Moduli stack of principal bundles

<h2 id="basic-properties">Basic properties</h2>
<p>It is known that 
  
    
      
        
          Bun
          
            G
          
        
        ⁡
        (
        X
        )
      
    
    {\displaystyle \operatorname {Bun} _{G}(X)}
  
 is a <a href="/facts/Smooth_stack/9mVm3h0g">smooth stack</a> of dimension 
  
    
      
        (
        g
        (
        X
        )
        −
        1
        )
        dim
        ⁡
        G
      
    
    {\displaystyle (g(X)-1)\dim G}
  
 where 
  
    
      
        g
        (
        X
        )
      
    
    {\displaystyle g(X)}
  
 is the genus of <i>X</i>. 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 <a href="/facts/Harder%E2%80%93Narasimhan_stratification/KqtuXeUI">Harder–Narasimhan stratification</a>), also for parahoric G over curve X see <a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> and for G only a flat group scheme of finite type over X see.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>  
</p><p>If <i>G</i> is a split reductive group, then the set of connected components 
  
    
      
        
          π
          
            0
          
        
        (
        
          Bun
          
            G
          
        
        ⁡
        (
        X
        )
        )
      
    
    {\displaystyle \pi _{0}(\operatorname {Bun} _{G}(X))}
  
 is in a natural bijection with the fundamental group 
  
    
      
        
          π
          
            1
          
        
        (
        G
        )
      
    
    {\displaystyle \pi _{1}(G)}
  
.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="the-atiyahbott-formula">The Atiyah–Bott formula</h2>
<p class="note">Main article: <a href="/facts/Atiyah%E2%80%93Bott_formula/jN0IWuB8">Atiyah–Bott formula</a></p>
<h2 id="behrends-trace-formula">Behrend's trace formula</h2>
<p class="note">See also: <a href="/facts/Weil_conjecture_on_Tamagawa_numbers/XP1Jeqf4">Weil conjecture on Tamagawa numbers</a> and <a href="/facts/Behrend%27s_formula/TUR2P2Id">Behrend's formula</a></p>
<p>This is a (conjectural) version of the <a href="/facts/Lefschetz_trace_formula/TEjN2DQ8">Lefschetz trace formula</a> for 
  
    
      
        
          Bun
          
            G
          
        
        ⁡
        (
        X
        )
      
    
    {\displaystyle \operatorname {Bun} _{G}(X)}
  
 when <i>X</i> is over a finite field, introduced by Behrend in 1993.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> It states:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> if <i>G</i> is a <a href="/facts/Smooth_scheme/Y76o7b9T">smooth</a> affine <a href="/facts/Group_scheme/bJW14FE2">group scheme</a> with semisimple connected <a href="/facts/Generic_fiber/3qaparob">generic fiber</a>, then
</p>

#
        
          Bun
          
            G
          
        
        ⁡
        (
        X
        )
        (
        
          
            F
          
          
            q
          
        
        )
        =
        
          q
          
            dim
            ⁡
            
              Bun
              
                G
              
            
            ⁡
            (
            X
            )
          
        
        tr
        ⁡
        (
        
          ϕ
          
            −
            1
          
        
        
          |
        
        
          H
          
            ∗
          
        
        (
        
          Bun
          
            G
          
        
        ⁡
        (
        X
        )
        ;
        
          
            Z
          
          
            l
          
        
        )
        )
      
    
    {\displaystyle \#\operatorname {Bun} _{G}(X)(\mathbf {F} _{q})=q^{\dim \operatorname {Bun} _{G}(X)}\operatorname {tr} (\phi ^{-1}|H^{*}(\operatorname {Bun} _{G}(X);\mathbb {Z} _{l}))}

<p>where (see also <a href="/facts/Behrend%27s_trace_formula/TUR2P2Id">Behrend's trace formula</a> for the details)
</p>
<ul><li><i>l</i> is a prime number that is not <i>p</i> and the ring 
  
    
      
        
          
            Z
          
          
            l
          
        
      
    
    {\displaystyle \mathbb {Z} _{l}}
  
 of <a href="/facts/L-adic_integers/SDgDbkLF">l-adic integers</a> is viewed as a subring of 
  
    
      
        
          C
        
      
    
    {\displaystyle \mathbb {C} }
  
.</li>
<li>
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
  
 is the <a href="/facts/Geometric_Frobenius/hC2AIwJ3">geometric Frobenius</a>.</li>
<li>
  
    
      
        #
        
          Bun
          
            G
          
        
        ⁡
        (
        X
        )
        (
        
          
            F
          
          
            q
          
        
        )
        =
        
          ∑
          
            P
          
        
        
          
            1
            
              #
              Aut
              ⁡
              (
              P
              )
            
          
        
      
    
    {\displaystyle \#\operatorname {Bun} _{G}(X)(\mathbf {F} _{q})=\sum _{P}{1 \over \#\operatorname {Aut} (P)}}
  
, the sum running over all isomorphism classes of <a href="/facts/Torsor_(algebraic_geometry)/nMKyR4eF">G-bundles</a> on <i>X</i> and convergent.</li>
<li>
  
    
      
        tr
        ⁡
        (
        
          ϕ
          
            −
            1
          
        
        
          |
        
        
          V
          
            ∗
          
        
        )
        =
        
          ∑
          
            i
            =
            0
          
          
            ∞
          
        
        (
        −
        1
        
          )
          
            i
          
        
        tr
        ⁡
        (
        
          ϕ
          
            −
            1
          
        
        
          |
        
        
          V
          
            i
          
        
        )
      
    
    {\displaystyle \operatorname {tr} (\phi ^{-1}|V_{*})=\sum _{i=0}^{\infty }(-1)^{i}\operatorname {tr} (\phi ^{-1}|V_{i})}
  
 for a <a href="/facts/Graded_vector_space/Ff1maywV">graded vector space</a> 
  
    
      
        
          V
          
            ∗
          
        
      
    
    {\displaystyle V_{*}}
  
, provided the <a href="/facts/Series_(mathematics)/yDnlsgIe">series</a> on the right absolutely converges.</li></ul>
<p><i>A priori,</i> 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.
</p>
<h2 id="notes">Notes</h2>

<ul><li>Heinloth, Jochen (2010), <a href="https://www.uni-due.de/~hm0002/Artikel/StacksCourse_v2.pdf">"Lectures on the moduli stack of vector bundles on a curve"</a> (PDF), in Schmitt, Alexander (ed.), <i>Affine flag manifolds and principal bundles</i>, Trends in Mathematics, Basel: Birkhäuser/Springer, pp. 123–153, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-0346-0288-4_4">10.1007/978-3-0346-0288-4_4</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-0346-0287-7, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3013029">3013029</a></li>
<li>J. Heinloth, A.H.W. Schmitt, The Cohomology Ring of Moduli Stacks of Principal Bundles over Curves, 2010 preprint, available at <a href="http://www.uni-essen.de/~hm0002/">http://www.uni-essen.de/~hm0002/</a>.</li>
<li>Gaitsgory, Dennis; Lurie, Jacob (2019), <a href="https://people.math.harvard.edu/~lurie/papers/tamagawa-abridged.pdf"><i>Weil's conjecture for function fields, Vol. 1</i></a> (PDF), Annals of Mathematics Studies, vol. 199, Princeton, NJ: Princeton University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-691-18214-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3887650">3887650</a></li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>C. Sorger, <a href="https://web.archive.org/web/20060605061258/http://users.ictp.it/~pub_off/lectures/lns001/Sorger/Sorger.pdf">Lectures on moduli of principal G-bundles over algebraic curves</a></li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Geometric_Langlands_conjectures/lq4JEUNV">Geometric Langlands conjectures</a></li>
<li><a href="/facts/Ran_space/jkSrhrYo">Ran space</a></li>
<li><a href="/facts/Moduli_stack_of_vector_bundles/eKzvcWo9">Moduli stack of vector bundles</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>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 <a href="https://web.archive.org/web/20130411033546/http://www.math.harvard.edu/~lurie/283notes/Lecture2-FunctionFields.pdf" target="_blank">https://web.archive.org/web/20130411033546/http://www.math.harvard.edu/~lurie/283notes/Lecture2-FunctionFields.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Heinloth 2010, Proposition 2.1.2 - Heinloth, Jochen (2010), "Lectures on the moduli stack of vector bundles on a curve" (PDF), in Schmitt, Alexander (ed.), Affine flag manifolds and principal bundles, Trends in Mathematics, Basel: Birkhäuser/Springer, pp. 123–153, doi:10.1007/978-3-0346-0288-4_4, ISBN 978-3-0346-0287-7, MR 3013029 <a href="https://www.uni-due.de/~hm0002/Artikel/StacksCourse_v2.pdf" target="_blank">https://www.uni-due.de/~hm0002/Artikel/StacksCourse_v2.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>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 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Heinloth 2010, Proposition 2.1.2 - Heinloth, Jochen (2010), "Lectures on the moduli stack of vector bundles on a curve" (PDF), in Schmitt, Alexander (ed.), Affine flag manifolds and principal bundles, Trends in Mathematics, Basel: Birkhäuser/Springer, pp. 123–153, doi:10.1007/978-3-0346-0288-4_4, ISBN 978-3-0346-0287-7, MR 3013029 <a href="https://www.uni-due.de/~hm0002/Artikel/StacksCourse_v2.pdf" target="_blank">https://www.uni-due.de/~hm0002/Artikel/StacksCourse_v2.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Behrend, Kai A. (1991), The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles (PDF) (PhD thesis), University of California, Berkeley <a href="http://www.math.ubc.ca/~behrend/thesis.pdf" target="_blank">http://www.math.ubc.ca/~behrend/thesis.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Gaitsgory & Lurie 2019, Chapter 5: The Trace Formula for BunG(X), p. 260 - Gaitsgory, Dennis; Lurie, Jacob (2019), Weil's conjecture for function fields, Vol. 1 (PDF), Annals of Mathematics Studies, vol. 199, Princeton, NJ: Princeton University Press, ISBN 978-0-691-18214-8, MR 3887650 <a href="https://people.math.harvard.edu/~lurie/papers/tamagawa-abridged.pdf" target="_blank">https://people.math.harvard.edu/~lurie/papers/tamagawa-abridged.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Moduli stack of principal bundles open-in-new

Moduli stack of principal bundles