Endomorphism Algebras of Hyperelliptic Jacobians and Finite Projective Lines Arsen Elkin and Yuri G. Zarhin

HYPERELLIPTIC JACOBIANS AND U3(2 m)

HOMOMORPHISMS OF ABELIAN VARIETIES by Yuri

— We study Galois properties of points of prime order on an abelian variety that imply the simplicity of its endomorphism algebra. Applications of these properties to hyperelliptic jacobians are discussed. Résumé (Homomorphismes des variétés abéliennes). — Nous étudions les propriétés galoisiennes des points d’ordre fini des variétés abéliennes qui impliquent la simplicité de leur algèbre d’end...

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 ...

Very Simple Representations: Variations on a Theme of Clifford

We discuss a certain class of absolutely irreducible group representations that behave nicely under the restriction to normal subgroups and subalgebras. Interrelations with doubly transitive permutation groups and hyperelliptic jacobians are discussed.

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 pro...