Numerical evidence for higher order Stark-type conjectures

Abstract

We give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Stark's conjecture over Q, Rubin's conjecture, Popescu's conjecture, and a conjecture due to Burns that constitutes a generalization of Brumer's classical conjecture on annihilation of class groups. Our approach is general and could be used for any abelian extension of number fields, independent of the signature and type of places (finite or infinite) that split completely in the extension.We then employ our techniques in the situation where K is a totally real, abelian, ramified cubic extension of a real quadratic field. We numerically verify the conjectures listed above for all fields K of this type with absolute discriminant less than 10 12 , for a total of 19197 examples. The places that split completely in these extensions are always taken to be the two real archimedean places of k and we are in a situation where all the S-truncated L-functions have order of vanishing at least two.

On special elements for $\mathbb{G}_m$

“…If G is abelian, this prediction combines with Theorem 4.3 (ii) and the containment (46) in Remark 4.4(i) to recover the central conjecture (Conjecture 4.3) of Livingstone Boomla and the first author in [9]. We recall that the latter conjecture itself constituted a refinement and extension of the conjectures formulated by Emmons and Popescu in [18] and by Vallières in [46] and can be investigated numerically using the methods developed by Bley in [1] and by McGown, Sands and Vallières in [33].…”

“…) . and this sequence combines with the surjection (34) and containment (33) to imply that the reduction of Theorem 3.9(iii) that is provided by Proposition 3.21 is valid. This completes the proof of Theorem 3.9(iii).…”

“…The key point in our proof of Theorem 3.9(iii) is that Proposition 3.25 below combines with [10, Thm. 3.17 (iii)] to imply that for any a p of δ(A (p) ) one has (33)…”