Elliptic curves with CM unveil a new phenomenon in the theory of large algebraic fields. Rather than drawing a line between and or and they give an example where the line goes beween and and another one where the line goes between and .
The aim of this paper is to clarify the ordinarity of cyclotomic function fields. In the previous work [J. Number Theory 133 (2013)], the author determined all monic irreducible polynomials m such that the maximal real subfield of the mth cyclotomic function field is ordinary. In this paper, we extend this result to the general case.
Let X ⊂ Pn be an integral and non-degenerate m-dimensional variety defined over R. For any P ∈ Pn(R) the real X-rank r x,R(P) is the minimal cardinality of S ⊂ X(R) such that P ∈ . Here we extend to the real case an upper bound for the X-rank due to Landsberg and Teitler.