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Let $X\to Y$ be a finite morphism which is Galois in the category of smooth projective connected curves over $\mathbf{C}$.

Assume $g=g(X) \geq 2$.

Is the degree of $X\to Y$ bounded by $84(g-1)$?

I think the answer is yes. In fact, the degree of $X\to Y$ is the cardinality of $\# \mathrm{Aut}(X/Y)$. This is bounded from above by $\# \mathrm{Aut}(X)$ and this is known to be bounded from above by $84(g-1)$ (if g>1).

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    In positive characteristic, the order of Aut(X) is bounded by $16g^4$ with a few exceptions (see works of Stichtenoth and of Singh in 1974 - 1975).2012-03-09

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