I want to use a picture of Columbo on a book cover, and his name in the title of the book, which refers to his manner of deductive exposure of guilty characters.

Who owns the rights?

As a reference to a public "character", I think this is kind of like using Darth Vader's name: Like a a book named "Evil like Vader" or maybe, using a Silence of the Lambs" reference: "A Hannibel Crime".

I think the name itself is not copyrighted and can be used as a title reference or part of a a comedy routine or published literary discussion. I just can't write a story about a Columbo-modeled character and sell it as Columbo entertainment.

Can someone advise me?

I can't give you legal advice on that, but if you look for "Columbo" in the book selections on Amazon, you should find numerous books with Columbo in the title -- not so much the books "starring" Columbo, which presumably are licensed differently and selectively, but rather the books that talk about Columbo without being actual "Columbo" stories. The authors or publishers of those may be able to give you some guidance, if you can contact them -- although I would caution that copyrighted images present different issues that just using the Columbo name with an *original* image. Good luck!