Hodge Groups of Certain Superelliptic Jacobians II (original) (raw)
2011, Mathematical Research Letters
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This paper focuses on the Hodge groups associated with certain superelliptic Jacobians, particularly extending results previously known for hyperelliptic cases to scenarios where the degree of the polynomial defining the curve is less than the prime power involved in the curve's definition. The authors explore the implications of these groups in relation to the structure of the Jacobian, revealing additional properties that lead to a deeper understanding of the Hodge groups in the context of superelliptic curves and their applications in the study of Mordell-Weil groups.
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Throughout this paper K is a field of characteristic zero, ¯ K its algebraic closure and Gal(K) = Aut(¯ K/K) the absolute Galois group of K. We write k for a field of characteristic zero and k(t) for the field of rational functions over k in independent variable t. If d is a positive integer then we write k(t 1 ,. .. t d) for the field of rational functions over k in d independent variables t 1 ,. .. t d. If X is an abelian variety over ¯ K then we write End(X) for the ring of all its ¯ K-endomorphisms and End 0 (X) for the corresponding Q-algebra End(X) ⊗ Q; the notation 1 X stands for the identity automorphism of X. If m is a positive integer then we write X m for the kernel of multiplication by m in X(¯ K). It is well-known [11] that X m is a free Z/mZ-module of rank 2dim(X). If X is defined over K then X m is a Galois submodule in X(¯ K). Definition 0.1. Suppose that K contains a field k and X is an abelian variety defined over K. We say that X is isotrivial over k if there exis...
Non-supersingular hyperelliptic jacobians
Bulletin de la Société mathématique de France, 2004
In his previous papers [25, 26, 28] the author proved that in characteristic = 2 the jacobian J(C) of a hyperelliptic curve C : y 2 = f (x) has only trivial endomorphisms over an algebraic closure Ka of the ground field K if the Galois group Gal(f) of the irreducible polynomial f (x) ∈ K[x] is either the symmetric group Sn or the alternating group An. Here n ≥ 9 is the degree of f. The goal of this paper is to extend this result to the case when either n = 7, 8 or n = 5, 6 and char(K) > 3.
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