Weight bounds for \((3,\gamma )\)-hyperelliptic curves

\((N,\gamma )\) $(N,\gamma )$ -hyperelliptic semigroups were introduced by Fernando Torres to encapsulate the most salient properties of Weierstrass semigroups associated with totally ramified points of N-fold covers of curves of genus \(\gamma \) $\gamma$ . Torres characterized \((2,\gamma )\) $(2,\gamma )$ -hyperelliptic semigroups of maximal weight whenever their genus is large relative to \(\gamma \) $\gamma$ . Here we do the same for \((3,\gamma )\) $(3,\gamma )$ -hyperelliptic semigroups, and we formulate a conjecture about the general case whenever \(N \ge 3\) $N\ge 3$ is prime.