2024/01/04 by Jędrzej Garnek, Garnek, Jędrzej
Engineering · Physics and Astronomy · #11F80 #14H40 #Algebraic Geometry (math.AG) #Control and Stability of Dynamical Systems #Dynamics and Control of Mechanical Systems #FOS: Mathematics #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.2401.02377
openalex publication_date 2024/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let J be the Jacobian of a superelliptic curve defined by the equation yℓ = f(x), where f is a separable polynomial of degree non-divisible by ℓ. In this article we study the "exponential" (i.e. ℓ-power) torsion of J. In particular, under some mild conditions on the polynomial f, we determine the image of the associated ℓ-adic representation up to the determinant. We show also that the image of the determinant is contained in an explicit \mathbb Zℓ-lattice with a finite index. As an application, we prove the Hodge, Tate and Mumford-Tate conjectures for a generic superelliptic Jacobian of the above type.