2023/12/25 by Yuri G. Zarhin, Zarhin, Yuri G.
Mathematics · #11G10 #11G30 #14H40 #14K05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2401.01365
openalex publication_date 2023/12/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ℓ be an odd prime and K a field of characteristic different from ℓ. Let K be an algebraic closure of K. Assume that K contains a primitive ℓth root of unity. Let n ≠ ℓ be another odd prime. Let f(x) and h(x) be degree n polynomials with coefficients in K and without repeated roots. Let us consider superelliptic curves Cf,ℓ: yℓ=f(x) and Ch,ℓ: yℓ=h(x) of genus (n-1)(ℓ-1)/2, and their jacobians J(f,ℓ) and J(h,ℓ), which are (n-1)(ℓ-1)/2-dimensional abelian varieties over K. Suppose that one of the polynomials is irreducible and the other reducible over K. We prove that if J(f,ℓ) and J(h,ℓ) are isogenous over K then both endomorphism algebras End0(J(f,ℓ)) and End0(J(h,ℓ)) contain an invertible element of multiplicative order n.