The endomorphism rings of jacobians of cyclic covers of the projective line

2004/02/18 by Yuri G. Zarhin, YURI G. ZARHIN · 5 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems #Meromorphic and Entire Functions

paper · doi:10.1017/s0305004103007102

Abstract

Abstract. Suppose K is a field of characteristic zero, Ka is its algebraic closure, f(x) ∈ K[x] is an irreducible polynomial of degree n ≥ 5, whose Galois group coincides either with the full symmetric group Sn or with the alternating group An. Let p be an odd prime, Z[ζp] the ring of integers in the pth cyclotomic field Q(ζp). Suppose C is the smooth projective model of the affine curve y p = f(x) and J(C) is the jacobian of C. We prove that the ring End(J(C)) of Ka-endomorphisms of J(C) is canonically isomorphic to Z[ζp]. 1.

Cited by

• Endomorphism algebras of factors of certain hypergeometric Jacobians
• Esnault-Levine-Wittenberg indices
• Centers of Hodge groups of superelliptic jacobians
• Cyclic covers of prime power degree, jacobians and endomorphisms
• Curves with prescribed symmetry and associated representations of\n mapping class groups
• Fixed points and homology of superelliptic Jacobians

Related

• The endomorphism rings of jacobians of cyclic covers of the projective line
• Cyclic covers of the projective line, their jacobians and endomorphisms
• Endomorphisms of ordinary superelliptic jacobians
• Cyclic covers of prime power degree, jacobians and endomorphisms
• Endomorphism Algebras of Abelian varieties with special reference to Superelliptic Jacobians