Let e be an elliptic curve over a finite field fq, where q pn and p. In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. Various elliptic curves 8 consider an elliptic curve e. Elliptic curve discrete logarithm problem ecdlp is the discrete logarithm problem for the group of points on an elliptic curve over a. We prove several results on torsion points and galois representations for complex multiplication cm elliptic curves over a number eld. The earliest results identifiable as galois cohomology had been known long before, in algebraic number theory and the arithmetic of elliptic curves. Basics of galois deformation theory and representation theory of profinite groups. Let eq be an elliptic curve and p be a prime number, and let g be the galois group of the extension of q obtained by adjoining the. For some large galois extensions lof k containing all ppower roots of unity, we show the vanishing of certain galois cohomology groups of lwith values. Elliptic cohomology prospects in mathematics durham. Some of the notes give complete proofs group theory, fields and galois theory, algebraic number theory, class field theory, algebraic geometry, while others are more in the nature of introductory overviews to a topic. Vanishing of some galois cohomology groups for elliptic curves. Elliptic curves can have points with coordinates in any.
It evolved into a joint project and contains many improvements. The book surveys some recent developments in the arithmetic of modular elliptic curves. The notes by tim dokchitser describe the proof, obtained by the author in a joint project with vladimir dokchitser, of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the tateshafarevich group. This will be an introductory course on the arithmetic of elliptic curves, concentrating on the study of the group of rational points. We determine all cases when the galois cohomology group h1g, ep does not vanish, and investigate the analogous question for epi when i1. The elliptic curve ef is said to be isotrivial if there is a finite galois. If l is a field, we shall write hil, m for the galois cohomology group. Several examples are given, and applications to modularity of galois representations are discussed. Conformal field theory and elliptic cohomology introduction.
This theory is then recast into the powerful and more general language of galois cohomology and descent theory. The concept of elliptic cohomology originates around. These are full notes for all the advanced graduatelevel courses i have taught since 1986. The first chapters, which grew out of tates haverford lectures, cover the arithmetic theory of elliptic curves over the field of rational numbers. Peter landweber, douglas ravenel, robert stong, periodic cohomology theories defined by elliptic curves, in haynes miller et. An introduction to the theory of elliptic curves pdf 104p covered topics are. Number fields generated by torsion points on elliptic curves. Elliptic curves with complex multiplication university of warwick. Since degree is 3, line through points p and p0 on e if p p0, use tangent at p has athird pointon e.
Vanishing of some galois cohomology groups for elliptic curves tyler lawson christian wuthrich september 23, 2015 abstract let eq be an elliptic curve and pbe a prime number, and let gbe the galois group of the extension of. Preface notation basic results from galois cohomology the iwasawa theory of the selmer group the euler characteristic formula numerical examples. We include an application to the verification of certain cases of the birch and swinnertondyer conjecture, and another application to. Elliptic curves, hilbert modular forms and galois deformations. Historically, elliptic cohomology arose from the study of elliptic genera. The purpose of this talk is 1 to introduce the notion and results of galois embedding, 2 and its application to elliptic curves and abelian surfaces. Pdf flavor symmetry and galois group of elliptic curves.
Galois cohomology and isomorphism classification of elliptic. A short course on galois cohomology william stein spring 2010. For this, we make explicit use of the correspondence between the twists and the first galois cohomology set with values in the automorphism group of the elliptic curve. Ranks of elliptic curves university of california, berkeley. Galois deformation, modular lifting and rt theorems ucla. It is related to elliptic curves and modular forms. This is done using the galois group of k s over k and its action on the automorphism group of the elliptic curve over k s. Arithmetic of elliptic curves through the ages school of. To that end, we will study elliptic curves over the rational numbers and endeavor to prove a weak version of the mordellweil theorem. For some large galois extensions lof k containing all ppower roots of unity, we show the vanishing of certain galois cohomology groups of.
Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. A conference on homotopy theory, june 1993, ams 1995. The current interest in qcurves, it is fair to say, began with. Husemollers text was and is the great first introduction to the world of elliptic curves and a good guide to the current research literature as well.
Galois cohomology study group university of warwick. Structure of tateshafarevich groups of elliptic curves over global. In our approach the flavor symmetry including the rparity has its origin in the galois group of elliptic curves with cm. The genesis of these notes was a series of four lectures given by the first author at the tata institute of fundamental research. Glv of a padic lie group, where v is a vector space over q p, we get a representation of the lie algebra of g, denoted lieg. This question has recently become much more pressing due to the work of mike hopkins and. Although the formal definition of an elliptic curve is fairly technical and requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry. As an application, we show that one interesting result on the number of twists of elliptic curves over finite fields also holds for some polarized abelian varieties of odd prime dimension over. Galois groups of number fields generated by torsion points of. The following are great sources for applying galois cohomology to elliptic curves. Going back to elliptic curves, lets briefly make k a global field with e an elliptic curve or abelian. Stolzteichner building on ideas of segal, relating equivariant versions of elliptic cohomology to loop groups, stolz and teichner. Galois embeddings of elliptic curves and abelian surfaces.
Tate, galois cohomology we say g acts trivially on a if. The gmodules form a category called gabwith morphisms being homomorphisms of groups which respect the gaction. Group cohomology, galois cohomology and elliptic curves yonatan harpaz 1 group cohomology let gbe a group. Introduction the purpose of the present paper is to address an old question posed by segal 37 to. We will be interested in studying elliptic curves over q or some number eld, and it will be useful to sometimes reduce them modulo some number. This includes selmer groups and tateshafarevich groups. A qcurve over k is an elliptic curve over k which is isogenous to all its galois conjugates.
Modular forms and galois cohomology cambridge studies in advanced mathematics 69 geometric modular forms and elliptic curves world scientific topics. Galois cohomology involves studying the group gby applying homo. Read download rational points on elliptic curves pdf pdf. Jul 10, 2017 as an application, we show that one interesting result on the number of twists of elliptic curves over finite fields also holds for some polarized abelian varieties of odd prime dimension over. Elliptic curves, second edition dale husemoller springer springer new york berlin heidelberg hong kong london milan paris tokyo. Our weapons of choice will be the cohomology of galois groups and the algebra of elliptic curves over the padic eld. Numerous and frequentlyupdated resource results are available from this search. A publication of the tata institute of fundamental research. Galois representations associated with elliptic curves jerome t. Galois groups of number fields generated by torsion points of elliptic curves volume 104 kay wingberg. Mathematics of the ussrsbornik galois cohomology and some questions of the theory of algorithms to cite this article. A brief introduction to galois representations attached to.
Galois representations and elliptic curves 3 from a representation g. This will be an introductory course on the arithmetic of elliptic curves, concentrating on the study of. In order to prove the mordell weil theorem, one breaks it in to parts, proving ekmek is nite, and then using descent. Elliptic curves m24 tom fisher elliptic curves are the rst nontrivial curves, and it is a remarkable fact that they have continuously been at the centre stage of mathematical research for centuries. Pdf galois cohomology of elliptic curves semantic scholar. The normal basis theorem implies that the first cohomology group of the additive group of l will vanish. Elliptic curves, the geometry of elliptic curves, the algebra of elliptic curves, elliptic curves over finite fields, the elliptic curve discrete logarithm problem, height functions, canonical heights on elliptic curves, factorization using elliptic curves, lseries. This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. Free elliptic curves books download ebooks online textbooks. By a gmodule we mean an abelian group awith an action of gon it i.
If we take g galkk, with k a galois extension of k of possibly in nite. Group cohomology, galois cohomology and elliptic curves. The proof of the theorem is in the section 4 of his paper. It places a special emphasis on the construction of rational points on elliptic curves, the birch and swinnertondyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. When z, q, qz are considered as gmodules, this is with the trivial action, unless stated otherwise. We prove several results on torsion points and galois representations for complex multiplication cm elliptic curves over a number eld containing the cm eld. For an in nite galois extension, one uses cocycles that come by in ation from nite galois subextensions. The fundamental idea of the iwasawa theory of elliptic curves, which grew out of iwasawas basic work on the ideal class groups of cyclotomic fields, is to study deep arithmetic questions about \e\ over \f\ via the study of coarser questions about the arithmetic of \e\ over various infinite extensions of \f\. Some of the notes give complete proofs group theory, fields and galois theory, algebraic number theory, class field theory, algebraic geometry, while others are more in the nature of.
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