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Topic: Modular group Gamma

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	Congruence subgroup - Wikipedia, the free encyclopedia
		In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries.
		Detailed information about the congruence subgroups of the modular group Gamma has proved basic in much research, in number theory, and in other areas such as monstrous moonshine.
		The modular group Λ (also called the theta subgroup) is another subgroup of the modular group Γ.
	en.wikipedia.org /wiki/Congruence_subgroup (565 words)

	Encyclopedia: Fuchsian group (Site not responding. Last check: 2007-11-05)
		A Fuchsian group is always a discrete group, and is a special case of a lattice in a semisimple Lie group.
		In mathematics, a discrete group is a group G equipped with the discrete topology.
		The group PSL(2,R) acts on H by linear fractional transformations: In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication.
	www.nationmaster.com /encyclopedia/Fuchsian-group (1105 words)

	[No title]
		The modular group is a group $\Gamma$, consisting of certain "fractional linear transformations" of the complex plane.
		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.
		Modular functions are, then, the automorphic functions on the upper half plane under the action of the modular group.
	pear.math.pitt.edu /mathzilla/Examples/Fermat/flt05.tex (3500 words)

	Talk:Modular group - Wikipedia, the free encyclopedia
		Also, technically, the modular group is PSL(2,Z), not SL(2,Z), although the distinction is usually glossed over in practice.
		Having said that, I have questioned the relevance of his SL(2,Z) to the modular* group anyway, as the modular group is isomorphic to PSL(2,Z) - see this section on his talk page.
		Recall that a matrix with entries in a commutative ring R is invertible iff the determinant is invertible in R.
	www.wikipedia.org /wiki/Talk:Modular_group_Gamma (832 words)

	Modular group - Wikipedia, the free encyclopedia
		It is for this reason that doubly-periodic functions, such as elliptic functions, possess a modular group symmetry.
		The action of the modular group on the rational numbers can most easily be understood by envisioning a square grid, with grid point (p,q) corresponding to the fraction p/q.
		In both cases, the numbers can be arranged to form a semigroup subset of the modular group.
	en.wikipedia.org /wiki/SL%282,Z%29 (1319 words)

	Modular group Gamma Info - Bored Net - Boredom (Site not responding. Last check: 2007-11-05)
		In mathematics, the modular group Γ is a matrix group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics.
		Definition The modular group Γ (Gamma) = SL(2,Z) is the 2-dimensional special linear group over the integers.
		For this reason, the group of orientation-preserving isometries of H is actually PSL(2,R), not SL(2,R).
	www.borednet.com /e/n/encyclopedia/m/mo/modular_group_gamma.html (327 words)

	The Modular Group and Fractals
		The goal here is to establish that these are once again given by the modular group, and that furthermore, these have utility in that they are automorphisms of the unit interval: they are both injective and surjective maps of the unit interval.
		Here, a modular form is a kind of analytic function that is closely related to elliptic functions, and has a certain type of modular group symmetry.
		The interior is compared visually to the Dedkind Eta (a modular form of weight 12) and the Weierstrass elliptic invariant g2 (a modular form of weight 2), as well as to sums built from the number-theoretic divisor function.
	www.linas.org /math/sl2z.html (1340 words)

	About the Modular Group Gamma_0(n) and its Normalizer in the Rational cases (Site not responding. Last check: 2007-11-05)
		About the Modular Group Gamma_0(n) and its Normalizer in the Rational cases
		(n), n in N, be the group of transformations z -> (az+b)/(cz+d), a,b,c,d in Z, ad-bc = 1, nc, of the upper half of the complex plane with corresponding field K(n) of invariant (modular) functions.
		A thorough investigation of these groups is presented including a formula for the length of a minimal generating system and a method to construct "canonical" fundamental regions.
	www.ece.cmu.edu /~pueschel/examples/fundreg/fundreg.html (155 words)

	Chapter 3 - The Minkowski Question Mark and the Modular Group (Site not responding. Last check: 2007-11-05)
		This group can be presented as a free group of a pair of generators, and there are many different possible choices of generators.
		of the modular group, and showed that it was a semigroup, and that it had a representation as a set of intervals.
		Finally, we note that the interval representation is a topology, and that it is not exactly a trivial topology for a subset of the modular group.
	www.linas.org /math/chap-minkowski/chap-minkowski.html (5943 words)

	[No title]
		What I termed the ``Shimura-Taniyama-Weil'' conjecture became known as the ``modular curve conjecture'' and then, from the summer of 1999, as the ``modular curve theorem'' after the work of Breuil, Conrad, Diamond, and Taylor in the same vein as the work of Wiles and Taylor for the ``semi-stable'' case.
		The linear group $\Gl{3}{k}$ acts on the space of cubics, and two cubic curves in $\Proj{2}$ that are related by this action are isomorphic.
		However the final calculation of a precise upper bound for the Selmer group in the semistable case (of the symmetric square representation associated to a modular form) is not yet complete as it stands.
	www.albany.edu /~hammond/gellmu/examples/f356g.glm (5880 words)

	A GELLMU Demonstration
		Elliptic curves are the “group objects” in the category of algebraic curves that reside in projective space: for each extension field K of k the set E(K) of “K-valued points” of E is an abelian group.
		A congruence subgroup of SL_{2}(Z) is a subgroup Gamma that contains one of the principal congruence subgroups; the principal congruence subgroup Gamma(N) of level N is the set of all elements gamma of SL_{2}(Z) that are congruent (mod N) to the identity matrix.
		The set of modular forms of given weight w forms a finite-dimensional vector space over C in which the set of cuspforms is a linear subspace of codimension bounded by the number of equivalence classes of cusps.
	www.albany.edu /~hammond/gellmu/examples/f356g.html (6263 words)

	Congruence subgroup (Site not responding. Last check: 2007-11-05)
		It can be posed in topological terms: if Γ is some arithmetic group, there is a topology on Γ for which a base ofneighbourhoods of {e} is the set of subgroups of finite index; and there is another topology defined in the same way using onlycongruence subsgroups.
		Therefore the problem can be posed as arelationship of two compact topological groups, with the question reduced to calculation of a possible kernel.
		The use of adele methods for automorphic representations (for example in the Langlands program) implicitly uses that kind of completion with respect to a congruence subgroup topology- for the reason that then all congruence subgroups can then be treated within a single group representation.
	www.therfcc.org /congruence-subgroup-218252.html (542 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		For this we study special eigenfunctions of the transfer operators with eigenvalues $\mp 1$, which are also solutions of the Lewis equations for the groups $\Gamma_{0}(n)$ and which are determined by eigenfunctions of the transfer operator for the modular group $\PSL (2,\mathbb{Z})$.
		It turns out that the sum of the components of these old solutions for the group $\Gamma_{0}(n)$ determine for any $n$ a solution of the Lewis equation for the modular group and hence also an eigenfunction of the transfer operator for this group.
		Indeed these operators are just the Hecke operators for the period functions of the modular group derived previously by Zagier and M\"uhlenbruch using the Eichler-Manin-Shimura correspondence between period polynomials and modular forms for the modular group.
	www.ma.utexas.edu /mp_arc/a/03-124 (222 words)

	NSDL Metadata Record -- $\Gamma(2)$ modular symmetry, renormalization, group flow and the quantum Hall effect
		We construct a family of holomorphic $\beta$-functions whose RG flow preserves the $\Gamma(2)$ modular symmetry and reproduces the observed stability of the Hall plateaus.
		The semi-circle law relating the longitudinal and Hall conductivities that has been observed experimentally is obtained from the integration of the RG equations for any permitted transition which can be identified from the selection rules encoded in the flow diagram.
		The existence of a crossing point occuring in the crossover of the permitted transitions is discussed.
	nsdl.org /mr/248888 (145 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		The modular group $\Gamma$ acts on the upper complex plane \[ \HH = \{z \in \C: \Im(z) > 0 \} \] by \[ gz = \frac{az+b}{cz+d} \] if $g = \bar{X}$, where \[ X = \begin{pmatrix} a & b\\ c & d \end{pmatrix}.
		\] \end{corollary} \begin{theorem} $\Gamma$ is freely-generated by the subgroups $C_2 = \Span{s},\; C_3 = \Span{u}$, ie each $g \in \Gamma$ is uniquely expressible in the form \[ g = u^{i_0}su^{i_1} \cdots u_{n-1}^{i_{n-1}} s u_n^{i_n}, \] where \[ 0 \le i_0, i_n \le 2,\; 1 \le i_j, i_n \le 2 \quad (0 < j < n).
		Let $\Gamma^+ \subset \Gamma$ correspond to the matrices with \emph{non-negative} entries: \[ \Gamma^+ = \{ \bar{X}: X = \begin{pmatrix} a & b\\c & d \end{pmatrix}: ad - bc = 1,\; a,b,c,d \ge 0.
	www.maths.tcd.ie /~tim/EllipticCurves/ch11.tex (851 words)

	[No title]
		These two permutations generate a subgroup $G=\langle g_0, g_1\rangle$ of the symmetric group $S^E$ of all permutations of $E$, called the {\sl monodromy group} of $\cal B$; the topological hypotheses imply that ${\cal G}$ has to be connected, so $G$ acts transitively on $E$.
		Conversely, every $2$-generator transitive group arises in this way from some bipartite map $\cal B$: the edges are the symbols permuted, the fl and white vertices correspond to the cycles of the two generators $g_0$ and $g_1$, and the faces correspond to the cycles of $g_{\infty}=(g_0g_1)^{-1}$.
		Similarly, the automorphism group of $\cal B$ is identified with the group of covering transformations of $\beta$.
	www.emis.ams.org /journals/SLC/wpapers/s35jones.tex (970 words)

	Klein Quartic Physics
		The tetrahedral rotation/reflection group [3,3] is isomorphic to the octahedral rotation group (4,3,2).
		kernel[s]; of a homomorphism, GAMMA(n) is a normal subgroup of GAMMA(1), and the factor group acts on the quotient surface as a group of automorphisms.
		The modular group GAMMA(1) is the triangle group (2,3,oo).
	www.valdostamuseum.org /hamsmith/KQphys.html (5084 words)

	gamma
		Gamma (Γ γ) is the 3rd letter of the Greek alphabet and has a numeric value of 3.
		gamma refers to the exponent in a power-law relationship between video or pixel values and the displayed brightness.
		In the game of Tetris, gamma is another name for the J tetromino.
	www.fact-library.com /gamma.html (172 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Geometrically, modularity means that the curve C can be parameterized by means of an algebraic map by one of the particular Riemann surfaces X_0(N) where N is an integer.
		This surface may be described as the result of taking the upper half plane H of the complex plane and glueing together any two points of H which are translates of each other under the action of the modular group Gamma_0(N).
		You may be familiar with the process of taking the plane and glueing together points whose coordinates differ by integers, resulting in the torus; well, this replaces the straight-edge fundamental domain of the group Z+Z by the loopy-edged fundamental domains of the Gamma_0(N)'s.
	www.math.niu.edu /~rusin/known-math/95/modular.djr (421 words)

	Articles - Prime geodesic (Site not responding. Last check: 2007-11-05)
		Given a Fuchsian group, that is, a discrete subgroup Î“ of PSL(2, R), Î“ acts on H via linear fractional transformation.
		Closed geodesics have been used to study Riemann surfaces; indeed, one of Riemann's original definitions of the genus of a surface was in terms of simple closed curves.
		Closed geodesics have been instrumental in studying the eigenvalues of Laplacian operators, arithmetic Fuchsian groups, and TeichmÃ¼ller spaces.
	www.multisection.com /articles/Prime_geodesic (588 words)

	"Permutation representations of the symmetry groups of regular hyperbolic tessellations" (Site not responding. Last check: 2007-11-05)
		Amalgamating a primitive orbit in a brother-sister pair.
		It is well known that the modular group, \Gamma = [ x,y
		It is an infinite Coxeter group generated by the reflections in the sides of the right hyperbolic triangle forming the fundamental region of the tessellation.
	users.wpi.edu /~hservat/mushtaq.html (183 words)

	Algebra and Number Theory Seminar
		The corresponding theta constants $\theta[\chi](0,\tau)$ can be used to study function theory on surfaces represented by the action of subgroups of the modular group $\Gamma = \mbox{ PSL}(2,{\bf Z})$ on the upper half plane.
		The linear group trinomial provides a mnemonic device for the recently discovered permutation polynomials of M\"uller-Cohen-Matthews, whereas the symplectic group equation generalizes them, thereby giving rise to strong genus zero coverings for characteristic two.
		Unipotent representations of the absolute Galois group of a number field naturally appear in the study of the $l$-adic realizations of a one-motive $M$.
	www.math.psu.edu /rvaughan/nthsemf00.html (1151 words)

	Gamma modular del grupo (Site not responding. Last check: 2007-11-05)
		En matemáticas, el grupo modular (gamma) es un grupo que es un objeto fundamental del estudio en teoría del número, geometría, álgebra, y muchas otras áreas de las matemáticas avanzadas.
		El grupo modular puede ser representado como grupo de transformaciones geométricas o como grupo de matrices.
		El grupo modular es importante porque forma a subgrupo del grupo de isometries del plano hiperbólico.
	www.yotor.net /wiki/es/ga/Gamma%20modular%20del%20grupo.htm (735 words)

	Efficient Average-Case Algorithms for the Modular Group - Cai, Fuchs, Kozen, Liu (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Abstract: The modular group occupies a central position in many branches of mathematical sciences.
		In this paper we give average polynomial-time algorithms for the unbounded and bounded membership problems for finitely generated subgroups of the modular group.
		1 Introduction 1.1 The Modular Group The modular group \Gamma is a remarkable mathematical object.
	citeseer.lcs.mit.edu /35940.html (608 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		In this paper, using the theory of modular forms, we prove seven identities of the following type: $$\sum_{k=0}^{[n/m]}\sigma_r(k)\sigma_s(n-mk)=P\sigma_{r+s+1}(n)+ Qn\sigma_{r+s-1}(n),\tag1 $$ which hold for every $n$ satisfying suitable congruences, for suitable integers $m\ge2$ and $r,s=1$ or 3, and for rationals $P$ and $Q$ (Theorem 2).
		1,$ the Eisenstein series $$E_{2k}(\tau):=1+\frac2{\zeta(1-2k)}\sum_{n=1}^{\infty}\sigma_{2k-1}(n)q^n$$ are modular forms of weight $2k$ for $\Gamma.$ The function $E_2(\tau):=1-24\sum_{n=1}^\infty\sigma_1(n)q^n$ is not a modular form, but is transformed under the action of $ SL(2,\Bbb Z)$ as follows: $$E_2\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{2}E_2(\tau)+\frac6 {\pi i}c(c\tau+d).$$ We shall also denote $E_{2k,m}(\tau)=E_{2k}(m\tau).$ For $k>1,$ the functions $E_{2k,m}$ are modular forms of weight $2k$ for $\Gamma_0(m).$ \subheading{3.
		With the same notation as above, for a modular form $f$ of weight $2k$ for $\Gamma_0(m)$ we have $$f\left(\frac{a}{c}\right)=\lim_{\tau\rightarrow i\infty} (c\tau+d)^{-2k}f\left(\frac{a\tau+b}{c\tau+d}\right)= \lim_{\varepsilon\rightarrow 0} (c\varepsilon)^{2k} f\left(\frac{a}{c}+\varepsilon\right).
	www.dm.unipi.it /gauss-pages/melfi/public_html/articoli/eger.tex (1672 words)

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