Roundabout Theatre Company said on Thursday that Rivera will star in a revival of “The Mystery of Edwin Drood” this fall at its Studio 54 theater under the direction of Scott Ellis. The Tony-winning play with music and lyrics by Rupert Holmes is loosely based on the book of the same name that Charles Dickens was working on when he died in 1870. The audience decides who the murderer is. Now 79, Rivera earned Tonys for “Kiss of the Spider Woman” and “The Rink” and has appeared in “Nine,” ‘‘Chicago” and “Bye Bye Birdie.” It’s the first time back on Broadway for Rivera since 2005 when she brought her retrospective “Chita Rivera: The Dancer’s Life.” Todd Haimes, the Roundabout’s artistic director, in a statement called the musical, which was last on Broadway from 1985-87, “a wonderfully offbeat show that gets the whole audience involved. And to be welcoming back our dear friend Chita Rivera is just the icing on the cake for this delightfully odd musical.” The Roundabout also announced two other productions: The Pulitzer Prize-winning “Picnic” by William Inge next winter at its American Airlines Theatre under the direction of Sam Gold, and the world premiere of “The Unavoidable Disappearance of Tom Durnin” by Steven Levenson next season at Roundabout’s off-Broadway space.