Cue Mr. Armstrong, for whom Mr. Jones is writing a film adaptation of the stage musical “American Idiot” (based on the Green Day album), and who happens to be a pop music scholar of sorts as well as a tireless tunesmith.

“Billie’s got this archive that he sits on — a bunch of songs that aren’t really appropriate for Green Day, and some of them sound like the Beatles,” Mr. Jones said recently, on a break from rehearsals in New Haven. “And he’s always writing new things.” Just a few days after their meeting, Mr. Jones received a demo recording of a brand-new tune from Mr. Armstrong, who warned him he’d eventually “have a plethora to choose from.”

Mr. Armstrong, who had relished the experience of seeing “American Idiot” adapted for the stage, and even joined the cast for several weeks on Broadway, said in a recent phone interview that he jumped at the chance to write more for the theater: “I got so inspired. I was listening to all those early Beatles records, and I read the book ‘Revolution in the Head’ and just started banging these songs out.”