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We study the arithmetic properties of hyperelliptic curves given by the affine equation by exploiting the structure of the automorphism groups. We show that these curves satisfy Lang’s conjecture about the covering radius (for some special covering maps).
Let C = (C, g^1/4 ) be a tetragonal curve. We consider the scrollar
invariants e1 , e2 , e3 of g^1/4 . We prove that if W^1/4 (C) is a non-singular variety,
then every g^1/4 ∈ W^1/4 (C) has the same scrollar invariants.
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