Non-Ample Fields

Abstract

It is sometimes more difficult to give examples of objects that do not have a certain property P than examples of objects that have that property. A standard method to do that is to prove that each object having the property P has another property P’ and then to look for an object that does not have the property P’. For example, by Corollary 5.3.3, every ample field is infinite. Hence, finite fields are not ample. More sophisticated examples of nonample fields are function fields of several variables over arbitrary fields (Theorem 6.1.8(a)). Likewise we prove that if E/K is a function field of one variable and F is the compositum of a directed family of finite extensions of E of bounded genus, then F is nonample (Theorem 6.1.8(b)). The proof uses elementary methods like the Riemann-Hurwitz genus formula. We have not been able to prove that number fields are nonample by elementary means. We have rather used in Proposition 6.2.5 the deep theorem of Faltings (formerly, Mordell’s conjecture).

Section 6.3 surveys concepts and results on Abelian varieties, Jacobian varieties, and homogeneous spaces (the latter is applied only in 11.5). Likewise, Section 6.4 surveys the very deep Mordell-Lang conjecture proved by Faltings and others. As a consequence we prove that the rational rank of every nonzero Abelian variety over an ample field of characteristic zero is infinite (Theorem 6.5.2). That result combined with a result of Kato-Rohrlich (Example 6.5.5) leads to examples of infinite algebraic extensions of number fields that are nonample. Finally, we prove that for each positive integer d there is a linearly disjoint sequence K ₁,K ₂,K ₃,… of extensions of ℚ of degree d whose compositum is nonample (Example 6.8.9). The proof is based on the concept of the “gonality” of a function field of one variable that we establish in Sections 6.6 and 6.7 as well as on a result of Frey (Lemma 6.8.7) based on the Mordell-Lang conjecture.