On some asymptotic formulas for curves in positive characteristic

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We prove some general asymptotic formula for the values of L-function of a sequence of constructible Qₗ-sheaves on curves over Fq with some good asymptotic properties. We also give the asymptotic formula for the number of points on the stack BunG for asymptotically exact sequence of curves and a split reductive group G. In the case of GLₙ we prove that the same formula holds if we take only semistable points in the account.