Wild and even points in global function fields

Abstract

We develop a criterion for a point of global function field to be a unique wild point of some self-equivalence of this field. We show that this happens if and only if the class of the point in the Picard group of the field is 2-divisible. Moreover, given a finite set of points, whose classes are 2-divisible in the Picard group, we show that there is always a self-equivalence of the field for which this is precisely the set of wild points. Unfortunately, for more than one point this condition is no longer a necessary one.

Wild sets in global function fields

“…The problem of divisibility in the Picard group of a curve has been investigated by numerous authors in recent years (let us mention [8] to give just one example). In [2] we proved that a singleton {p} is a wild set if and only if the class of p is 2-divisible in Pic X. Below we investigate the same question for arbitrary divisors.…”

“…This result was generalized in our earlier paper to global function fields (see Theorem 4.3), providing a complete characterization of wild sets of rank 0. However, it was already remarked in [2] that these are not all possible wild set of a global function field. The purpose of the present paper is to describe wild of arbitrary rank.…”

Incidence relation for primes of a global function field

“…Take two irreducible polynomials f, g ∈ F q [t] with coordinates in a finite field of odd characteristic. The quadratic reciprocity law asserts that f g g f = (−1) (q−1)(deg f deg g)/ 2 . This leads to a definition of a relation on the set of irreducible polynomials.…”

“…Let K = F q (X) be an arbitrary global function field of characteristic = 2. A generalization of the relation to the set of points (primes) of K, whose classes are 2-divisible in the Picard group Pic X, was introduced in [2] and further investigated in [3,1]. We should emphasize the fact that on such points the relation is defined canonically.…”

“…Proof. By[2, Proposition 3.4] the class of p is 2-divisible in Pic X if and only if E X = ∆ X\{p} . Hence the first assertion follows immediately from the preceding proposition.…”