The complex AGM, periods of elliptic curves over C and complex elliptic logarithms

We give an account of the complex Arithmetic-Geometric Mean (AGM), as first studied by Gauss, together with details of its relationship with the theory of elliptic curves over \C, their period lattices and complex parametrisation. As an application, we present efficient methods for computing bases for the period lattices and elliptic logarithms of points, for arbitrary elliptic curves defined over \C. Earlier authors have only treated the case of elliptic curves defined over the real numbers; here, the multi-valued nature of the complex AGM plays an important role. Our method, which we have implemented in both \Magma\ and \Sage, is illustrated with several examples using elliptic curves defined over number fields with real and complex embeddings.Comment: The addional file elog_ex.sage contains a Sage script for the examples in the last section of the paper, and the file elog_ex.out contains the result of running that script with Sage version 5.

Efficient Solution of Rational Conics

We present efficient algorithms for solving Legendre equations over Q (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known

On elliptic curves of prime power conductor over imaginary quadratic fields with class number one

The main result of this paper is to extend from Q to each of the nine imaginary quadratic fields of class number one a result of Serre (1987) and Mestre-Oesterlé (1989), namely that if E is an elliptic curve of prime conductor then either E or a 2-, 3- or 5-isogenous curve has prime discriminant. For four of the nine fields, the theorem holds with no change, while for the remaining five fields the discriminant of a curve with prime conductor is either prime or the square of a prime. The proof is conditional in two ways: first that the curves are modular, so are associated to suitable Bianchi newforms; and second that a certain level-lowering conjecture holds for Bianchi newforms. We also classify all elliptic curves of prime power conductor and non-trivial torsion over each of the nine fields: in the case of 2-torsion, we find that such curves either have CM or with a small finite number of exceptions arise from a family analogous to the Setzer-Neumann family over Q

Computing the endomorphism ring of an elliptic curve over a number field

We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the j-invariant of an elliptic curve with complex multiplication (CM), and if so, the associated CM discriminant. More generally, given an elliptic curve E over a number field, one can use them to compute the endomorphism ring of E. Our algorithms admit simple implementations that are asymptotically and practically faster than existing approaches.Comment: updated and extended Table 1; 20 page

Clinical implications of acquired braf inhibitors resistance in melanoma

Understanding the role of mitogen-activated protein kinase (MAPK) pathway-activating mutations in the development and progression of melanoma and their possible use as therapeutic targets has substantially changed the management of this neoplasm, which, until a few years ago, was burdened by severe mortality. However, the presence of numerous intrinsic and extrinsic mechanisms of resistance to BRAF inhibitors compromises the treatment responses\u2019 effectiveness and durability. The strategy of overcoming these resistances by combination therapy has proved successful, with the additional benefit of reducing side effects derived from paradoxical activation of the MAPK pathway. Furthermore, the use of other highly specific inhibitors, intermittent dosing schedules and the association of combination therapy with immune checkpoint inhibitors are promising new therapeutic strategies. However, numerous issues related to dose, tolerability and administration sequence still need to be clarified, as is to be expected from currently ongoing trials. In this review, we describe the clinical results of using BRAF inhibitors in advanced melanoma, with a keen interest in strategies aimed at overcoming resistance

What is the probability that a random integral quadratic form in nn variables has an integral zero?

We show that the density of quadratic forms in $n$ variables over $\mathbb Z_p$ that are isotropic is a rational function of $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. When real quadratic forms in $n$ variables are distributed according to the Gaussian Orthogonal Ensemble (GOE) of random matrix theory, we determine explicitly the probability that a random such real quadratic form is isotropic (i.e., indefinite). As a consequence, for each $n$, we determine an exact expression for the probability that a random integral quadratic form in $n$ variables is isotropic (i.e., has a nontrivial zero over $\mathbb Z$), when these integral quadratic forms are chosen according to the GOE distribution. In particular, we find an exact expression for the probability that a random integral quaternary quadratic form has an integral zero; numerically, this probability is approximately $98.3\%$.Comment: 17 pages. This article supercedes arXiv:1311.554

The proportion of plane cubic curves over Q that everywhere locally have a point

We show that the proportion of plane cubic curves over Qpℚp that have a Qpℚp-rational point is a rational function in pp, where the rational function is independent of pp, and we determine this rational function explicitly. As a consequence, we obtain the density of plane cubic curves over Qℚ that have points everywhere locally; numerically, this density is shown to be ≈97.3%≈97.3%

Materiales y tecnologías en la Arquitectura Modernista: casos de estudio de decoración de fachadas en Italia, Portugal y Polonia persiguiendo una restauración racional

The results of a diagnostic survey on the materials of representative Art Nouveau buildings in Italy, Portugal and Poland are here presented and compared, as a contribution to their understanding and, hence, to support compatible restoration. In particular, the facade decorations were investigated for the appraisal of their materials and technologies, often neglected in current maintenance/restoration works and so cancelled, leading to a severe loss in architectural image. The ongoing diagnostic campaign, in collaboration among different universities, is aimed to set up a database on materials and technologies of Art Nouveau facade decorations at a European scale, as a technical-scientific background for the highlighting of preservation guidelines

09221 Abstracts Collection -- Algorithms and NumberTheory

From 24.05. to 29.05.2009, the Dagstuhl Seminar 09221 ``Algorithms and Number Theory \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available