On the Variation of the Rank of Jacobian Varieties on Unramified Abelian Towers over Number Fields
نویسنده
چکیده
Let C be a smooth projective curve defined over a number field k, X/k(C) a smooth projective curve of positive genus, JX the Jacobian variety of X and (τ, B) the k(C)/k-trace of JX . We estimate how the rank of JX(k(C))/τB(k) varies when we take an unramified abelian cover π : C ′ → C defined over k.
منابع مشابه
Number of points on abelian and Jacobian varieties over finite fields
We give upper and lower bounds for the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.
متن کاملThe Minimum and Maximum Number of Rational Points on Jacobian Surfaces over Finite Fields
We give some bounds on the numbers of rational points on abelian varieties and jacobians varieties over finite fields. The main result is that we determine the maximum and minimum number of rational points on jacobians varieties of dimension 2.
متن کاملAbelian Étale Coverings of Modular Curves over Local Fields
We relate a part of the abelian étale fundamental group of curves over local fields to the component group of the Néron model of the jacobian. We apply the result to the modular curve X0(p)/Qp to show that the unramified abelian covering X1(p) → X0(p) (Shimura covering) uses up all the possible ramification over the special fiber of X0(p).
متن کاملAbelian Étale Coverings of Modular Curves over Local Fields
We relate a part of the abelian étale fundamental group of curves over local fields to the component group of the Néron model of the jacobian. We apply the result to the modular curve X0(p)/Qp to show that the unramified abelian covering X1(p) → X0(p) (Shimura covering) uses up all the possible ramification over the special fiber of X0(p).
متن کاملThe Rank of Elliptic Surfaces in Unramified Abelian Towers
Let E → C be an elliptic surface defined over a number field K. For a finite covering C → C defined over K, let E ′ = E ×C C be the corresponding elliptic surface over C. In this paper we give a strong upper bound for the rank of E (C/K) in the case of unramified abelien coverings C → C and under the assumption that the Tate conjecture is true for E /K. In the case that C is an elliptic curve a...
متن کامل