Hyperelliptic Jacobians and Projective Linear Galois Groups

In [9] the author proved that in characteristic 0 the jacobian J(C) = J(Cf ) of a hyperelliptic curve C = Cf : y 2 = f(x) has only trivial endomorphisms over an algebraic closure Ka of the ground field K if the Galois group Gal(f) of the irreducible polynomial f ∈ K[x] is “very big”. Namely, if n = deg(f) ≥ 5 and Gal(f) is either the symmetric group Sn or the alternating group An then the ring End(J(Cf )) of Ka-endomorphisms of J(Cf ) coincides with Z. Later the author [10] proved that End(J(Cf )) = Z for an infinite series of Gal(f) = L2(2 ) := PSL2(F2r ) and n = 2 r + 1 (with r ≥ 3 and dim(J(Cf )) = 2 ) or when Gal(f) is the Suzuki group Sz(2) and n = 2 + 1 (with dim(J(Cf )) = 2 ). He also proved the same assertion when n = 11 or 12 and Gal(f) is the Mathieu group M11 or M12. (In those cases J(Cf ) has dimension 5.) We refer the reader to [7], [8], [4], [5], [6], [9], [10] for a discussion of known results about, and examples of, hyperelliptic jacobians without complex multiplication. In the present paper we prove that End(J(Cf ) = Z when the set R = Rf of roots of f could be identified with the (m − 1)-dimensional projective space P(Fq) over a finite field Fq of odd characteristic in such a way that Gal(f), viewed as a permutation group of Rf , becomes either the projective linear group PGL(m,Fq) or the projective special linear group Lm(q) := PSL(m,Fq). Here we assume that m > 2. In this case