Hyperelliptic jacobians without complex multiplication in positive characteristic

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We prove that in odd characteristic the jacobian of a hyperelliptic curve y²=f(x) has no nontrivial endomorphisms over an algebraic closure of the ground field if the Galois group of the polynomial f of even degree is ``very big". The case of characteristic zero was previously treated by the author (Math. Res. Letters 7(2000), 123--132).