Hyperelliptic Jacobians without Complex Multiplication

has only trivial endomorphisms over an algebraic closure of the ground field K if the Galois group Gal(f) of the polynomial f ∈ K[x] is “very big”. More precisely, if f is a polynomial of degree n ≥ 5 and Gal(f) is either the symmetric group Sn or the alternating group An then End(J(C)) = Z. Notice that it easily follows that the ring of K-endomorphisms of J(C) coincides with Z and the real problem is how to prove that every endomorphism of J(C) is defined over K. There some results of this type in the literature. Previously Mori [7], [8] has constructed explicit examples (in all characteristic) of hyperelliptic jacobians without nontrivial endomorphisms. In particular, he provided examples over Q with semistable Cf and big (doubly transitive) Gal(f) [8]. The semistability of Cf implies the semistability of J(Cf ) and, thanks to a theorem of Ribet [11], all endomorphisms of J(Cf ) are defined over Q. (Applying to Cf/Q the Shafarevich conjecture [14] (proven by Fontaine [3] and independently by Abrashkin [1], [2]) and using Lemma 4.4.3 and arguments on p. 42 of [12], one may prove that the Galois group Gal(f) of the polynomial f involved is S2g+1 where deg(f) = 2g + 1.) André ([6], pp. 294-295) observed that results of Katz ([4], [5]) give rise to examples of hyperelliptic jacobians J(Cf ) over the field of rational function C(z) with End(J(Cf )) = Z. Namely, one may take f(x) = h(x) − z where h(x) ∈ C[x] is a Morse function. In particular, this explains Mori’s example [7] y = x − x+ z