Two-dimensional families of hyperelliptic jacobians with big monodromy

2013/10/24 by Yuri G. Zarhin, Zarhin, Yuri G.
Mathematics · #11G10 #11G30 #14H40 #14K05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1310.6532

openalex publication_date 2013/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a global field of characteristic different from 2 and u(x)∈ K[x] be an irreducible polynomial of even degree 2g≥ 6, whose Galois group over K is either the full symmetric group S_2g or the alternating group A_2g. We describe explicitly how to choose (infinitely many) pairs of distinct elements t₁, t₂ of K such that the g-dimensional jacobian of a hyperelliptic curve y²=(x-t₁)(x-t₂))u(x) has no nontrivial endomorphisms over an algebraic closure of K and has big ℓ-adic monodromy.

Citations

• Homomorphisms of abelian varieties

Related

• Non-supersingular Hyperelliptic jacobians
• Hyperelliptic jacobians and \U₃(2^m)
• Hyperelliptic jacobians without complex multiplication, doubly transitive permutation groups and projective representations
• Hyperelliptic jacobians without complex multiplication and Steinberg representations in positive characteristic
• Hyperelliptic jacobians and modular representations