Endomorphisms of ordinary superelliptic jacobians

2019/08/29 by Zarhin, Yuri G.
#11G10 #11G30 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 14H40 #Secondary 14K05

paper · doi:10.48550/arxiv.1908.11715

Abstract

Let K be a field of prime characteristic p, n>4 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group is either the full symmetric group S_n or the alternating group A_n. Let l be an odd prime different from p, Z[ζ_l] the ring of integers in the lth cyclotomic field, C_f,l:y^l=f(x) the corresponding superelliptic curve and J(C_f,l) its jacobian. We prove that the ring of all endomorphisms of J(C_f,l) coincides with Z[ζ_l] if J(C_f,l) is an ordinary abelian variety and (l,n)≠ (5,5).

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