Vanishing of some galois cohomology groups for elliptic curves. 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. This will be an introductory course on the arithmetic of elliptic curves, concentrating on the study of. Peter landweber, douglas ravenel, robert stong, periodic cohomology theories defined by elliptic curves, in haynes miller et. Basics of galois deformation theory and representation theory of profinite groups. 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. We prove several results on torsion points and galois representations for complex multiplication cm elliptic curves over a number eld containing the cm eld. This will be an introductory course on the arithmetic of elliptic curves, concentrating on the study of the group of rational points. 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.
Several examples are given, and applications to modularity of galois representations are discussed. Pdf flavor symmetry and galois group of elliptic curves. 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. In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology.
Let e be an elliptic curve over a finite field fq, where q pn and p. Pdf galois cohomology of elliptic curves semantic scholar. Elliptic curves, hilbert modular forms and galois deformations. Galois cohomology study group university of warwick. Modular forms and galois cohomology cambridge studies in advanced mathematics 69 geometric modular forms and elliptic curves world scientific topics.
Galois cohomology and isomorphism classification of elliptic. Elliptic curves, second edition dale husemoller springer springer new york berlin heidelberg hong kong london milan paris tokyo. Ranks of elliptic curves university of california, berkeley. 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. 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. Greenberg, introduction to the iwasawa theory of elliptic curves. Number fields generated by torsion points on elliptic curves. Stolzteichner building on ideas of segal, relating equivariant versions of elliptic cohomology to loop groups, stolz and teichner. Elliptic curves can have points with coordinates in any. This question has recently become much more pressing due to the work of mike hopkins and. 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. A conference on homotopy theory, june 1993, ams 1995.
This theory is then recast into the powerful and more general language of galois cohomology and descent theory. Our weapons of choice will be the cohomology of galois groups and the algebra of elliptic curves over the padic eld. To that end, we will study elliptic curves over the rational numbers and endeavor to prove a weak version of the mordellweil theorem. 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. We determine all cases when the galois cohomology group h1g, ep does not vanish, and investigate the analogous question for epi when i1. This includes selmer groups and tateshafarevich groups. In more recent times, the study of elliptic curves see 2 has. Galois embeddings of elliptic curves and abelian surfaces. 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. Free elliptic curves books download ebooks online textbooks. For some large galois extensions lof k containing all ppower roots of unity, we show the vanishing of certain galois cohomology groups of. Group cohomology, galois cohomology and elliptic curves yonatan harpaz 1 group cohomology let gbe a group. These are full notes for all the advanced graduatelevel courses i have taught since 1986. 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.
If we take g galkk, with k a galois extension of k of possibly in nite. It is related to elliptic curves and modular forms. The genesis of these notes was a series of four lectures given by the first author at the tata institute of fundamental research. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel.
A brief introduction to galois representations attached to. The earliest results identifiable as galois cohomology had been known long before, in algebraic number theory and the arithmetic of elliptic curves. The first chapters, which grew out of tates haverford lectures, cover the arithmetic theory of elliptic curves over the field of rational numbers. It evolved into a joint project and contains many improvements.
The proof of the theorem is in the section 4 of his paper. The elliptic curve ef is said to be isotrivial if there is a finite galois. 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. Elliptic cohomology prospects in mathematics durham. Introduction the purpose of the present paper is to address an old question posed by segal 37 to. When z, q, qz are considered as gmodules, this is with the trivial action, unless stated otherwise. Group cohomology, galois cohomology and elliptic curves. The gmodules form a category called gabwith morphisms being homomorphisms of groups which respect the gaction. Elliptic curves with complex multiplication university of warwick. 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.
Galois representations and elliptic curves 3 from a representation g. The normal basis theorem implies that the first cohomology group of the additive group of l will vanish. Galois groups of number fields generated by torsion points of. Numerous and frequentlyupdated resource results are available from this search.
Historically, elliptic cohomology arose from the study of elliptic genera. Galois representations associated with elliptic curves jerome t. Conformal field theory and elliptic cohomology introduction. The following are great sources for applying galois cohomology to elliptic curves. Structure of tateshafarevich groups of elliptic curves over global. An introduction to the theory of elliptic curves pdf 104p covered topics are. The answer is given in terms of a certain first galois cohomology set which is closely related to quadratic extensions of the field k. Galois cohomology involves studying the group gby applying homo. Arithmetic of elliptic curves through the ages school of. The current interest in qcurves, it is fair to say, began with. Preface notation basic results from galois cohomology the iwasawa theory of the selmer group the euler characteristic formula numerical examples.
The book surveys some recent developments in the arithmetic of modular elliptic curves. We include an application to the verification of certain cases of the birch and swinnertondyer conjecture, and another application to. This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. Galois deformation, modular lifting and rt theorems ucla. Read download rational points on elliptic curves pdf pdf. Various elliptic curves 8 consider an elliptic curve e. Softcover version available from bookstores worldwide.
The concept of elliptic cohomology originates around. In our approach the flavor symmetry including the rparity has its origin in the galois group of elliptic curves with cm. A publication of the tata institute of fundamental research. A short course on galois cohomology william stein spring 2010. In order to prove the mordell weil theorem, one breaks it in to parts, proving ekmek is nite, and then using descent. 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. Galois group galkk acts on all elm, so there is the natural continuous representation ladic representation. If l is a field, we shall write hil, m for the galois cohomology group. 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. Mathematics of the ussrsbornik galois cohomology and some questions of the theory of algorithms to cite this article. 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. 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.
A qcurve over k is an elliptic curve over k which is isogenous to all its galois conjugates. Going back to elliptic curves, lets briefly make k a global field with e an elliptic curve or abelian. 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. 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. Elliptic curve discrete logarithm problem ecdlp is the discrete logarithm problem for the group of points on an elliptic curve over a. By a gmodule we mean an abelian group awith an action of gon it i. Tate, galois cohomology we say g acts trivially on a if. We prove several results on torsion points and galois representations for complex multiplication cm elliptic curves over a number eld. For an in nite galois extension, one uses cocycles that come by in ation from nite galois subextensions. Fermats method of descent, plane curves, the degree of a morphism, riemannroch space, weierstrass equations, the group law, the invariant differential, formal groups, elliptic curves over local fields, kummer theory, mordellweil, dual isogenies and the weil pairing, galois cohomology, descent by cyclic isogeny. Since degree is 3, line through points p and p0 on e if p p0, use tangent at p has athird pointon e. Galois groups of number fields generated by torsion points of elliptic curves volume 104 kay wingberg.
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