The Experts below are selected from a list of 330 Experts worldwide ranked by ideXlab platform
Karl Rubin - One of the best experts on this subject based on the ideXlab platform.
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finding large selmer rank via an arithmetic theory of local constants
Annals of Mathematics, 2007Co-Authors: Barry Mazur, Karl RubinAbstract:We obtain lower bounds for Selmer ranks of elliptic curves over dihedral Extensions of number fields. Suppose K/k is a quadratic Extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let κ - denote the maximal abelian p-Extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(κ-/K) by inversion). We prove (under mild hypotheses on p) that if the Zp-rank of the pro-p Selmer group Sp(E/K) is odd, then rank Zp S p (E/F) > [F: K] for every Finite Extension F of K in κ - .
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finding large selmer rank via an arithmetic theory of local constants
arXiv: Number Theory, 2005Co-Authors: Barry Mazur, Karl RubinAbstract:We obtain lower bounds for Selmer ranks of elliptic curves over dihedral Extensions of number fields. Suppose $K/k$ is a quadratic Extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $F$ denote the maximal abelian $p$-Extension of $K$ that is unramified at all primes where $E$ has bad reduction and that is Galois over $k$ with dihedral Galois group (i.e., the generator $c$ of $Gal(K/k)$ acts on $Gal(F/K)$ by -1). We prove (under mild hypotheses on $p$) that if the rank of the pro-$p$ Selmer group $S_p(E/K)$ is odd, then the rank of $S_p(E/L)$ is at least $[L:K]$ for every Finite Extension $L$ of $K$ in $F$.
Kim Min-soo - One of the best experts on this subject based on the ideXlab platform.
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InFinite order linear differential equation satisfied by $p$-adic Hurwitz-type Euler zeta functions
2021Co-Authors: Kim Min-sooAbstract:In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $\zeta(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder considered the question of whether $\zeta(s)$ satisfies a non-algebraic differential equation and showed that it formally satisfies an inFinite order linear differential equation. Recently, Prado and Klinger-Logan extended Van Gorder's result to show that the Hurwitz zeta function $\zeta(s,a)$ is also formally satisfies a similar differential equation \begin{equation*}\label{HurDE} T\left[\zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{equation*} But unfortunately in the same paper they proved that the operator $T$ applied to Hurwitz zeta function $\zeta(s,a)$ does not converge at any point in the complex plane $\mathbb{C}$. In this paper, by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's operator $T,$ we establish an analogue of Prado and Klinger-Logan's differential equation satisfied by $\zeta_{p,E}(s,a)$ which is the $p$-adic analogue of the Hurwitz-type Euler zeta functions \begin{equation*}\label{HEZ} \zeta_E(s,a)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+a)^s}. \end{equation*} In contrast with the complex case, due to the non-archimedean property, the operator $T_{p}^{a}$ applied to the $p$-adic Hurwitz-type Euler zeta function $\zeta_{p,E}(s,a)$ is convergent $p$-adically in the area of $s\in\mathbb{Z}_{p}$ with $s\neq 1$ and $a\in K$ with $|a|_{p}>1,$ where $K$ is any Finite Extension of $\mathbb{Q}_{p}$ with ramification index over $\mathbb{Q}_{p}$ less than $p-1.$Comment: 18 pages. Final version. Dedicated to the memory of Prof. David Goss (1952-2017
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InFinite order linear differential equation satisfied by $p$-adic Hurwitz-type Euler zeta functions
2020Co-Authors: Kim Min-sooAbstract:In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function $\zeta(s)$ is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder considered the question of whether $\zeta(s)$ satisfies a non-algebraic differential equation and showed that it formally satisfies an inFinite order linear differential equation. Recently, Prado and Klinger-Logan extended Van Gorder's result to show that the Hurwitz zeta function $\zeta(s,a)$ is also formally satisfies a similar differential equation \begin{equation*}\label{HurDE} T\left[\zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{equation*} But unfortunately in the same paper they proved that the operator $T$ applied to Hurwitz zeta function $\zeta(s,a)$ does not converge at any point in the complex plane $\mathbb{C}$. In this paper, by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder's operator $T,$ we establish an analogue of Prado and Klinger-Logan's differential equation satisfied by $\zeta_{p,E}(s,a)$ which is the $p$-adic analogue of the Hurwitz-type Euler zeta functions \begin{equation*}\label{HEZ} \zeta_E(s,a)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+a)^s}. \end{equation*} In contrast with the complex case, due to the non-archimedean property, the operator $T_{p}^{a}$ applied to the $p$-adic Hurwitz-type Euler zeta function $\zeta_{p,E}(s,a)$ is convergent $p$-adically in the area of $s\in\mathbb{Z}_{p}$ with $s\neq 1$ and $a\in K$ with $|a|_{p}>1,$ where $K$ is any Finite Extension of $\mathbb{Q}_{p}$ with ramification index over $\mathbb{Q}_{p}$ less than $p-1.$Comment: 16 pages, revised version. We would like to thank Professor Lawrence C. Washington for pointing out a gap in the proof of Lemma 3.1 of the original manuscript and for his helpful suggestion
W Qian - One of the best experts on this subject based on the ideXlab platform.
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the use of Finite Extension strain energy release rates in fracture of interfacial cracks
International Journal of Solids and Structures, 1997Co-Authors: W QianAbstract:Abstract The existing solutions for interfacial cracks in bimaterial media obtained from the contact model and oscillatory model were compared. The oscillatory near tip stress field was found to agree very well with that of the contact model except for the extremely small contact zone. Using the oscillatory solution. Mode I and Mode II “strain energy release rates” for Finite crack Extensions were obtained in terms of the stress intensity factors and the assumed crack Extension Aa. Finite elements in conjunction with the crack closure method were used to calculate these “strain energy release rates” from which accurate stress intensity factors were obtained. An alternative and efficient method based on crack surface displacement ratio was also introduced to obtain stress intensity factors. Non-oscillatory (Δα-independent) Mode I and Mode II “strain energy release rates” were proposed to provide an alternate measure of fracture mode mixity or to be used as a fracture criterion for interfacial cracks.
Franzviktor Kuhlmann - One of the best experts on this subject based on the ideXlab platform.
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every place admits local uniformization in a Finite Extension of the function field
Advances in Mathematics, 2009Co-Authors: Hagen Knaf, Franzviktor KuhlmannAbstract:We prove that every place P of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a Finite Extension F of F. We show that F|F can be chosen to be Galois, after a Finite purely inseparable Extension of the ground field K. Instead of being Galois, the Extension can also be chosen such that the induced Extension FP|FP of the residue fields is purely inseparable and the value group of F only gets divided by the residue characteristic. If F lies in the completion of an Abhyankar place, then no Extension of F is needed. Our proofs are based solely on valuation theoretical theorems, which are of particular importance in positive characteristic. They are also applicable when working over a subring R⊂K and yield similar results if R is regular and of dimension smaller than 3.
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every place admits local uniformization in a Finite Extension of the function field
arXiv: Algebraic Geometry, 2007Co-Authors: Hagen Knaf, Franzviktor KuhlmannAbstract:We prove that every place P of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a Finite Extension E of F. We show that E|F can be chosen to be Galois, after a Finite purely inseparable Extension of the ground field K. Instead of being Galois, the Extension can also be chosen such that the induced Extension EP|FP of the residue fields is purely inseparable and the value group of F only gets divided by the residue characteristic. If F lies in the completion of an Abhyankar place, then no Extension of F is needed. Our proofs are based solely on valuation theoretical theorems, which are of particular importance in positive characteristic. They are also applicable when working over a subring R of K and yield similar results if R is regular and of dimension smaller than 3.
Barry Mazur - One of the best experts on this subject based on the ideXlab platform.
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finding large selmer rank via an arithmetic theory of local constants
Annals of Mathematics, 2007Co-Authors: Barry Mazur, Karl RubinAbstract:We obtain lower bounds for Selmer ranks of elliptic curves over dihedral Extensions of number fields. Suppose K/k is a quadratic Extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let κ - denote the maximal abelian p-Extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(κ-/K) by inversion). We prove (under mild hypotheses on p) that if the Zp-rank of the pro-p Selmer group Sp(E/K) is odd, then rank Zp S p (E/F) > [F: K] for every Finite Extension F of K in κ - .
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finding large selmer rank via an arithmetic theory of local constants
arXiv: Number Theory, 2005Co-Authors: Barry Mazur, Karl RubinAbstract:We obtain lower bounds for Selmer ranks of elliptic curves over dihedral Extensions of number fields. Suppose $K/k$ is a quadratic Extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $F$ denote the maximal abelian $p$-Extension of $K$ that is unramified at all primes where $E$ has bad reduction and that is Galois over $k$ with dihedral Galois group (i.e., the generator $c$ of $Gal(K/k)$ acts on $Gal(F/K)$ by -1). We prove (under mild hypotheses on $p$) that if the rank of the pro-$p$ Selmer group $S_p(E/K)$ is odd, then the rank of $S_p(E/L)$ is at least $[L:K]$ for every Finite Extension $L$ of $K$ in $F$.