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Search Results: 1 - 10 of 452169 matches for " J. Arias-de-Reyna "
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X-Ray of Riemann zeta-function
J. Arias-de-Reyna
Mathematics , 2003,
Abstract: We present drawings on the complex plane of the lines Im(zeta(s))=0 and Re(zeta(s))=0. This allow to illustrate many properties of the zeta function of Riemann. This is an expository paper. It does not pretend to prove any new result about the zeta function. But it gives a new perspective to many known results. Also it may prove useful to show to students of Complex Variables or Analytic Number Theory. (This was my initial motivation to make the drawings).
Complex-Valued Gaussian Processes for Regression: A Widely Non-Linear Approach
Rafael Boloix-Tortosa,Eva Arias-de-Reyna,F. Javier Payan-Somet,Juan J. Murillo-Fuentes
Computer Science , 2015,
Abstract: In this paper we propose a novel Bayesian kernel based solution for regression in complex fields. We develop the formulation of the Gaussian process for regression (GPR) to deal with complex-valued outputs. Previous solutions for kernels methods usually assume a complexification approach, where the real-valued kernel is replaced by a complex-valued one. However, based on the results in complex-valued linear theory, we prove that both a kernel and a pseudo-kernel are to be included in the solution. This is the starting point to develop the new formulation for the complex-valued GPR. The obtained formulation resembles the one of the widely linear minimum mean-squared (WLMMSE) approach. Just in the particular case where the outputs are proper, the pseudo-kernel cancels and the solution simplifies to a real-valued GPR structure, as the WLMMSE does into a strictly linear solution. We include some numerical experiments to show that the novel solution, denoted as widely non-linear complex GPR (WCGPR), outperforms a strictly complex GPR where a pseudo-kernel is not included.
Automorphic Galois representations and the inverse Galois problem
Sara Arias-de-Reyna
Mathematics , 2013,
Abstract: A strategy to address the inverse Galois problem over Q consists of exploiting the knowledge of Galois representations attached to certain automorphic forms. More precisely, if such forms are carefully chosen, they provide compatible systems of Galois representations satisfying some desired properties, e.g. properties that reflect on the image of the members of the system. In this article we survey some results obtained using this strategy.
Formal groups, supersingular abelian varieties and tame ramification
Sara Arias-de-Reyna
Mathematics , 2009,
Abstract: Let us consider an abelian variety defined over $\mathbb{Q_{\ell}}$ with good supersingular reduction. In this paper we give explicit conditions that ensure that the action of the wild inertia group on the $\ell$-torsion points of the variety is trivial. Furthermore we give a family of curves of genus 2 such that their Jacobian surfaces have good supersingular reduction and satisfy these conditions. We address this question by means of a detailed study of the formal group law attached to abelian varieties.
A glimpse inside the mathematical kitchen
Juan Arias-de-Reyna,Jan van de Lune
Mathematics , 2010, DOI: 10.7153/jmi-05-30
Abstract: We prove the inequality sum_{k=1}^infty (-1)^{k+1} r^k cos(k*phi) (k+2)^{-1} < sum_{k=1}^infty(-1)^{k+1} r^k (k+2)^{-1} for 0 < r <= 1 and 0 < phi < pi. For the case r = 1 we give two proofs. The first one is by means of a general numerical technique (maximal slope principle) for proving inequalities between elementary functions. The second proof is fully analytical. Finally we prove a general rearrangement theorem and apply it to the remaining case 0 < r < 1. Some of these inequalities are needed for obtaining general sharp bounds for the errors committed when applying the Riemann-Siegel expansion of Riemann's zeta function.
Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion
Sara Arias-de-Reyna,Wojciech Gajda,Sebastian Petersen
Mathematics , 2010,
Abstract: In this paper we prove the Geyer-Jarden conjecture on the torsion part of the Mordell-Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on l-torsion points, for almost all primes l, contains the full symplectic group.
Tame Galois realizations of GSp_4(F_l) over Q
Sara Arias-de-Reyna,Núria Vila
Mathematics , 2009,
Abstract: In this paper we obtain realizations of the 4-dimensional general symplectic group over a prime field of characteristic $\ell>3$ as the Galois group of a tamely ramified Galois extension of $\mathbb{Q}$. The strategy is to consider the Galois representation $\rho_{\ell}$ attached to the Tate module at $\ell$ of a suitable abelian surface. We need to choose the abelian varieties carefully in order to ensure that the image of $\rho_{\ell}$ is large and simultaneously maintain a control on the ramification of the corresponding Galois extension. We obtain an explicit family of curves of genus 2 such that the Galois representation attached to the $\ell$-torsion points of their Jacobian varieties provide tame Galois realizations of the desired symplectic groups.
Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image
Sara Arias-de-Reyna,Luis Dieulefait,Gabor Wiese
Mathematics , 2012,
Abstract: This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part is concerned with symplectic Galois representations having a huge residual image, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. A key ingredient is a classification of symplectic representations whose image contains a nontrivial transvection: these fall into three very simply describable classes, the reducible ones, the induced ones and those with huge image. Using the idea of an (n,p)-group of Khare, Larsen and Savin we give simple conditions under which a symplectic Galois representation with coefficients in a finite field has a huge image. Finally, we combine this classification result with the main result of the first part to obtain a strenghtened application to the inverse Galois problem.
Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations
Sara Arias-de-Reyna,Luis Dieulefait,Gabor Wiese
Mathematics , 2012,
Abstract: This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem.
Abelian varieties over number fields, tame ramification and big Galois image
Sara Arias-de-Reyna,Christian Kappen
Mathematics , 2012,
Abstract: Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally polarized abelian variety A of dimension n over F such that the resulting \ell-torsion representation \rho_{A,\ell} from G_F to GSp(A[\ell](\bar{F})) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(\mathbb{F}_\ell) as the Galois group of a finite tame extension of number fields F'/F such that F is unramified above \ell.
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