TY - JOUR

T1 - Non-isogenous elliptic curves and hyperelliptic jacobians

AU - Zarhin, Yuri G.

N1 - Funding Information:
I was partially supported by Simons Foundation Collaboration grant # 585711. Part of this work was done during my stay in 2022 at the Max-Planck Institut für Mathematik (Bonn, Germany), whose hospitality and support are gratefully acknowledged.
Publisher Copyright:
© 2023 International Press of Boston, Inc.. All rights reserved.

PY - 2023

Y1 - 2023

N2 - Let K be a field of characteristic different from 2, K̄ its algebraic closure. Let n ≥ 3 be an odd prime such that 2 is a primitive root modulo n. Let f(x) and h(x) be degree n polynomials with coefficients in K and without repeated roots. Let us consider genus (n − 1)/2 hyperelliptic curves Cf : y2 = f(x) and Ch : y2 = h(x), and their jacobians J(Cf) and J(Ch), which are (n − 1)/2-dimensional abelian varieties defined over K. Suppose that one of the polynomials is irreducible and the other reducible. We prove that if J(Cf) and J(Ch) are isogenous over K̄ then both jacobians are abelian varieties of CM type with multiplication by the field of nth roots of 1. We also discuss the case when both polynomials are irreducible while their splitting fields are linearly disjoint. In particular, we prove that if char(K) = 0, the Galois group of one of the polynomials is doubly transitive and the Galois group of the other is a cyclic group of order n, then J(Cf) and J(Ch) are not isogenous over K̄.

AB - Let K be a field of characteristic different from 2, K̄ its algebraic closure. Let n ≥ 3 be an odd prime such that 2 is a primitive root modulo n. Let f(x) and h(x) be degree n polynomials with coefficients in K and without repeated roots. Let us consider genus (n − 1)/2 hyperelliptic curves Cf : y2 = f(x) and Ch : y2 = h(x), and their jacobians J(Cf) and J(Ch), which are (n − 1)/2-dimensional abelian varieties defined over K. Suppose that one of the polynomials is irreducible and the other reducible. We prove that if J(Cf) and J(Ch) are isogenous over K̄ then both jacobians are abelian varieties of CM type with multiplication by the field of nth roots of 1. We also discuss the case when both polynomials are irreducible while their splitting fields are linearly disjoint. In particular, we prove that if char(K) = 0, the Galois group of one of the polynomials is doubly transitive and the Galois group of the other is a cyclic group of order n, then J(Cf) and J(Ch) are not isogenous over K̄.

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U2 - 10.4310/mrl.2023.v30.n1.a11

DO - 10.4310/mrl.2023.v30.n1.a11

M3 - Article

AN - SCOPUS:85164508253

SN - 1073-2780

VL - 30

SP - 267

EP - 294

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 1

ER -