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Jonathan W. Sands - One of the best experts on this subject based on the ideXlab platform.
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VALUES AT s = -1 oF L-FUNCTIONS FOR MULTI-QUADRATIC ExtensionS oF NUMBER FIELDS, AND THE FITTING IDEAL oF THE TAME KERNEL
International Journal of Number Theory, 2009Co-Authors: Jonathan W. SandsAbstract:Fix a Galois Extension oF totally real number Fields such that the Galois group G has exponent 2. Let S be a Finite set oF primes oF F containing the inFinite primes and all those which ramiFy in , let denote the primes oF lying above those in S, and let denote the ring oF -integers oF . We then compare the Fitting ideal oF as a ℤ[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order oF ℚ[G], and hence in ℤ[1/2][G]. Results in ℤ[G] are obtained under the assumption oF the Birch–Tate conjecture, especially For biquadratic Extensions, where we compute the index oF the higher Stickelberger ideal. We Find a suFFicient condition For the Fitting ideal to contain the higher Stickelberger ideal in the case where is a biquadratic Extension oF F containing the First layer oF the cyclotomic ℤ2-Extension oF F, and describe a class oF biquadratic Extensions oF F = ℚ that satisFy this condition.
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VALUES AT s = -1 oF L-FUNCTIONS FOR MULTI-QUADRATIC ExtensionS oF NUMBER FIELDS, AND THE FITTING IDEAL oF THE TAME KERNEL
International Journal of Number Theory, 2009Co-Authors: Jonathan W. SandsAbstract:Fix a Galois Extension [Formula: see text] oF totally real number Fields such that the Galois group G has exponent 2. Let S be a Finite set oF primes oF F containing the inFinite primes and all those which ramiFy in [Formula: see text], let [Formula: see text] denote the primes oF [Formula: see text] lying above those in S, and let [Formula: see text] denote the ring oF [Formula: see text]-integers oF [Formula: see text]. We then compare the Fitting ideal oF [Formula: see text] as a ℤ[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order oF ℚ[G], and hence in ℤ[1/2][G]. Results in ℤ[G] are obtained under the assumption oF the Birch–Tate conjecture, especially For biquadratic Extensions, where we compute the index oF the higher Stickelberger ideal. We Find a suFFicient condition For the Fitting ideal to contain the higher Stickelberger ideal in the case where [Formula: see text] is a biquadratic Extension oF F containing the First layer oF the cyclotomic ℤ2-Extension oF F, and describe a class oF biquadratic Extensions oF F = ℚ that satisFy this condition.
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Values at s=-1 oF L-Functions For multi-quadratic Extensions oF number Fields, and the Fitting ideal oF the tame kernel
arXiv: Number Theory, 2007Co-Authors: Jonathan W. SandsAbstract:Fix a Galois Extension E/F oF totally real number Fields such that the Galois group G has exponent 2. Let S be a Finite set oF primes oF F containing the inFinite primes and all those which ramiFy in E, let S_E denote the primes oF E lying above those in S, and let O_E^S denote the ring oF S_E-integers oF E. We then compare the Fitting ideal oF K_2(O_E^S) as a Z[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order oF Q[G], and hence in Z[1/2][G]. Results in Z[G] are obtained under the assumption oF the Birch-Tate conjecture, especially For biquadratic Extensions, where we compute the index oF the higher Stickelberger ideal. We Find a suFFicient condition For the Fitting ideal to contain the higher Stickelberger ideal in the case where E is a biquadratic Extension oF F containing the First layer oF the cyclotomic Z_2-Extension oF F, and describe a class oF biquadratic Extensions oF F=Q that satisFy this condition.
Emmanuel N. Saridakis - One of the best experts on this subject based on the ideXlab platform.
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Nonminimal torsion-matter coupling Extension oF F(T) gravity
Physical Review D, 2014Co-Authors: Tiberiu Harko, Francisco S. N. Lobo, Giovanni Otalora, Emmanuel N. SaridakisAbstract:We construct an Extension oF F(T) gravity with the inclusion oF a non-minimal torsion-matter coupling in the action. The resulting theory is a novel gravitational modiFication, since it is diFFerent From both F(T) gravity, as well as From the non-minimal curvature-matter-coupled theory. The cosmological application oF this new theory proves to be very interesting. In particular, we obtain an eFFective dark energy sector whose equation-oF-state parameter can be quintessence or phantom-like, or exhibit the phantom-divide crossing, while For a large range oF the model parameters the Universe results in a de Sitter, dark-energy-dominated, accelerating phase. Additionally, we can obtain early-time inFlationary solutions too, and thus provide a uniFied description oF the cosmological history.
Tiberiu Harko - One of the best experts on this subject based on the ideXlab platform.
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Nonminimal torsion-matter coupling Extension oF F(T) gravity
Physical Review D, 2014Co-Authors: Tiberiu Harko, Francisco S. N. Lobo, Giovanni Otalora, Emmanuel N. SaridakisAbstract:We construct an Extension oF F(T) gravity with the inclusion oF a non-minimal torsion-matter coupling in the action. The resulting theory is a novel gravitational modiFication, since it is diFFerent From both F(T) gravity, as well as From the non-minimal curvature-matter-coupled theory. The cosmological application oF this new theory proves to be very interesting. In particular, we obtain an eFFective dark energy sector whose equation-oF-state parameter can be quintessence or phantom-like, or exhibit the phantom-divide crossing, while For a large range oF the model parameters the Universe results in a de Sitter, dark-energy-dominated, accelerating phase. Additionally, we can obtain early-time inFlationary solutions too, and thus provide a uniFied description oF the cosmological history.
Rumen Dimitrov - One of the best experts on this subject based on the ideXlab platform.
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A class oF $${\Sigma _{3}^{0}}$$ modular lattices embeddable as principal Filters in $${\mathca
Archive for Mathematical Logic, 2008Co-Authors: Rumen DimitrovAbstract:Let I _0 be a a computable basis oF the Fully eFFective vector space V _∞ over the computable Field F . Let I be a quasimaximal subset oF I _0 that is the intersection oF n maximal subsets oF the same 1-degree up to *. We prove that the principal Filter $${\mathcal{L}^{\ast}(V,\uparrow )}$$ oF V = cl ( I ) is isomorphic to the lattice $${\mathcal{L}(n, \overline{F})}$$ oF subspaces oF an n -dimensional space over $${\overline{F}}$$ , a $${\Sigma _{3}^{0}}$$ Extension oF F . As a corollary oF this and the main result oF Dimitrov (Math Log 43:415–424, 2004) we prove that any Finite product oF the lattices $${(\mathcal{L}(n_{i}, \overline{F }_{i}))_{i=1}^{k}}$$ is isomorphic to a principal Filter oF $${\mathcal{ L}^{\ast}(V_{\inFty})}$$ . We thus answer Question 5.3 “What are the principal Filters oF $${\mathcal{L}^{\ast}(V_{\inFty}) ?}$$ ” posed by Downey and Remmel (Computable algebras and closure systems: coding properties, handbook oF recursive mathematics, vol 2, pp 977–1039, Stud Log Found Math, vol 139, North-Holland, Amsterdam, 1998) For spaces that are closures oF quasimaximal sets.
Tara L. Smith - One of the best experts on this subject based on the ideXlab platform.
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Decomposition oF Witt rings and Galois groups
Canadian Journal of Mathematics, 1995Co-Authors: Ján Mináč, Tara L. SmithAbstract:The intriguing relation between the theory oF quadratic Forms and Galois theory has been oF interest For a long time. (See For example [Wi:1936], [Wr:1979], [Wr:1983], [Wr:1985], [JWr:1989], [AEJ:1984], among others.) However, recently the connection between the Witt ring structure For a Field F (oF characteristic not 2) and its Galois groups was made quite precise. IF L is a Galois Field Extension oF F , with Gal(L/F ) ∼= G, we call L a G-Extension oF F . Given a Field F , with charF 6= 2, one can consider the Field Extension F /F which is the compositum over F oF all Z/2Z, Z/4Z-, and D4-Extensions oF F . (Here D4 denotes the dihedral group oF order 8.) One can then show that the Galois group GF oF F (3) over F , hereaFter reFerred to as the W-group oF F , is determined by W (F ), and that GF determines W (F ) except in the case when the level s(F ) oF F is ≤ 2 and the Form 〈1, 1〉 is universal. (In this paper basic knowledge oF quadratic Form theory and proFinite groups will be assumed. See [La:1973] or [Sc:1985] For the Former and [N:1971] and [Se:1965] For the latter. Throughout we will assume all Fields to be oF characteristic not 2.) In other words, knowledge oF GF is essentially equivalent to knowledge oF W (F ). (See [MiSm:1993], [:1990], [MiSp:1995], [Sm:1988], [Sp:1987].) This relationship between a speciFic Galois group oF F and W (F ) opened a new way oF attacking some questions in quadratic From theory and posed other new questions. In particular, it allows one to use the techniques oF inverse Galois theory to study some classical problems in the theory oF Witt rings. One oF the most outstanding problems is the characterization oF Witt rings in the category oF all rings. In spite oF many eFForts, very little is known. Indeed, we know only
