On the Rank of Abelian Varieties over Function Fields
نویسنده
چکیده
Let C be a smooth projective curve defined over a number field k, A/k(C) an abelian variety and (τ, B) the k(C)/k-trace of A. We estimate how the rank of A(k(C))/τB(k) varies when we take a finite Galois k-cover π : C → C defined over k.
منابع مشابه
L-functions with Large Analytic Rank and Abelian Varieties with Large Algebraic Rank over Function Fields
The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch and Swinnerton-Dyer, Bloch, and Beilinson relate the orders of vanishing of ...
متن کاملA Rank Inequality for the Tate Conjecture over Global Function Fields
We present an observation of D. Ramakrishnan concerning the Tate Conjecture for varieties over a global function field (i.e., the function field of a smooth projecture curve over a finite field), which was pointed out during a lecture given at the AIM’s workshop on the Tate Conjecture in July 2007. The result is perhaps “known to the experts,” but we record it here, as it does not appear to be ...
متن کاملThe Growth of the Rank of Abelian Varieties upon Extensions
We study the growth of the rank of elliptic curves and, more generally, Abelian varieties upon extensions of number fields. First, we show that if L/K is a finite Galois extension of number fields such that Gal(L/K) does not have an index 2 subgroup and A/K is an Abelian variety, then rkA(L)− rkA(K) can never be 1. We obtain more precise results when Gal(L/K) is of odd order, alternating, SL2(F...
متن کاملMordell-Weil growth for GL2-type abelian varieties over Hilbert class fields of CM fields
Let A be a modular abelian variety of GL2-type over a totally real field F of class number one. Under some mild assumptions, we show that the Mordell-Weil rank of A grows polynomially over Hilbert class fields of CM extensions of F .
متن کاملAnderson T-motives Are Analogs of Abelian Varieties with Multiplication by Imaginary Quadratic Fields
An analogy between abelian Anderson T-motives of rank r and dimension n , and abelian varieties over C with multiplication by an imaginary quadratic field K, of dimension r and of signature (n, r − n), permits us to get two elementary results in the theory of abelian varieties. Firstly, we can associate to this abelian variety a (roughly speaking) K-vector space of dimension r in C. Secondly, i...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004