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11
Verifying a padic abelian Stark conjecture at s = 1
 J. NUMBER THEORY
, 2004
"... In a previous paper [13], the second author developed a new approach to the abelian padic Stark conjecture at s = 1 and stated related conjectures. The aim of the present paper is to develop and apply techniques to numerically investigate one of these – the ‘Weak Refined Combined Conjecture ’ – in ..."
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In a previous paper [13], the second author developed a new approach to the abelian padic Stark conjecture at s = 1 and stated related conjectures. The aim of the present paper is to develop and apply techniques to numerically investigate one of these – the ‘Weak Refined Combined Conjecture ’ – in fifteen cases.
Computing the Hilbert class field of real quadratic fields
 Math. Comp
"... Abstract. Using the units appearing in Stark’s conjectures on the values of Lfunctions at s = 0, we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field. Let k be a real quadratic field of discriminant dk, sothatk = Q ( √ dk), and let ω ..."
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Abstract. Using the units appearing in Stark’s conjectures on the values of Lfunctions at s = 0, we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field. Let k be a real quadratic field of discriminant dk, sothatk = Q ( √ dk), and let ω denote an algebraic integer such that the ring of integers of k is Ok: = Z + ωZ. An important invariant of k is its class group Clk, which is, by class field theory, associated to an Abelian extension of k, the socalled Hilbert class field, denoted by Hk. This field is characterized as the maximal Abelian extension of k which is unramified at all (finite and infinite) places. Its Galois group is isomorphic to the class group Clk; hence the degree [Hk: k] istheclassnumberhk. There now exist very satisfactory algorithms to compute the discriminant, the ring of integers and the class group of a number field, and especially of a quadratic field (see [3] and [16]). For the computation of the Hilbert class field, however, there exists an efficient version only for complex quadratic fields, using complex multiplication (see [18]), and a general method for all number fields, using Kummer
Noncommutative tori, real multiplication and line bundles
, 2006
"... This thesis explores an approach to Hilbert's twelfth problem for real quadratic number fields, concerning the determination of an explicit class field theory for such fields. The basis for our approach is a paper by Manin proposing a theory of Real Multiplication realising such an explicit the ..."
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This thesis explores an approach to Hilbert's twelfth problem for real quadratic number fields, concerning the determination of an explicit class field theory for such fields. The basis for our approach is a paper by Manin proposing a theory of Real Multiplication realising such an explicit theory, analogous to the theory of Complex Multiplication associated to imaginary quadratic fields. Whereas elliptic curves play the leading role in the latter theory, objects known as Noncommutative Tori are the subject of Manin's dream. In this thesis we study a family of topological spaces known as Quantum Tori that arise naturally from Manin's approach. Our aim throughout this thesis is to show that these nonHausdorff spaces have an algebraic character, which is unexpected through their definition, though entirely consistent with their envisioned role in Real Multiplication. Chapter 1 is a general introduction to the problem, providing a historical and technical background to the motivation behind this thesis. Chapter 2 deals with the problem of defining continuous maps between Quantum Tori using ideas from
The BrumerStark Conjecture in some families of extensions of specified degree
 Math. Comp
, 2004
"... Abstract. As a starting point, an important link is established between Brumer’s conjecture and the BrumerStark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if K/F is an abelian extension of relative degree 2p, p an odd prime, w ..."
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Abstract. As a starting point, an important link is established between Brumer’s conjecture and the BrumerStark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if K/F is an abelian extension of relative degree 2p, p an odd prime, we prove the lpart of the BrumerStark conjecture for all odd primes l � = p with F belonging to a wide class of base fields. In the same setting, we study the 2part and ppart of BrumerStark with no special restriction on F and are left with only two welldefined specific classes of extensions that elude proof. Extensive computations were carried out within these two classes and a complete numerical proof of the BrumerStark conjecture was obtained in all cases. 0. Overview and results An important conjecture due to Brumer predicts that specific group ring elements constructed from the values of partial zetafunctions at s = 0 annihilate the ideal class groups of certain number fields. Recent progress has been made on this conjecture ([Gr1], [Wi]) and this will be used to obtain new results on the related
Higher derivatives of Lseries associated to real quadratic fields
, 2006
"... This text is a modified version of a chapter in a PhD thesis [21] submitted to Nottingham University in September 2006, which studied an approach to Hilbert’s twelfth problem inspired by Manin’s proposed theory of Real Multiplication [7]. In [20] we defined and studied a nontrivial notion of line bu ..."
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This text is a modified version of a chapter in a PhD thesis [21] submitted to Nottingham University in September 2006, which studied an approach to Hilbert’s twelfth problem inspired by Manin’s proposed theory of Real Multiplication [7]. In [20] we defined and studied a nontrivial notion of line bundles over Quantum Tori. In this text we study sections of these line bundles leading to a study concerning theta functions for Quantum Tori. We prove the existence of such meromorphic theta functions, and view their application in the context of Stark’s conjectures and Hilbert’s twelfth problem. Generalising the work of Shintani, we show that (modulo a Conjecture 5.7) we can write the derivatives of Lseries associated to Real Quadratic Fields in terms of special values of theta functions over Quantum Tori.
STARK’S CONJECTURE OVER COMPLEX CUBIC NUMBER FIELDS
"... Abstract. Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark’s conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. P ..."
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Abstract. Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark’s conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the FinckePohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Stark’s conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units. 1.
Numerical Verification of the StarkChinburg Conjecture for Some Icosahedral Representations
, 2004
"... In this paper, we give fourteen examples of icosahedral representations for which we have numerically verified the StarkChinburg conjecture. ..."
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In this paper, we give fourteen examples of icosahedral representations for which we have numerically verified the StarkChinburg conjecture.
INDEX FORMULAE FOR STARK UNITS AND THEIR SOLUTIONS
 PACIFIC JOURNAL OF MATHEMATICS
, 2013
"... Let K/k be an abelian extension of number fields with a distinguished place of k that splits totally in K. In that situation, the abelian rank one Stark conjecture predicts the existence of a unit in K, called the Stark unit, constructed from the values of the Lfunctions attached to the extension. ..."
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Let K/k be an abelian extension of number fields with a distinguished place of k that splits totally in K. In that situation, the abelian rank one Stark conjecture predicts the existence of a unit in K, called the Stark unit, constructed from the values of the Lfunctions attached to the extension. In this paper, assuming the Stark unit exists, we prove index formulae for it. In a second part, we study the solutions of the index formulae and prove that they admit solutions unconditionally for quadratic, quartic and sextic (with some additional conditions) cyclic extensions. As a result we deduce a weak version of the conjecture (“up to absolute values”) in these cases and precise results on when the Stark unit, if it exists, is a square.
Abstract
, 2008
"... In a previous paper [13], the second author developed a new approach to the abelian padic Stark conjecture at s = 1 and stated related conjectures. The aim of the present paper is to develop and apply techniques to numerically investigate one of these – the ‘Weak Refined Combined Conjecture ’ – in ..."
Abstract
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In a previous paper [13], the second author developed a new approach to the abelian padic Stark conjecture at s = 1 and stated related conjectures. The aim of the present paper is to develop and apply techniques to numerically investigate one of these – the ‘Weak Refined Combined Conjecture ’ – in fifteen cases. 1