Topic
Stark conjectures
About: Stark conjectures is a research topic. Over the lifetime, 43 publications have been published within this topic receiving 689 citations.
Papers
TL;DR: In this article, it was shown that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s = 0 of derivatives of the Dirichlet L-functions associated to K/k.
Abstract: Let K/k be a finite abelian extension of global fields. We prove that a natural equivariant leading term conjecture implies a family of explicit congruence relations between the values at s=0 of derivatives of the Dirichlet L-functions associated to K/k. We also show that these congruences provide a universal approach to the ‘refined abelian Stark conjectures’ formulated by, inter alia, Stark, Gross, Rubin, Popescu and Tate. We thereby obtain the first proofs of, amongst other things, the Rubin–Stark conjecture and the ‘refined class number formulas’ of both Gross and Tate for all extensions K/k in which K is either an abelian extension of ℚ or is a function field.
65 citations
TL;DR: In this article, it was shown how to construct explicit generators for specific ray class fields of K using a numerical method that arose in the study of complete sets of equiangular lines in Cd, known in quantum information as symmetric informationally complete measurements.
Abstract: Let K be a real quadratic field. For certain K with sufficiently small discriminant we produce explicit unit generators for specific ray class fields of K using a numerical method that arose in the study of complete sets of equiangular lines in Cd (known in quantum information as symmetric informationally complete measurements or sics). The construction in low dimensions suggests a general recipe for producing unit generators in infinite towers of ray class fields above arbitrary K and we summarise this in a conjecture. Such explicit generators are notoriously difficult to find, so this recipe may be of some interest. In a forthcoming paper we shall publish promising results of numerical comparisons between the logarithms of these canonical units and the values of L-functions associated to the extensions, following the programme laid out in the Stark Conjectures.
61 citations
TL;DR: In this article, a universal theory of refined Stark conjectures is presented, including a full proof in several important cases, and explain the connection to previous conjectures of Bloch and Kato, of Lichtenbaum and of Serre and Tate.
Abstract: We present a universal theory of refined Stark conjectures. We give evidence in support of these explicit and very general conjectures, including a full proof in several important cases, and explain the connection to previous conjectures of Bloch and Kato, of Lichtenbaum and of Serre and Tate. We also deduce a wide range of unconditional consequences of these results concerning the annihilation, as Galois modules, of ideal class groups by explicit elements constructed from the values of higher order derivatives of (non-abelian) Artin L-series.
51 citations
TL;DR: In this article, a conjecture which generalizes Darmon's refined class number formula was formulated, and relations between this conjecture and the equivariant leading term conjecture of Burns were discussed.
Abstract: We formulate a conjecture which generalizes Darmon's "refined class number formula". We discuss relations between our conjecture and the equivariant leading term conjecture of Burns. As an application, we give another proof of the "except 2-part" of Darmon's conjecture, which was first proved by Mazur and Rubin.
30 citations
TL;DR: For certain real quadratic fields $K$ with sufficiently small discriminant, the authors showed that the logarithm of these canonical units are related to the values of $L$-functions associated to the extensions.
Abstract: For certain real quadratic fields $K$ with sufficiently small discriminant we produce explicit unit generators for specific ray class fields of $K$ using a numerical method that arose in the study of complete sets of equiangular lines in $\mathbb{C}^d$ (known in quantum information as symmetric informationally complete measurements or SICs). The construction in low dimensions suggests a general recipe for producing unit generators in infinite towers of ray class fields above arbitrary real quadratic $K$, and we summarise this in a conjecture. There are indications [19,20] that the logarithms of these canonical units are related to the values of $L$-functions associated to the extensions, following the programme laid out in the Stark Conjectures.
26 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 4 |
| 2019 | 5 |
| 2018 | 3 |
| 2017 | 1 |
| 2016 | 2 |
| 2014 | 3 |