After defeating the Coral Siren, you will head back to the hatch closest to Tree Town. There will be a scientist at the entrance who will task you with finding six mermaid shells. They are marked on your map, and to get that far down, you need to go to Arena Town's Sunken Ruin entrance. Continue down until you reach the place where Shantae fell in the beginning of the game, directly under the entrance pit and having to go around first.
From here, head left to find a place to dig with your new Gastro Drill. Continue to head left and down to reach the ocean floor area. All six shells are shining sections of dirt to drill through. Collect all six, and when you get back to the green section of the map, head left and around to reach a warp zone that will take you back to Tree Town. Backtrack to the hatch and down to turn in the quest and obtain the Lucky Amulet.