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	Modular function - Wikipedia, the free encyclopedia
		It can be shown that every modular function can be expressed as a rational function of Klein's absolute invariant j(τ), and that every rational function of j(τ) is a modular function; furthermore, all analytic modular functions are modular forms, although the converse does not hold.
		If a modular function f is not identically 0, then it can be shown that the number of zeroes of f is equal to the number of poles of f in the closure of the fundamental region R
		There are a number of other usages of the term modular function, apart from this classical one; for example, in the theory of Haar measures, it is a function Δ(g) determined by the conjugation action.
	en.wikipedia.org /wiki/Modular_function (257 words)

	Springer Online Reference Works
		The study of modular functions began in the 19th century in connection with the study of elliptic functions and preceded the appearance of the general theory of automorphic functions.
		Modular functions have also been applied in the study of conformal mapping; boundary properties of analytic functions and cluster sets (cf.
		The modular group (1) is then replaced by the modular group of automorphisms of the unit disc.
	eom.springer.de /m/m064430.htm (682 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.
	cgd.best.vwh.net /home/flt/flt05.htm (2993 words)

	Modular Function
		Every rational function of Klein's absolute invariant J is a modular function, and every modular function can be expressed as a rational function of J (Apostol 1997, p.
		Modular functions are special cases of modular forms, but not vice versa.
		An important property of modular functions is that if f is modular and not identically 0, then the number of zeros of f is equal to the number of poles of f in the closure of the fundamental region
	users.skynet.be /fa956617/math/topics/ModularFunction.html (169 words)

	The Miracle of Theta Functions (Site not responding. Last check: 2007-10-31)
		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)

	Luboš Motl's reference frame: Modular functions & elliptic genus
		This trace is a function of "q=exp(2.pi.i.tau)" where "tau" is the complex structure of the torus.
		Recall that the holomorphic modular functions (without singularities) are polynomials of the modular functions of weight 4 and 6, respectively.
		For example, the allowed weight zero modular functions include "G4^3 / G6^2" and similar expressions as long as you know that "4 x 3" as well as "6 x 2" are equal to "12".
	motls.blogspot.com /2006/09/modular-functions-elliptic-genus.html (1210 words)

	Modular Fractal Measures (Site not responding. Last check: 2007-10-31)
		Modular forms are analytic functions that are closely related to elliptic functions, and have a particular kind of modular group symmetry.
		A particular example of this is the twenty-fourth root of the modular discriminant, known as the Dedekind eta.
		Notably, modular forms already have the modular group symmetry, and so already have the symmetry properties of the final curve that we are searching for.
	www.linas.org /math/fdist/fdist.html (1399 words)

	Arithmetic cycles and automorphic forms (Site not responding. Last check: 2007-10-31)
		For instance, on any Hilbert modular surface, there is a rational function, defined over the underlying real quadratic field, which is the lifting of the classical $j$-invariant.
		For example, the modularity of the Mordell-Weil component is derived using Borcherds results on the generators and relations for certain spaces of modular forms and his construction of meromorphic modular forms on $\mathcal M$.
		The Deligne-Rapoport model of the reduction of a modular curve at a prime with "\Gamma_0(p)"-level structure is a key tool in the study of the arithmetic of modular forms.
	www.math.mcgill.ca /goren/WorkshopAbstracts.html (2852 words)

	Elliptic and Modular Functions
		More information on elliptic functions can be found for example in Chandrasekharan [Cha85], and for modular functions and their use see Koblitz [Kob84].
		Given a lattice L = [a, b] in the complex plane, this function returns the value of the elliptic j-invariant of L. This is the j-invariant of tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
		Given a pair L = [a,b] of complex numbers generating a lattice in C, return the q-series expansion of the discriminant Delta(q) evaluated at q = e^(2pi itau) where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
	www.math.lsu.edu /magma/text582.htm (1211 words)

	Modular Programming
		"Modular Programming" is the act of designing and writing programs as interactions among functions that each perform a single well-defined function, and which have minimal side-effect interaction between them.
		"Modular Programming" discourages the use of control variables and flags in parameters; their presence tends to indicate that the caller needs to know too much about how the function is implemented.
		To be reasonably cohesive, modular functions should not be excessively large; functions with over a page of code will be frowned upon.
	c2.com /cgi/wiki?ModularProgramming (570 words)

	Amazon.com: Modular Functions and Dirichlet Series in Number Theory (Graduate Texts in Mathematics): Books: Tom M. ... (Site not responding. Last check: 2007-10-31)
		Most of this book is devoted to a classical treatment of elliptic and modular functions with some of their number-theoretic applications.
		Among the major topics covered are Rademacher's convergent series for the partition modular function, Lehner's congruences for the Fourier coefficients of the modular function j, and Hecke's theory of entire forms with multiplicative Fourier coefficients.
		It is an excellent intro to modular forms, modular functions, the j-invariant, the Weierstrass elliptic functions and the Hecke operators, in the context of the modular group SL(2,Z).
	www.amazon.com /exec/obidos/tg/detail/-/0387971270?v=glance (891 words)

	kirupa.com - Making Functions More Modular
		The trick is simple: instead of using many if-else-statements we use a function referrer to determine the next action.
		This is quite useful, as the lookup of local variables is faster than the function lookup of the Math class.
		Well, we call the function that is referred to by action(), and this is initially the function stand().
	www.kirupa.com /developer/actionscript/modular.htm (499 words)

	Amazon.com: Abelian Varieties with Complex Multiplication and Modular Functions: Books: Goro Shimura (Site not responding. Last check: 2007-10-31)
		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.amazon.com /exec/obidos/tg/detail/-/0691016569?v=glance (956 words)

	Elliptic and Modular Functions
		This is a modular function of weight 0 whose Fourier expansion starts with j(s)=e^(-2pi i s) + 744e(2pi i s) + 196884e(2pi i s) +...
		Given a point t in the upper half plane and a positive integer p, return the normalized q-series expansion of the discriminant Delta(q) evaluated at t to precision p.
		Given a pair L = [a,b] of complex numbers generating a lattice in C, return the normalized q-series expansion of the discriminant Delta(q) evaluated at tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
	www.umich.edu /~gpcc/scs/magma/text570.htm (1235 words)

	Construction of a class of modular functions and forms., Marvin Isadore Knopp
		Construction of a class of modular functions and forms., Marvin Isadore Knopp
		Construction of a class of modular functions and forms.
		II : Construction of a class of modular functions and forms.
	projecteuclid.org /getRecord?id=euclid.pjm/1103037552 (137 words)

	Math Forum - Ask Dr. Math
		There I encountered "Modular functions." They said there were functions over the complex area with an incredible amount of symmetry.
		Date: 06/23/97 at 09:00:46 From: Doctor Anthony Subject: Re: Modular functions Dear Mart, Modular funcions are functions with super-symmetry, which means they can be transfomed in an infinity of different ways and yet remain unaltered.
		Elliptic equations are of the form y^2 = x^3 + ax^2 + bx + c with a, b, c integers, and we require integer solutions for x and y.
	mathforum.org /library/drmath/view/52239.html (223 words)

	bibliography for automorphic and modular forms, L-functions, representations, and number theory
		Piatetski-Shapiro, 'On zeta functions of infinite-dimensional representations', Mat.
		[Rankin 1939] R. Rankin, `Contributions to the theory of Ramanujan's function $\tau(n)$ and similar arithmetic functions, I', Proc.
		[Satake 1966] I. Satake, `Spherical functions and Ramanujan conjecture', in Proc.
	www.math.umn.edu /~garrett/m/b/bib.html (3638 words)

	Citebase - Traces of CM values of modular functions (Site not responding. Last check: 2007-10-31)
		Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps.
		Using the theta correspondence, we generalize this result to traces of CM values of (weakly holomorphic) modular functions on modular curves of arbitrary genus.
		We also study the theta lift for the weight 0 Eisenstein series for SL(2,Z) and realize a certain generating series of arithmetic intersection numbers as the derivative of Zagier's Eisenstein series of weight 3/2.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0408406 (166 words)

	Inverting modular-arithmetic functions (Site not responding. Last check: 2007-10-31)
		We can use what we know about division to find the inverse of some functions.
		The rule says to multiply the input by 3 and then add 1.
		Since 3 does not have a multiplicative inverse modulo 15, it's not so clear you can properly reverse the process.
	www.cs.brown.edu /courses/cs007/modmult/node3.html (145 words)

	Controller offers micro-modular PLC functions., EZAutomation
		EZTouchPLC integrates a micro modular PLC base with either 32 I/O (for 6" units) or 64 I/O (for 8, 10 and 15" units) available.
		The I/O utilized in the base are snap-in EZI/O modules (starting at $19 for 8 Digital out) which can be chosen from a large variety of options including Analog I/O, High Speed Counter, Relay, and AC/DC combo modules.
		Dec 7, 2005 - Compact PLC is designed for outdoor applications.
	news.thomasnet.com /fullstory/469793 (922 words)

	Citebase - Traces of CM values of modular functions and related topics (Site not responding. Last check: 2007-10-31)
		Citebase - Traces of CM values of modular functions and related topics
		Traces of CM values of modular functions and related topics
		Authors: Bruinier, Jan H. The purpose of this note is to report on recent joint work with J. Funke, P. Jenkins, and K. Ono on the traces of CM values of modular functions and some applications.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0510291 (129 words)

	Wiley::Topics in Complex Function Theory, Volume 3, Abelian Functions and Modular Functions of Several Variables
		Wiley::Topics in Complex Function Theory, Volume 3, Abelian Functions and Modular Functions of Several Variables
		Wiley > Mathematics & Statistics > Algebra > Complex & Functional Analysis > Topics in Complex Function Theory, Volume 3, Abelian Functions and Modular Functions of Several Variables
		Topics in Complex Function Theory, Volume 3, Abelian Functions and Modular Functions of Several Variables
	www.wiley.com /WileyCDA/WileyTitle/productCd-0471504017.html (147 words)

	R: Various modular functions (Site not responding. Last check: 2007-10-31)
		Modular functions including Klein's modular function J (aka Dedekind's Valenz function J, aka the Klein invariant function, aka Klein's absolute invariant), the lambda function, and Delta
		meaning to use the theta function expansion, and
		Last updated with Webcuts support: Sun Mar 12 12:42:38 MET 2006
	www.matematik.lu.se /help/R/.R/library/elliptic/html/J.html (54 words)

	Shimura, G.: Abelian Varieties with Complex Multiplication and Modular Functions.
		Shimura, G.: Abelian Varieties with Complex Multiplication and Modular Functions.
		F.A.Q. Abelian Varieties with Complex Multiplication and Modular Functions
		The Zeta Function of an Abelian Variety with Complex Multiplication
	pup.princeton.edu /titles/6242.html (289 words)

	CiteULike: Multipliers of a Family of Almost Modular Functions (Site not responding. Last check: 2007-10-31)
		CiteULike: Multipliers of a Family of Almost Modular Functions
		Multipliers of a Family of Almost Modular Functions
		Note: You or your institution must have access rights to this article.
	www.citeulike.org /article/14904 (102 words)

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