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We prove the generalised Tate conjecture for H^3 of products of ellipticcurves over finite fields, by slightly modifying an argument of M. Spiessconcerning the Tate conjecture. We prove it fully if the elliptic curves runamong at most 3 isogeny classes. We also show how things become more intricatefrom H^4 onwards, for more that 3 isogeny classes.

On the generalised Tate conjecture for products of elliptic curves over finite fields