Tame torsion and the tame inverse Galois problem

Abstract Fix a positive integer g and squarefree m . We prove the existence of genus curve $$C/{\mathbb {Q}}$$ C / Q such that mod representation its Jacobian is tame. The method to analyse period matrices hyperelliptic Mumford curves, which could be independent interest. As an application, we study tame version inverse Galois problem for symplectic matrix groups over finite fields.