Endomorphism algebras of hyperelliptic jacobians and finite projective lines

TL;DR

This paper proves that hyperelliptic Jacobians are absolutely simple under specific conditions on the polynomial defining the curve, involving degree, irreducibility, and Galois group properties.

Contribution

It establishes new criteria for the absolute simplicity of hyperelliptic Jacobians based on the degree and Galois group of the defining polynomial.

Findings

Jacobian is absolutely simple for degree q+1 with q ≡ 5 mod 8

Polynomial f(x) is irreducible over characteristic zero fields

Galois group of f(x) is isomorphic to L_2(q)

Abstract

We prove that the jacobian of a hyperelliptic curve y^2=f(x) is absolutely simple if deg(f)=q+1 where q is a power prime congruent to 5 modulo 8, the polynomial f(x) is irreducible over the ground field of characteristic zero and its Galois group is L_2(q).