A Lindemann-Weierstrass theorem for semi-abelian varieties over function fields

A Lindemann-weierstrass Theorem for Semi-abelian Varieties over Function Fields

We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of a Q-linearly independent set of algebraic numbers are algebraically independent), replacing Qalg by C(t)alg, and Gm by an arbitrary commutative algebraic group over C(t)alg without unipotent quotients. Both the formulations of our results and the methods have a differential algebraic flavour.

N ov 2 00 8 A Lindemann - Weierstrass theorem for semiabelian varieties over function fields ∗

We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of a Q-linearly independent set of algebraic numbers are algebraically independent), replacing Qalg by C(t)alg, and Gm by a semiabelian variety over C(t) alg. Both the formulations of our results and the methods are differential algebraic in nature.

Tate’s Isogeny Theorem for Abelian Varieties over Finite Fields

Control Theorems for Abelian Varieties over Function Fields

We prove a function field version of Mazur’s control theorems for abelian varieties over Zp-extensions.

Abelian varieties over finite fields

A. Weil proved that the geometric Frobenius π = Fa of an abelian variety over a finite field with q = pa elements has absolute value √ q for every embedding. T. Honda and J. Tate showed that A 7→ πA gives a bijection between the set of isogeny classes of simple abelian varieties over Fq and the set of conjugacy classes of q-Weil numbers. Higher-dimensional varieties over finite fields, Summer s...