Hyperelliptic jacobians without complex multiplication, doubly transitive permutation groups and projective representations

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In his previous paper (Math. Res. Letters 7(2000), 123--132) the author proved that in characteristic zero the jacobian J(C) of a hyperelliptic curve C: y²=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(x) ∈ K[x] is either the symmetric group Sₙ or the alternating group Aₙ. Here n>4 is the degree of f. In the next paper (Progress in Math. 195(2001), 473--490) we extended this result to the case of certain``smaller'' Galois groups. In particular, we treated the infinite series n=2ʳ+1, Gal(f)=L₂(2ʳ). The case of small Mathieu groups Mₙ (with n=11,12) was also treated. In this paper we do the case of large Mathieu groups Mₙ (with n=22,23,24). We also treat the infinite series Gal(f)=Lₘ(2ʳ) (with m>2 except the cases (m,r)=(3,2) or (4,1)), assuming that the set R of roots of f can be identified with the corresponding projective space Pm-1)(F₂ᵣ) over the finite field F₂ᵣ of characteristic 2 in such a way that the Galois action on R becomes the natural action of Lₘ(2ʳ) on the projective space.