:10.1093/imrn/rnz223, MR 4338216; see Theorem 2.5
  4. ^ Heinloth 2010, Proposition 2.1.2
  5. ^ Behrend, Kai A. (1991), The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles (PDF) (PhD thesis), University of California, Berkeley
  6. ^ Gaitsgory & Lurie 2019, Chapter 5: The Trace Formula for BunG(X), p. 260

References

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Further reading

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See also

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