Introduction 2 1. The Group Law on Pell Conics and Elliptic Curves 2 1.1. Group Law on Conics 2 1.2. Group Law on Elliptic curves 3 2. The Group Structure 3 2.1. Finite Fields 3 2.2. p-adic Numbers 3 2.3. Integral and Rational Points 4 3. Applications 4 3.1. Primality Tests 4 3.2. Factorization Methods 5 4. 2-Descent 5 4.1. Selmer and Tate-Shafarevich Group 5 4.2. Heights 6 5. Analytic Methods ...

The well-known Birch and Swinnerton-Dyer conjecture gives a deep connection between the leading coefficient of the L-series and the arithmetic properties of an abelian variety. Both are very important and subtle. This paper is part of an effort to compute the analytic side explicitly in a special case. Indeed, we are interested in the central derivative of certain algebraic Hecke L-series, rela...

The condensed tannin (CT) content of eight shrub species (Cytisus purgans, Cytisus scoparius, Genista florida, Genista occidentalis, Calluna vulgaris, Erica arborea, Erica australis and Juniperus communis) from a grazing mountain area of northern Spain was analysed and related to various indicators of nutritive value. Chemical composition, in vitro gas production, OM degradation and true DM dig...

Alluaudomyia canariensis Szadziewski & Dominiak sp. nov. from the Canary Islands and A. wyskokensis Szadziewski & Dominiak sp. nov. from Poland and Ukraine are described and illustrated. The genus Alluaudomyia is reported from the Canary Islands for the first time. The article is supplemented with a checklist and an identification key for the species so far recorded from Europe and the Canary I...

Let X be a del Pezzo surface of degree 5 defined over a field F . A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every Del Pezzo surface of degree 5 is rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree 5 over a field F are precisely the F -forms of the moduli space M0,5 of stable curves of genus 0 with 5 marked points. Suppose ...

Let X be a del Pezzo surface of degree 5 defined over a field F . A theorem of Yu. I. Manin and P. Swinnerton-Dyer asserts that every Del Pezzo surface of degree 5 is rational. In this paper we generalize this result as follows. Recall that del Pezzo surfaces of degree 5 over a field F are precisely the F -forms of the moduli space M0,5 of stable curves of genus 0 with 5 marked points. Suppose ...

This is the first of two interconnected parts: Part I contains the geometric theory of generalized modular forms and their connections with the cooperation algebra for elliptic cohomology, E``∗E``, while Part II is devoted to the more algebraic theory associated with Hecke algebras and stable operations in elliptic cohomology. We investigate the structure of the stable operation algebra E``∗E``...