Geodesic
Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers
Canadian journal of mathematics, Tome 61 (2009) no. 1, pp. 3-28

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Let $X$ be a smooth projective geometrically connected curve over a finite field with function field $K$ . Let $g$ be a connected semisimple group scheme over $X$ . Under certain hypotheses we prove the equality of two numbers associated with $g$ . The first is an arithmetic invariant, its Tamagawa number. The second is a geometric invariant, the number of connected components of the moduli stack of $g$ -torsors on $X$ . Our results are most useful for studying connected components as much is known about Tamagawa numbers. 
Behrend, Kai; Dhillon, Ajneet. Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers. Canadian journal of mathematics, Tome 61 (2009) no. 1, pp. 3-28. doi: 10.4153/CJM-2009-001-5
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