Most Hyperelliptic Curves Have Big Monodromy

ρf,` : Gal(k̄/k)→ Aut(Jf [`](k̄)) ' GL2g(F`) on the `-torsion points of Jf is as big as possible for almost all primes `, if the following two (sufficient) conditions hold: (1) the endomorphism ring of Jf is Z; (2) for some prime ideal p ⊂ Zk, the fiber over p of the Néron model of Cf is a smooth curve except for a single ordinary double point. These conditions can be translated concretely in terms of the polynomial f , and are implied by: (1’) the Galois group of the splitting field of f is the full symmetric group Sn (this is due to a result of Zarhin [Z], which shows that this condition implies (1)); (2’) for some prime ideal p ⊂ Zk, f factors in Fp = Zk/pZk as f = f1f2 where fi ∈ Fp[X] are relatively prime polynomials such that f1 = (X−α) for some α ∈ Fp and f2 is squarefree of degree n− 2; indeed, this implies (2). In this note, we show that, in some sense, “most” polynomials f satisfy these two conditions, hence “most” jacobians of hyperelliptic curves have maximal monodromy modulo all but finitely many primes (which may, a priori, depend on the polynomial, of course!). More precisely, for k and Zk as above, let us denote Fn = {f ∈ Zk[X] | f is monic of degree n},