Download or read book Elliptic Curves and Big Galois Representations written by Daniel Delbourgo and published by Cambridge University Press. This book was released on 2008-07-31 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: Describes the arithmetic of modular forms and elliptic curves; self-contained and ideal for both graduate students and professional number theorists.

Download or read book Geometric Modular Forms and Elliptic Curves written by Haruzo Hida and published by World Scientific. This book was released on 2012 with total page 468 pages. Available in PDF, EPUB and Kindle. Book excerpt: 1. An algebro-geometric tool box. 1.1. Sheaves. 1.2. Schemes. 1.3. Projective schemes. 1.4. Categories and functors. 1.5. Applications of the key-lemma. 1.6. Group schemes. 1.7. Cartier duality. 1.8. Quotients by a group scheme. 1.9. Morphisms. 1.10. Cohomology of coherent sheaves. 1.11. Descent. 1.12. Barsotti-Tate groups. 1.13. Formal scheme -- 2. Elliptic curves. 2.1. Curves and divisors. 2.2. Elliptic curves. 2.3. Geometric modular forms of level 1. 2.4. Elliptic curves over C. 2.5. Elliptic curves over p-adic fields. 2.6. Level structures. 2.7. L-functions of elliptic curves. 2.8. Regularity. 2.9. p-ordinary moduli problems. 2.10. Deformation of elliptic curves -- 3. Geometric modular forms. 3.1. Integrality. 3.2. Vertical control theorem. 3.3. Action of GL(2) on modular forms -- 4. Jacobians and Galois representations. 4.1. Jacobians of stable curves. 4.2. Modular Galois representations. 4.3. Fullness of big Galois representations -- 5. Modularity problems. 5.1. Induced and extended Galois representations. 5.2. Some other solutions. 5.3. Modularity of Abelian Q-varieties

Download or read book Abelian l Adic Representations and Elliptic Curves written by Jean-Pierre Serre and published by CRC Press. This book was released on 1997-11-15 with total page 203 pages. Available in PDF, EPUB and Kindle. Book excerpt: This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one

Download or read book Mod 4 Galois Representations and Elliptic Curves written by Christopher Holden and published by . This book was released on 2008 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book On Elliptic Units and P adic Galois Representations Attached to Elliptic Curves written by Alvaro Lozano-Robledo and published by . This book was released on 2005 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Computational Aspects of Modular Forms and Galois Representations written by Bas Edixhoven and published by Princeton University Press. This book was released on 2011-05-31 with total page 438 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.

Download or read book Galois Representations of Elliptic Curves and Abelian Entanglements written by and published by . This book was released on 2015 with total page 106 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Galois Representations Associated to Torsion Points of Elliptic Curves written by Theodore Hwa and published by . This book was released on 2003 with total page 70 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Elliptic Curves Modular Forms Fermat s Last Theorem written by John Coates and published by International Press of Boston. This book was released on 1995 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book The Large Sieve and Galois Representations written by David James Zywina and published by . This book was released on 2008 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book A First Course in Modular Forms written by Fred Diamond and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 462 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers elliptic curves as complex tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner theory; Hecke eigenforms and their arithmetic properties; the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory. The authors assume no background in algebraic number theory and algebraic geometry. Exercises are included.

Download or read book Elliptic Curves Over Finite Fields and Their L Torsion Galois Representations written by Michael Baker and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Let $q$ and $\ell$ be distinct primes. Given an elliptic curve $E$ over $\mathbf{F}_q$, we study the behaviour of the 2-dimensional Galois representation of $\mathrm{Gal}(\overline{\mathbf{F}_q}/\mathbf{F}_q) \cong \widehat{\mathbf Z}$ on its $\ell$-torsion subgroup $E[\ell]$. This leads us to the problem of counting elliptic curves with prescribed $\ell$-torsion Galois representations, which we answer for small primes $\ell$ by counting rational points on suitable modular curves. The resulting exact formulas yield expressions for certain sums of Hurwitz class numbers.

Download or read book Elliptic Curves and Related Topics written by H. Kisilevsky and published by American Mathematical Soc.. This book was released on 1994-01-01 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book represents the proceedings of a workshop on elliptic curves held in St. Adele, Quebec, in February 1992. Containing both expository and research articles on the theory of elliptic curves, this collection covers a range of topics, from Langlands's theory to the algebraic geometry of elliptic curves, from Iwasawa theory to computational aspects of elliptic curves. This book is especially significant in that it covers topics comprising the main ingredients in Andrew Wiles's recent result on Fermat's Last Theorem.

Download or read book Galois Representations Associated to Torsion Points of Elliptic Curves written by Seong Eun Jung and published by . This book was released on 2018 with total page 46 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Galois representations from non torsion points on elliptic curves written by Matthew Hughes and published by . This book was released on 2013 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Elliptic Curves and Icosahedral Galois Representations written by Edray Herber Goins and published by . This book was released on 1999 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Algorithms for Modular Elliptic Curves Full Canadian Binding written by J. E. Cremona and published by CUP Archive. This book was released on 1997-05-15 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents an extensive set of tables giving information about elliptic curves.