A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so

Abstract

We show that a positive proportion of cubic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof involves the comparison of 2-descent and 3-descent in a certain family of Mordell curves E k : y 2 = x 3 + k. As a byproduct of our methods, we show that, for every r ≥ 0, a positive proportion of curves E k have Tate-Shafarevich group with 3-rank at least r.

A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so

“…Indeed, given a cubic field K, there might not be any quartic field L with cubic resolvent field K such that Disc(L) = Disc(K). However, we have that Disc(L u )/Disc(K) is uniformly bounded; we thus refine our proof in [1] to prove that a positive proportion of the (possibly non-maximal) cubic resolvent rings of O Lu are non-monogenic and yet have no local obstruction to being monogenic.…”

“…In a previous paper [1], we proved that a positive proportion of cubic fields are not monogenic despite having no local obstruction to being monogenic. The purpose of this paper is to prove the analogous result for quartic fields.…”

“…However, there are many binary number fields that are not monogenic. In fact, every ring of rank n ≤ 3 is binary, despite the fact that many cubic fields are not monogenic [1]. On the other hand, for n ≥ 4, it is natural to conjecture that, when ordered by discriminant, 100% of number fields of degree n are not binary, though it was not previously known that even a positive proportion of degree n number fields are not binary.…”

“…If (R, ω) is an oriented ring of rank n, then (R, ω) has no local obstruction to being binary if the oriented ring (R ⊗ Z Z p , ω ⊗ 1) of rank n over Z p is binary for all primes p. 1 If R is an (unoriented) ring of rank n, we say that R has no local obstruction to being binary if there exists an orientation ω such that (R, ω) has no local obstruction to being binary. Since, for a ring of rank n over Z, there are exactly two possible choices of orientation, this definition naturally extends the earlier-stated notion for a ring of rank n to have no local obstruction to being monogenic.…”

“…It is thus natural to try and prove Theorems 1 and 2 by considering the family {K} of non-monogenic cubic fields from [1], and for each such K to construct a quartic field L such that the cubic resolvent ring of O L is O K . As such an L would not be binary and have Disc(L) = Disc(K), it would follow that a positive proportion of quartic fields are not binary (and hence not monogenic) when ordered by discriminant.…”