"AUTOREM textu písničky Závidím je český herec, textař a hudebník Jiří Grossmann (1947 – 1971)."

My question is simply why "autorem" is in instrumental sg. but not nominative sg.? 😳

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When it's in instrumental, it puts the emphasis on the name, which also becomes the subject of the sentence this way. "Autor textu písničky je ...... Jiří Grossmann," is also correct, but I hear the "autorem" form more often in formal occasions like on tv or at a concert. But the fact is that although autorem fits better, I'd almost always say just "autor" when speaking.
I hope it helped