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Ronald Van Luijk - One of the best experts on this subject based on the ideXlab platform.
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density of Rational Points on del pezzo surfaces of degree one
Advances in Mathematics, 2014Co-Authors: Cecilia Salgado, Ronald Van LuijkAbstract:We state conditions under which the set S(k) of k-Rational Points on a del Pezzo surface S of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic fibration S -> P-1 induced by the anticanonical map has a nodal fiber over a k-Rational Point of P-1. It also suffices to require the existence of a Point in S(k) that does not lie on six exceptional curves of S and that has order 3 on its fiber of the elliptic fibration. This allows us to show that within a parameter space for del Pezzo surfaces of degree 1 over R, the set of surfaces S defined over Q for which the set S(Q) is Zariski dense, is dense with respect to the real analytic topology. We also include conditions that may be satisfied for every del Pezzo surface S and that can be verified with a finite computation for any del Pezzo surface S that does satisfy them. (c) 2014 Elsevier Inc. All rights reserved.
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density of Rational Points on del pezzo surfaces of degree one
arXiv: Algebraic Geometry, 2012Co-Authors: Cecilia Salgado, Ronald Van LuijkAbstract:We state conditions under which the set S(k) of k-Rational Points on a del Pezzo surface S of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic fibration over the projective line induced by the anticanonical map has a nodal fiber over a k-Rational Point. It also suffices to require the existence of a Point in S(k) that does not lie on six exceptional curves of S and that has order 3 on its fiber of the elliptic fibration. This allows us to show that within a parameter space for del Pezzo surfaces of degree 1 over the field of real numbers, the set of surfaces S defined over the field Q of Rational numbers for which the set S(Q) is Zariski dense, is dense with respect to the real analytic topology. We also include conditions that may be satisfied for every del Pezzo surface S and that can be verified with a finite computation for any del Pezzo surface S that does satisfy them.
V E Kravtsov - One of the best experts on this subject based on the ideXlab platform.
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commensurability effects in one dimensional anderson localization anomalies in eigenfunction statistics
Annals of Physics, 2011Co-Authors: V E Kravtsov, V I YudsonAbstract:Abstract The one-dimensional (1d) Anderson model (AM), i.e. a tight-binding chain with random uncorrelated on-site energies, has statistical anomalies at any Rational Point f = 2 a λ E , where a is the lattice constant and λE is the de Broglie wavelength. We develop a regular approach to anomalous statistics of normalized eigenfunctions ψ(r) at such commensurability Points. The approach is based on an exact integral transfer-matrix equation for a generating function Φr(u, ϕ) (u and ϕ have a meaning of the squared amplitude and phase of eigenfunctions, r is the position of the observation Point). This generating function can be used to compute local statistics of eigenfunctions of 1d AM at any disorder and to address the problem of higher-order anomalies at f = p q with q > 2. The descender of the generating function P r ( ϕ ) ≡ Φ r ( u = 0 , ϕ ) is shown to be the distribution function of phase which determines the Lyapunov exponent and the local density of states. In the leading order in the small disorder we derived a second-order partial differential equation for the r-independent (“zero-mode”) component Φ(u, ϕ) at the E = 0 ( f = 1 2 ) anomaly. This equation is nonseparable in variables u and ϕ. Yet, we show that due to a hidden symmetry, it is integrable and we construct an exact solution for Φ(u, ϕ) explicitly in quadratures. Using this solution we computed moments Im = N〈∣ψ∣2m〉 (m ⩾ 1) for a chain of the length N → ∞ and found an essential difference between their m-behavior in the center-of-band anomaly and for energies outside this anomaly. Outside the anomaly the “extrinsic” localization length defined from the Lyapunov exponent coincides with that defined from the inverse participation ratio (“intrinsic” localization length). This is not the case at the E = 0 anomaly where the extrinsic localization length is smaller than the intrinsic one. At E = 0 one also observes an anomalous enhancement of large moments compatible with existence of yet another, much smaller characteristic length scale.
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commensurability effects in one dimensional anderson localization anomalies in eigenfunction statistics
arXiv: Disordered Systems and Neural Networks, 2010Co-Authors: V E Kravtsov, V I YudsonAbstract:The one-dimensional (1d) Anderson model (AM) has statistical anomalies at any Rational Point $f=2a/\lambda_{E}$, where $a$ is the lattice constant and $\lambda_{E}$ is the de Broglie wavelength. We develop a regular approach to anomalous statistics of normalized eigenfunctions $\psi(r)$ at such commensurability Points. The approach is based on an exact integral transfer-matrix equation for a generating function $\Phi_{r}(u, \phi)$ ($u$ and $\phi$ have a meaning of the squared amplitude and phase of eigenfunctions, $r$ is the position of the observation Point). The descender of the generating function ${\cal P}_{r}(\phi)\equiv\Phi_{r}(u=0,\phi)$ is shown to be the distribution function of phase which determines the Lyapunov exponent and the local density of states. In the leading order in the small disorder we have derived a second-order partial differential equation for the $r$-independent ("zero-mode") component $\Phi(u, \phi)$ at the $E=0$ ($f=\frac{1}{2}$) anomaly. This equation is nonseparable in variables $u$ and $\phi$. Yet, we show that due to a hidden symmetry, it is integrable and we construct an exact solution for $\Phi(u, \phi)$ explicitly in quadratures. Using this solution we have computed moments $I_{m}=N $ ($m\geq 1$) for a chain of the length $N \rightarrow \infty$ and found an essential difference between their $m$-behavior in the center-of-band anomaly and for energies outside this anomaly. Outside the anomaly the "extrinsic" localization length defined from the Lyapunov exponent coincides with that defined from the inverse participation ratio ("intrinsic" localization length). This is not the case at the $E=0$ anomaly where the extrinsic localization length is smaller than the intrinsic one.
V I Yudson - One of the best experts on this subject based on the ideXlab platform.
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commensurability effects in one dimensional anderson localization anomalies in eigenfunction statistics
Annals of Physics, 2011Co-Authors: V E Kravtsov, V I YudsonAbstract:Abstract The one-dimensional (1d) Anderson model (AM), i.e. a tight-binding chain with random uncorrelated on-site energies, has statistical anomalies at any Rational Point f = 2 a λ E , where a is the lattice constant and λE is the de Broglie wavelength. We develop a regular approach to anomalous statistics of normalized eigenfunctions ψ(r) at such commensurability Points. The approach is based on an exact integral transfer-matrix equation for a generating function Φr(u, ϕ) (u and ϕ have a meaning of the squared amplitude and phase of eigenfunctions, r is the position of the observation Point). This generating function can be used to compute local statistics of eigenfunctions of 1d AM at any disorder and to address the problem of higher-order anomalies at f = p q with q > 2. The descender of the generating function P r ( ϕ ) ≡ Φ r ( u = 0 , ϕ ) is shown to be the distribution function of phase which determines the Lyapunov exponent and the local density of states. In the leading order in the small disorder we derived a second-order partial differential equation for the r-independent (“zero-mode”) component Φ(u, ϕ) at the E = 0 ( f = 1 2 ) anomaly. This equation is nonseparable in variables u and ϕ. Yet, we show that due to a hidden symmetry, it is integrable and we construct an exact solution for Φ(u, ϕ) explicitly in quadratures. Using this solution we computed moments Im = N〈∣ψ∣2m〉 (m ⩾ 1) for a chain of the length N → ∞ and found an essential difference between their m-behavior in the center-of-band anomaly and for energies outside this anomaly. Outside the anomaly the “extrinsic” localization length defined from the Lyapunov exponent coincides with that defined from the inverse participation ratio (“intrinsic” localization length). This is not the case at the E = 0 anomaly where the extrinsic localization length is smaller than the intrinsic one. At E = 0 one also observes an anomalous enhancement of large moments compatible with existence of yet another, much smaller characteristic length scale.
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commensurability effects in one dimensional anderson localization anomalies in eigenfunction statistics
arXiv: Disordered Systems and Neural Networks, 2010Co-Authors: V E Kravtsov, V I YudsonAbstract:The one-dimensional (1d) Anderson model (AM) has statistical anomalies at any Rational Point $f=2a/\lambda_{E}$, where $a$ is the lattice constant and $\lambda_{E}$ is the de Broglie wavelength. We develop a regular approach to anomalous statistics of normalized eigenfunctions $\psi(r)$ at such commensurability Points. The approach is based on an exact integral transfer-matrix equation for a generating function $\Phi_{r}(u, \phi)$ ($u$ and $\phi$ have a meaning of the squared amplitude and phase of eigenfunctions, $r$ is the position of the observation Point). The descender of the generating function ${\cal P}_{r}(\phi)\equiv\Phi_{r}(u=0,\phi)$ is shown to be the distribution function of phase which determines the Lyapunov exponent and the local density of states. In the leading order in the small disorder we have derived a second-order partial differential equation for the $r$-independent ("zero-mode") component $\Phi(u, \phi)$ at the $E=0$ ($f=\frac{1}{2}$) anomaly. This equation is nonseparable in variables $u$ and $\phi$. Yet, we show that due to a hidden symmetry, it is integrable and we construct an exact solution for $\Phi(u, \phi)$ explicitly in quadratures. Using this solution we have computed moments $I_{m}=N $ ($m\geq 1$) for a chain of the length $N \rightarrow \infty$ and found an essential difference between their $m$-behavior in the center-of-band anomaly and for energies outside this anomaly. Outside the anomaly the "extrinsic" localization length defined from the Lyapunov exponent coincides with that defined from the inverse participation ratio ("intrinsic" localization length). This is not the case at the $E=0$ anomaly where the extrinsic localization length is smaller than the intrinsic one.
Michael Stoll - One of the best experts on this subject based on the ideXlab platform.
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most odd degree hyperelliptic curves have only one Rational Point
Annals of Mathematics, 2014Co-Authors: Bjorn Poonen, Michael StollAbstract:Consider the smooth projective models C of curves y 2 = f(x) with f(x) 2 Z[x] monic and separable of degree 2g + 1. We prove that for g 3, a positive fraction of these have only one Rational Point, the Point at innity. We prove a lower bound on this fraction that tends to 1 as g! 1. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves. The method can be summarized as follows: using p-adic analysis and an idea of McCallum, we develop a reformulation of Chabauty’s method that shows that certain computable conditions imply #C(Q) = 1; on the other hand, using further p-adic analysis, the theory of arithmetic surfaces, a new result on torsion Points on hyperelliptic curves, and crucially the Bhargava{Gross theorems on the average number and equidistribution of nonzero 2-Selmer group elements, we prove that these conditions are often satised for p = 2.
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most odd degree hyperelliptic curves have only one Rational Point
arXiv: Number Theory, 2013Co-Authors: Bjorn Poonen, Michael StollAbstract:Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one Rational Point, the Point at infinity. We prove a lower bound on this fraction that tends to 1 as g tends to infinity. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves. The method can be summarized as follows: using p-adic analysis and an idea of McCallum, we develop a reformulation of Chabauty's method that shows that certain computable conditions imply #C(Q)=1; on the other hand, using further p-adic analysis, the theory of arithmetic surfaces, a new result on torsion Points on hyperelliptic curves, and crucially the Bhargava-Gross equidistribution theorem for nonzero 2-Selmer group elements, we prove that these conditions are often satisfied for p=2.
Cecilia Salgado - One of the best experts on this subject based on the ideXlab platform.
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density of Rational Points on del pezzo surfaces of degree one
Advances in Mathematics, 2014Co-Authors: Cecilia Salgado, Ronald Van LuijkAbstract:We state conditions under which the set S(k) of k-Rational Points on a del Pezzo surface S of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic fibration S -> P-1 induced by the anticanonical map has a nodal fiber over a k-Rational Point of P-1. It also suffices to require the existence of a Point in S(k) that does not lie on six exceptional curves of S and that has order 3 on its fiber of the elliptic fibration. This allows us to show that within a parameter space for del Pezzo surfaces of degree 1 over R, the set of surfaces S defined over Q for which the set S(Q) is Zariski dense, is dense with respect to the real analytic topology. We also include conditions that may be satisfied for every del Pezzo surface S and that can be verified with a finite computation for any del Pezzo surface S that does satisfy them. (c) 2014 Elsevier Inc. All rights reserved.
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density of Rational Points on del pezzo surfaces of degree one
arXiv: Algebraic Geometry, 2012Co-Authors: Cecilia Salgado, Ronald Van LuijkAbstract:We state conditions under which the set S(k) of k-Rational Points on a del Pezzo surface S of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic fibration over the projective line induced by the anticanonical map has a nodal fiber over a k-Rational Point. It also suffices to require the existence of a Point in S(k) that does not lie on six exceptional curves of S and that has order 3 on its fiber of the elliptic fibration. This allows us to show that within a parameter space for del Pezzo surfaces of degree 1 over the field of real numbers, the set of surfaces S defined over the field Q of Rational numbers for which the set S(Q) is Zariski dense, is dense with respect to the real analytic topology. We also include conditions that may be satisfied for every del Pezzo surface S and that can be verified with a finite computation for any del Pezzo surface S that does satisfy them.
