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Topic: Arithmetic of elliptic curves

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	PlanetMath: the arithmetic of elliptic curves
		The theory of elliptic curves is a very rich mix of algebraic geometry and number theory (arithmetic geometry).
		The conductor of an elliptic curve is an integer quantity that measures the arithmetic complexity of the curve (the entry contains examples).
		This is version 11 of the arithmetic of elliptic curves, born on 2005-03-01, modified 2006-11-09.
	planetmath.org /encyclopedia/ArithmeticOfEllipticCurves.html (577 words)

	Elliptic Curves and Elliptic Functions
		The case of elliptic curves in the complex numbers is especially interesting, not only because of the algebraic completeness of C, but also because of the rich analytic theory that exists for complex functions.
		This mapping is, in effect, a parameterization of the elliptic curve by points in a "fundamental parallelogram" in the complex plane.
		Although the discriminant of a defining polynomial isn't an invariant of an elliptic curve, it is close.
	cgd.best.vwh.net /home/flt/flt03.htm (3513 words)

	MA426 Elliptic Curves
		Elliptic curves are interesting geometrically because they are the first instances of equations whose solution set cannot be parametrised by rational functions.
		Elliptic curves link number theory, algebraic geometry and complex analysis, and have applications to factorization of integers (a very hard problem for large integers), cryptography and coding theory.
		Elliptic curves were moreover prominent in Andrew Wiles' famous proof of Fermat's Last Theorem, and are at the forefront of research in number theory and related subjects.
	www.maths.warwick.ac.uk /undergrad/pydc/mauve/mauve-MA426.html (496 words)

	Elliptic curve - Wikipedia, the free encyclopedia
		One finds that elliptic curves correspond to embeddings of the torus into the complex projective plane; such embeddings generalize to arbitrary fields, and so it is said that elliptic curves are non-singular projective algebraic curves of genus 1 over a field K, together with a distinguished point defined over K.
		Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof, by Andrew Wiles (assisted by Richard Taylor), of Fermat's last theorem.
		Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1 with a given point defined over K.
	en.wikipedia.org /wiki/Elliptic_curves (1566 words)

	Elliptic Curves (Spring semester 2005)
		An elliptic curve may be defined as a plane non-singular cubic curve together with a point lying on the curve.
		J.H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag
		Given an elliptic curve E over a finite field F, a natural question is to ask for the number of F-rational points of E. This can in principle be computed by evaluating a lot of Legrende symbols, but this leads to an algorithm which is exponential in the input size.
	www.math.leidenuniv.nl /~ekkelkam/elliptic_curves (856 words)

	Elliptic curves (Site not responding. Last check: 2007-10-11)
		Arithmetic of elliptic curves with complex multiplication (lecture notes by Karl Rubin).
		Beppo Levi and the arithmetic of elliptic curves, by Norbert Schappacher.
		On an elliptic analogue of Zagier's conjecture, Jörg Wildeshaus.
	www.fermigier.com /fermigier/elliptic.html.en (746 words)

	PlanetMath: rank of an elliptic curve
		"rank of an elliptic curve" is owned by alozano.
		See Also: elliptic curve, height function, Mordell-Weil theorem, Selmer group, Mazur's theorem on torsion of elliptic curves, Nagell-Lutz theorem, the arithmetic of elliptic curves
		This is version 10 of rank of an elliptic curve, born on 2003-08-04, modified 2004-03-14.
	planetmath.org /encyclopedia/RankOfAnEllipticCurve.html (238 words)

	Courbes elliptiques (Site not responding. Last check: 2007-10-11)
		Arithmetic of elliptic curves with complex multiplication (notes de cours de Karl Rubin).
		On the conjecture of Birch and Swinnerton-Dyer for elliptic curves, by Cristian D. Gonzalez-Aviles.
		The average rank of an algebraic family of elliptic curves, J.
	www.fermigier.com /fermigier/elliptic.html (750 words)

	Summer School on Elliptic and Hyperelliptic Curve Cryptography
		For elliptic curves we explain Schoof's algorithm as a method to count points on curves over prime fields and consider p-adic methods like AGM which are more efficient in the case of small characteristic fields.
		Elliptic curves I, Roger Oyono, was given as a flboard presentation.
		Silverman, "The Arithmetic of Elliptic Curves", Springer Verlag.
	www.hyperelliptic.org /tanja/conf/summerschool06 (1102 words)

	The Case for Elliptic Curve Cryptography
		Elliptic curve cryptosystems also are more computationally efficient than the first generation public key systems, RSA and Diffie-Hellman.
		Despite the many advantages of elliptic curves and despite the adoption of elliptic curves by many users, many vendors and academics view the intellectual property environment surrounding elliptic curves as a major roadblock to their implementation and use.
		In the elliptic curve case, there is actually one additional bit that needs to be transmitted in each direction which allows the recovery of both the x and y coordinates of an elliptic curve point.
	www.nsa.gov /ia/industry/crypto_elliptic_curve.cfm (1818 words)

	Current Projects of Barry Mazur
		Elliptic curves and class field theory, appeared in the Proceedings of the International Congress of Mathematicians, ICM 2002, Beijing, Ta Tsien Li, ed., vol II.
		This is a survey of open problems regarding, for the most part, the (p-adic) anti-cyclotomic arithmetic of elliptic curves, in view of the recent breakthroughs due to Cornut and Vatsal, building on the work of many other people, including Kolyvagin.
		This is a fuller account of our theory of "organizations" of the p-adic anti-cyclotomic arithmetic of an elliptic curve E over a quadratic field K. It is the text of a lecture given by me at the Barcelona conference.
	www.math.harvard.edu /~mazur/projects.html (1234 words)

	Elliptic curves with H. A. Verrill
		An elliptic curve is (given by) a cubic (with a point).
		Elliptic curves are the simplest possible curves after lines and conics.
		Curves more complicated than elliptic curves (e.g., defined by polynomials of higher degree than 3) are generally difficult to understand.
	www.math.lsu.edu /~verrill/teaching/math7280 (1726 words)

	ALGORITHMS FOR MODULAR ELLIPTIC CURVES: INTRODUCTION (Site not responding. Last check: 2007-10-11)
		First, we describe in detail an algorithm based on modular symbols for computing modular elliptic curves: that is, one-dimensional factors of the Jacobian of the modular curve $X_0(N)$, which are attached to certain cusp forms for the congruence subgroup $\Gamma_0(N)$.
		This means that the values we compute are guaranteed to be correct, and eliminates the uncertainty previously existing as to whether the curves we obtain by rounding the computed values are the modular elliptic curves they are supposed to be.
		As with the modular symbol algorithms, we have rewritten all the elliptic curve algorithms in C++.
	www.maths.nott.ac.uk /personal/jec/book/chapter1.html (3376 words)

	Math 3065: Topics in Elliptic Curves, Spring 2005
		A second apparently unrelated area where elliptic curves have had a major impact is cryptography, where cryptosystems based on elliptic curves over finite fields have some significant advantages over older public key cryptosystems.
		The bible for elliptic curves in number theory is Silverman's two-volume Graduate Texts in Mathematics series: The Arithmetic of Elliptic Curves (GTM 106) and Advanced Topics in Elliptic Curves (GTM 151); the second volume is beyond the scope of the course.
		Elliptic curve cryptography is introduced in the last chapter, along with information about elliptic-curve based primality tests and factorization methods.
	www.pitt.edu /AFShome/d/i/dickinsm/public/html/3065-052 (1015 words)

	Prof Graham R Everest, MTH, UEA
		For an excellent introduction to elliptic curves, consult Joe Silverman's two books The Arithmetic of Elliptic Curves and Advanced Topics in the Arithmetic of Elliptic Curves.
		Let E denote an elliptic curve defined over an algebraic number field K. As usual, we denote the addition on the group E(K) of K-rational points of E by + and the group identity as O. Then O is the point at infinity on the projective curve.
		It is known that the global height of a K-rational point on an elliptic curve is invariant under isomorphism.
	www.mth.uea.ac.uk /~h090/elp.html (1048 words)

	[No title]
		Lattices in (and the Field of Elliptic Functions EL A lattice L in the complex plane is the set of all integral linear combinations of two complex numbers (1 and (2, where (1 and (2 are linearly independent.
		Accordingly, an elliptic curve (being of degree 3) intersects a line (being of degree 1) in three points (counting multiplicities) in the complex projective plane.
		Arithmetic on Elliptic Curves There are many interesting results on the arithmetic on elliptic curves, two of the most important of which are the Mordell-Weil Theorem and the Siegel Theorem.
	www.ms.uky.edu /~uwenagel/ALG-GEOM-04/watson.doc (1428 words)

	Taniyama–Shimura theorem - Wikipedia, the free encyclopedia
		The Taniyama–Shimura theorem (also called the modularity theorem) establishes an important connection between arithmetic of elliptic curves over rational numbers and modular forms, analytic objects of the 19th century mathematics, which are certain periodic holomorphic functions investigated in number theory.
		The remaining cases (of elliptic curve not with semistable reduction) were computed by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.
		The curve we obtain by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve (but not in general isomorphic to it).
	en.wikipedia.org /wiki/Taniyama-Shimura_conjecture (862 words)

	Home Page of Joseph Silverman
		Arithmetic Geometry, co-editor with Gary Cornell, Springer-Verlag, 1986.
		The Ubiquity of Elliptic Curves, Expanded version of MAA Invited Lecture presented at the Baltimore meeting, January 2003.
		An Introduction to the Theory of Elliptic Curves, Summer School on Computational Number Theory and Applications to Cryptography, University of Wyoming, June/July 2006.
	www.math.brown.edu /~jhs (429 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		I know that rho is used >as a coefficient of an elliptic curve over the complex plane, but I'd like >to know how to map or transform that into a curve over a specific field in >GF(2^n) (assuming all parameters are correct for doing that of course).
		An elliptic curve giving any specific j-invariant can be found defining the curve to be y^2=4x^3-cx-c and then solving for j in the corresponding equation.
		The coefficients a and b of the curve are related to the lattice by the relation given through the expansion of the p-function...
	www.maths.tcd.ie /~tim/EllipticCurves/nn (406 words)

	Amazon.com: The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics): Books: Joseph H. Silverman (Site not responding. Last check: 2007-10-11)
		The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study.
		The theory of elliptic curves has to rank as one of the most fascinating fields in all of mathematics.
		The author reserves the cases of elliptic curves in characteristics 2 and 3 to the appendix.
	www.amazon.com /Arithmetic-Elliptic-Curves-Graduate-Mathematics/dp/0387962034 (2156 words)

	AU: The Arithmetic of Elliptic Curves (Site not responding. Last check: 2007-10-11)
		Elliptic curves are introduced from an algebraic geometric point of view.
		After a brief review of the basics of algebraic curves we proceed to the geometry of elliptic curves, the group law on elliptic curves and the determination of the ring of endomorphisms.
		Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer, GTM, 1986.
	www.imf.au.dk /courses/arithmellipticurves (542 words)

	Open Questions: Elliptic Curves and Modular Forms
		The subject of elliptic curves is a lot like a major city, in which many highways and railroad lines converge, the airport serves as a hub of major airlines, and a seaport connects with important inland waterways.
		Elliptic curves can be thought of in many different ways, but perhaps the simplest and most intuitive is in terms of plane curves.
		Now, we have already seen that an elliptic curve as a complex torus is essentially determined by the period lattice of the ℘ function that parameterizes the curve.
	www.openquestions.com /oq-ma017.htm (18524 words)

	Elliptic Curves
		Elliptic Functions and Elliptic Integrals by Viktor Prasolov and Yuri Solovyev (nice introduction to elliptic curves, functions and integrals).
		Connell's Handbook of elliptic curves is an ambitious project and still uncomplete.
		Miles Reid has given a course on elliptic curves that is currently being TeXed.
	www.rzuser.uni-heidelberg.de /~hb3/elleng.html (928 words)

	ELLIPTIC CURVES AND IWASAWA THEORY SPRING 2006
		Elliptic curves, modular forms and cryptography (Allahabad, 2000), 63--72, Hindustan Book Agency, New Delhi, 2003.
		Rubin, Karl Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer.
		Arithmetic theory of elliptic curves (Cetraro, 1997), 167--234, Lecture Notes in Math., 1716, Springer, Berlin, 1999.
	www.math.uiuc.edu /~duursma/ECandIwTh.html (205 words)

	Elliptic Curve Cryptography
		Elliptic curve public-key cryptosystems - an introduction, E. De Win and B. Preneel, State of the Art in Applied Cryptography, Springer-Verlag, LNCS 1528, pp.131-141, 1998.
		Elliptic curve discrete logarithms and the index calculus, Joseph H. Silverman and Joe Suzuki, Proc.
		Elliptic Curve Cryptography on a Palm OS Device, A. Weimerskirch, C. Paar, and S. Chang Shantz, To appear in proc.
	cnscenter.future.co.kr /crypto/algorithm/ecc.html (3215 words)

	MATH 5590: Elliptic Curves and Applications to Cryptography, Spring 2005 (Site not responding. Last check: 2007-10-11)
		Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1994.
		Principal topics include elliptic curves over the rationals, elliptic curves over finite fields, the group law, torsion points, the Weil pairing, heights on elliptic curves, the Mordell-Weil Theorem, and elliptic curves over the complex numbers.
		It is essential to have a basic understanding of the arithmetic of elliptic curves in order to solve one of the many open problems.
	www.uwyo.edu /astein/05math5590 (391 words)

	Open Directory - Science: Math: Number Theory: Elliptic Curves and Modular Forms (Site not responding. Last check: 2007-10-11)
		Counting Points on Elliptic Curves - Robert Harley, Pierrick Gaudry, François Morain and Mireille Fouquet have established new records for point counting in characteristic 2, using a new algorithm by to Takakazu Satoh.
		Elliptic Curves and Formal Groups - Lecture notes from a seminar J. Lubin, J.-P. Serre and J. Tate.
		Elliptical Curve Cryptography - Explains the difference between an elliptical curve and an ellipse.
	dmoz.org /Science/Math/Number_Theory/Elliptic_Curves_and_Modular_Forms (780 words)

	Arithmetic of Elliptic Curves M390C, Fernando Rodriguez Villegas (Site not responding. Last check: 2007-10-11)
		Arithmetic of Elliptic Curves M390C, Fernando Rodriguez Villegas
		The Arithmetic of Elliptic Curves, by Joe Silverman, Springer Verlag.
		Great notes of a course on elliptic curves given by J.
	www.ma.utexas.edu /users/villegas/F00/elliptic-curves.html (167 words)

	Elliptic Curves - Cambridge University Press (Site not responding. Last check: 2007-10-11)
		The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, originated by Abel, Gauss, Jacobi, and Legendre.
		This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves.
		Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics, with many exercises with hints scattered throughout the text.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521582288 (239 words)

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