Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Numerical evidence for higher order Stark-type conjectures


Authors: Kevin J. McGown, Jonathan W. Sands and Daniel Vallières
Journal: Math. Comp. 88 (2019), 389-420
MSC (2010): Primary 11R42; Secondary 11R27
DOI: https://doi.org/10.1090/mcom/3337
Published electronically: April 12, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $ \mathbb{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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11R42, 11R27

Retrieve articles in all journals with MSC (2010): 11R42, 11R27


Additional Information

Kevin J. McGown
Affiliation: Department of Mathematics and Statistics, California State University, Chico, California 95929
Email: kmcgown@csuchico.edu

Jonathan W. Sands
Affiliation: Department of Mathematics, University of Vermont, Burlington, Vermont 05401
Email: Jonathan.Sands@uvm.edu

Daniel Vallières
Affiliation: Department of Mathematics and Statistics, California State University, Chico, California 95929
Email: dvallieres@csuchico.edu

DOI: https://doi.org/10.1090/mcom/3337
Received by editor(s): June 2, 2017
Received by editor(s) in revised form: October 23, 2017
Published electronically: April 12, 2018
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society