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Topic: Elliptic modular function

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	Elliptic function -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		The elliptic functions can be seen as analogs of the (Function of an angle expressed as a ratio of the sides of right-angled triangle containing the angle) trigonometric functions (which have a single period only).
		Elliptic functions are the (The function obtained by expressing the dependent variable of one function as the independent variable of another; f and g are inverse functions if f(x)=y and g(y)=x) inverse functions of (additional info and facts about elliptic integral) elliptic integrals, which is how they were introduced historically.
		The (additional info and facts about Weierstrass elliptic function) Weierstrass elliptic function is the prototypical elliptic function, and in fact, the field of elliptic functions with respect to a given lattice is generated by and its derivative.
	www.absoluteastronomy.com /encyclopedia/e/el/elliptic_function.htm (880 words)

	Why 10 dimensions? - Wikipedia, the free encyclopedia
		Modular functions are a subclass of the more general modular forms.
		Modular functions are used in the mathematical analysis of Riemann surfaces.
		When the Ramanujan function is generalized, 24 is replaced by 8 (8 + 2 = 10) for fermion strings.
	en.wikipedia.org /wiki/Why_10_dimensions (1261 words)

	Elliptic Curves and Modular Functions
		Modular and elliptic functions are both special cases of the concept of an automorphic function, which is a meromorphic function of 1 or more complex variables defined on a particular complex manifold and invariant under a particular group of analytic transformations (symmetries) of the manifold.
		the modular functions, are essentially the meromorphic functions on D considered as a Riemann surface by its isomorphism with H/ We have stressed this idea of symmetry because of the way it relates the analytic and geometric properties of an object like a Riemann surface to the algebraic properties of a group.
		Modular functions are, then, the automorphic functions on the upper half plane under the action of the modular group.
	www.mbay.net /~cgd/flt/flt05.htm (2994 words)

	Moduli space - Wikipedia, the free encyclopedia
		The use of the term modulus here for such a parameter goes back to the same source as in modular form: a modular form in general is some kind of differential form (or tensor density, since the forms come with a 'weight') on a moduli space, that is, a space whose co-ordinates are the moduli.
		In the case of elliptic curves, there is one modulus, so moduli spaces are algebraic curves.
		The case of elliptic curves has been thoroughly studied, because of the great interest of the modular equations in this case.
	en.wikipedia.org /wiki/Moduli_space (765 words)

	Elliptic curves (Site not responding. Last check: 2007-10-09)
		Modularity of a family of elliptic curves, by Fred Diamond, Kenneth Kramer.
		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)

	ELLIPTIC CURVES IN NATURE (Site not responding. Last check: 2007-10-09)
		Elliptic curves are identified by Cremona's scheme: the conductor, followed by a letter that names the isogeny class, followed by a number that indexes curves within this class.
		As was true for the elliptic modular curves of level 11, we see (mutatis mutandis) that the X_1 curve has a 7-torsion point, but here it's rational over K, not Q. Over K the curves attain their potentially good reduction at 3, and thus have conductor 2 as expected.
		An elliptic curve over Q has the desired property if and only if its j-invariant is the value of that function at some point x in P^1(Q) other than the two zeros x=1, x=-1 of the function.
	modular.fas.harvard.edu /tables/nature (5086 words)

	PlanetMath: elliptic function
		Remark: An elliptic function which is holomorphic is constant.
		-function (see elliptic curve) is an elliptic function, probably the most important.
		This is version 4 of elliptic function, born on 2003-07-22, modified 2003-08-04.
	planetmath.org /encyclopedia/EllipticFunction.html (161 words)

	The Miracle of Theta Functions
		Modular functions are functions which are meromorphic in H, the upper half of the complex plane, and which are invariant under a group of linear fractional transformations, G, in the sense that
		Thus the behaviour of a modular function is uniquely determined by its behaviour on a fundamental region.
		Modular functions are, in a sense, an extension of elliptic (or doubly periodic) functions --- functions such as sn which are invariant under linear transformations and which arise naturally in the inversion of elliptic integrals.
	www.cecm.sfu.ca /organics/papers/borwein/paper/html/node12.html (740 words)

	Elliptic Curves and Elliptic Functions
		For every algebraic function, it is possible to construct a specific surface such that the function is "single-valued" on the surface as a domain of definition.
		Classically, such doubly periodic functions were called elliptic functions, since they occurred in the elliptic integrals which represent the arc length of an ellipse.
		Further, the definition of an elliptic curve requires that there are no repeated roots of the polynomial in x, and this may fail to be true when reducing mod p for some primes.
	cgd.best.vwh.net /home/flt/flt03.htm (3513 words)

	Indian Mathematical Genius and Mystic - Kaku Hyperspace Ramanujan (Site not responding. Last check: 2007-10-09)
		These modular functions are as mysterious as the man who investigated them, the mystic from the East.
		One function, which appears again and again in the theory of modular functions, is today called the Ramanujan function in his honor.
		It is precisely these magic numbers appearing in the elliptic modular function that determines the dimension of space-time to be 10.' 173 'In the final analysis, the origin of the 10-dimensional theory is as mysterious as Ramanujan himself.
	www.talkabouthealthnetwork.com /group/alt.yoga/messages/63513.html (2453 words)

	Omega
		Every rational function of J is a modular function, and every modular function can be expressed as a rational function of J. The Fourier series of J, modulo a constant multiplicative factor, is called the j-function.
		On the Modular Function and Its Importance for Arithmetic -- Paula Cohen (2000).
		It was mainly Taniyama who connected the elliptic and modular functions, and his result was used by Wiles to prove Fermat.
	mywebpages.comcast.net /dantsmith/nexu38.htm (11764 words)

	Project-Team-tanc
		This is due, on the one hand, to the multitude of algorithmic improvements introduced in [29], on the other hand, to the lack of logarithmic factors and better constants.
		High precision evaluation of such functions is at the core of algorithms to compute class polynomials (used in complex multiplication) or modular polynomials (used in the SEA elliptic curve point counting algorithm).
		Exploiting the deep connection between the arithmetic-geometric mean (AGM) and a special kind of modular forms known as theta constants, he devised an algorithm based on Newton iterations and the AGM that has quasi-optimal complexity.
	www.inria.fr /rapportsactivite/RA2004/tanc2004/uid15.html (830 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		I knew that elliptic functions were involved in the general solution of a quintic (fifth degree) equation, but I was not aware that similar functions suffice to solve all polynomial equations.
		Modular functions are also distantly related, via the topic of "moonshine" and via the "Happy Family" of the Monster group and the Miracle Octad Generator of R. Curtis, to my own work on symmetries of 4x4 matrices.
		Klein's method of solving the quintic in terms of hypergeometric functions in one variable can be extended to the sextic, but for higher order polynomials, either hypergeometric functions in several variables or 'Siegel functions' must be used (Belardinelli 1960, King 1996, Chow 1999).
	www.xanga.com /item.aspx?user=m759&tab=weblogs&uid=54606233 (763 words)

	Table of contents for Library of Congress control number 2001046415
		Holomorphic Functions, the Cauchy-Riemann Equations, and Harmonic Functions 14 §1.5.
		The Elliptic Modular Function and Picard's Theorem 314 §10.6.
		Schlicht Functions and the Bieberbach Conjecture 384 §13.2.
	www.loc.gov /catdir/toc/fy031/2001046415.html (548 words)

	L-functions and elliptic curves
		For many such series of interest, the corresponding function can be extended to a meromorphic function on the whole plane by a process known as "analytic continuation".
		In fact, it is a rather difficult result established by Langlands and Deligne in 1972 that if E is an elliptic curve with a parameterization by modular functions as defined previously (i.
		Since this result demonstrates the functional equation for L(E,s), it verifies the Hasse-Weil conjecture when E is a modular curve (i.
	www.mbay.net /~cgd/flt/flt06.htm (2077 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.
		It follows that all elliptic functions are of the form g(℘(z)) + ℘′(z)h(℘(z)), where g(t) and h(t) are quotients of polynomials in the indeterminate t.
		Modular functions have been studied very extensively in their own right, apart from their relation to elliptic curves, since they have many applications in number theory and other parts of mathematics.
	www.openquestions.com /oq-ma017.htm (18524 words)

	Elliptic modular units, by Joerg Wildeshaus (Site not responding. Last check: 2007-10-09)
		Given an elliptic curve E -----> B over a base B with zero section i, we denote, letting E' := E - i(B), by L(E) the Q-vector space with basis ({s}, s \in E'(B)).
		Also, if B is the spectrum of a local field, then the absolute value of \phi should be expressible in terms of the local Néron height function.
		We also connect the values of \phi on specific elements of ker(d) to modular, and to elliptic units.
	front.math.ucdavis.edu /ANT/0078 (101 words)

	Earliest Known Uses of Some of the Words of Mathematics (M) (Site not responding. Last check: 2007-10-09)
		MODULAR CURVE appears in 1878 in J. Smith, "On the modular curves," Rep.
		Modular equation is found in 1844 in "Investigation of the Transformation of Certain Elliptic Functions," by Arthur Cayley in Philosophical Magazine, vol.
		According to Klein, this was the origin of the general name modular functions for functions with this or similar invariance [William C. Waterhouse].
	members.aol.com /jeff570/m.html (8972 words)

	Elliptic Curves
		Elliptic Functions and Elliptic Integrals by Viktor Prasolov and Yuri Solovyev (nice introduction to elliptic curves, functions and integrals).
		He also has notes on modular forms and modular functions.
		Connell's Handbook of elliptic curves is an ambitious project and still uncomplete.
	www.rzuser.uni-heidelberg.de /~hb3/elleng.html (928 words)

	Wolfram Research, Inc.
		for a discussion of argument conventions for elliptic functions.
		is an invariant of elliptic curves, equal to
		thus related can be regarded as coming from different possible pairs of periods for the same elliptic curve (that is, different fundamental parallelograms).
	documents.wolfram.com /v3/RefGuide/KleinInvariantJ.html (73 words)

	Underground Number Theory Seminar (Site not responding. Last check: 2007-10-09)
		Hecke was able to show that these functions had an analytic continuation to C and satisfied a functional equation using an enormously complicated application of generalized theta functions.
		In his PhD thesis Tate realized these L-functions as the integrals of certain nice functions over the ideles of K, in one fell swoop he was able to prove the analytic continuation and functional equation of these L-functions.
		Finally, we will be able to give explicit and computable formulas for the value at s=1 of the L-function of modular elliptic curves over certain totally real fields.
	www.cco.caltech.edu /~dw/maths/seminarlast.html (1199 words)

	PLANTSCAPE
		Starting with a modular form on the upper half plane (which can have poles at the cusps) satisfying some extra conditions, Borcherds constructs modular forms on the orthogonal groups, whose singularities are explicitly determined, and with an explicit product expansion.
		To conclude, as the automorphism group of a distinguished conformal field theory in the critical dimension 26, the monster is fundamentally related to string theory of theoretical physics.
		The subject Functional Analysis started around the beginning of this century, inspired by a desire to have a unified framework in which the two notions of continuity and linearity that arise in diverse contexts could be discussed abstractly.
	www.iisc.ernet.in /~currsci/dec25/articles6.htm (4992 words)

	Publisher description for Library of Congress control number 97008673 (Site not responding. Last check: 2007-10-09)
		A similar theory can be developed for special values of elliptic or elliptic modular functions, and is called complex multiplication of such functions.
		This subject is closely connected with the zeta function of an abelian variety, which is also covered as a main theme in the book.
		In particular, this is the first book in which the topics of various algebraic relations among the periods of abelian integrals, as well as the special values of theta and Siegel modular functions, are treated extensively.
	www.loc.gov /catdir/description/prin021/97008673.html (271 words)

	The 222-infinity Orbifold
		The most crucial step was in using the inverse of the elliptic modular function which maps the upper half plane onto the modular triangle with angles
		Unfortunately this happened to be a very slow way to calculate the function (and its inverse) on the computer.
		I had to precalculate a huge table of values of this function (this took three days of computer computations) and then to use Lagrange cubic interpolation to find the value of the function at specific points.
	www.geom.uiuc.edu /apps/pinball/poster/section3.4.html (399 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		This is the first part of the Analysis sequence for newly arriving graduate students in Mathematics.
		In fact, complex-differentiable functions are so rigid that the entire behavior of such a function is completely determined if you know its values even on a tiny open set.
		All the usual functions - polynomials, rational functions, exponential, trigonometric functions, and so on - are differentiable in the complex sense.
	www.math.psu.edu /roe/501/home.html (1009 words)

	ADE
		This is a singularity of the simplest kind which can be "desingularized" into a set of exceptional fibres which are complex projective lines intersecting as the dual of the Dynkin graph of finite type.
		By moonshine, this means that each node of these extended graphs can be labelled by a modular function given by the moonshine correspondence, with the elliptic modular function j(z) corresponding to the identity of M and so to the affine node of the E8 Dynkin graph.
		Adjacency is not understood but there should be a mechanism for obtaining the neighbours of a moonshine modular function occurring here.
	math.ucr.edu /home/baez/ADE.html (1245 words)

	Hyperspace Clues to the Psychology of the Pattern that Connects in the light of 81 Tao Te Ching insights
		The explanation is associated with the modular functions identified by Srinivasa Ramanujan (1887-1920) and named after him.
		It is precisely these magic numbers appearing in the elliptic modular function that determines the dimension of space-time to be ten.
		172) acknowledges that as mysterious as are the modular functions was the self-taught Ramanujan: "...the strangest man in all of mathematics, probably in the entire history of science.
	www.laetusinpraesens.org /docs00s/taote5.php (9387 words)

	How to Compute the Coefficients of the Elliptic Modular Function j(z) (ResearchIndex)
		Abstract: We discuss various methods to compute the Fourier coefficients of the elliptic modular function j(z).
		3 a class of non-linear functional equations connected with mo..
		2 cients of the modular invariant J (context) - Rademacher, coe - 1938
	citeseer.ist.psu.edu /baier03how.html (269 words)

	Dr. George DeRise's Homepage (Site not responding. Last check: 2007-10-09)
		The right side is the sum of the first two dimensions of the Monster Sporadic Group M, the left side is the third coefficient in the Laurent expansion of j, the elliptic modular function.
		The elliptic modular function j(τ) is defined as:
		j is, in a sense, the simplest non constant modular function; any other modular function will be a rational function in j(τ).
	www.tncc.cc.va.us /faculty/derise/misc2.html (3619 words)

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