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Topic: Hecke operators

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		Vertex operators for J and H that add a row of size 3 to a polynomial indexed by a two column shape
		Symmetric function operators to add a row or a sequence of columns to the Schur symmetric functions.
		Zabrocki, Vertex operators for standard bases of the symmetric functions.
	garsia.math.yorku.ca /MPWP/maplefuncs.html (555 words)

	Hecke biography
		Hecke was awarded his doctorate at Göttingen in 1910 for a dissertation Zur Theorie der Modulfunktionen von zwei Variablen und ihrer Anwendung auf die Zahlentheorie which had been supervised by Hilbert.
		Hecke remained at Göttingen where he was appointed as an assistant to Hilbert and Klein.
		Probably Hecke's most important work was in 1936 with his discovery of the properties of the algebra of Hecke operators and of the Euler products associated with them.
	www-history.mcs.st-and.ac.uk /history/Biographies/Hecke.html (811 words)

	NationMaster - Encyclopedia: Hecke operator (Site not responding. Last check: )
		In mathematics, in particular in the theory of modular forms, a Hecke operator is a certain kind of 'averaging' operator that plays a significant role in the structure of vector spaces of modular forms (and more general automorphic representations).
		The theory of Hecke operators on modular forms is often said to have been founded by Mordell in a paper on the special cusp form of Ramanujan, ahead of the general theory given by Erich Hecke.
		In the classical elliptic modular form theory it is shown that the Hecke operators are a C-star algebra with respect to the Peterson inner product; and that therefore the spectral theory implies that there is a basis of modular forms that are eigenfunctions for all Hecke operators.
	www.nationmaster.com /encyclopedia/Hecke-operator (599 words)

	Matches for: Author/Editor=(Zhuravlev_V_G) (Site not responding. Last check: )
		The concept of Hecke operators was so simple and natural that, soon after Hecke's work, scholars made the attempt to develop a Hecke theory for modular forms, such as Siegel modular forms.
		As this theory developed, the Hecke operators on spaces of modular forms in several variables were found to have arithmetic meaning.
		Hecke operators are considered principally as an instrument for studying the multiplicative properties of the Fourier coefficients of modular forms.
	www.mathaware.org /bookstore?arg9=V._G._Zhuravlev&fn=100&l=20&pg1=CN&r=1&s1=Zhuravlev%5FV%5FG (223 words)

	Hecke biography
		In 1915 Hecke was appointed to a chair at the University of Basel but, three years later, he returned to a chair of mathematics at Göttingen.
		Courant then accepted the offer of Hecke's chair at Göttingen while Berlin, having been refused by their third choice Herglotz, tried to entice Hecke to leave Hamburg and accept the chair at Berlin.
		Hecke, however, was happy with his new post at Hamburg and turned down the offer from Berlin.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Hecke.html (811 words)

	Math Forum - Ask Dr. Math
		The eigenvector is a vector that remains invariant (except perhaps in magnitude) under the operation of the transformation matrix.
		Hecke Operators: The space of modular forms M = M(N) is finite dimensional.
		There are special operators called Hecke operators (denoted by T(p), one for each prime p) on this space.
	www.mathforum.org /library/drmath/view/51584.html (246 words)

	IHES PREPRINT M/03/19 (Site not responding. Last check: )
		In this article we report on a surprising relation between the transfer operators for the congruence subgroups $\Gamma_{0}(n)$ and the Hecke operators on the space of period functions for the modular group $\PSL (2,\mathbb{Z})$.
		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 $\PSL (2,\mathbb{Z})$.
		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.ihes.fr /PREPRINTS/M03/Resu/resu-M03-19.html (235 words)

	Hecke Operators
		A Brandt module M is equipped with a family of linear Hecke operators which act on it.
		Basis(M), but the Hecke operators of the ambient module may also be applied to elements of submodules.
		parameter set to false, the Hecke operators are determined by means of enumeration of ideals in a p-neighbouring operation analogous to the method of graphs approach of Mestre and Oesterl'e [Mes86].
	www.msri.org /about/computing/docs/magma/html/text1364.htm (149 words)

	[No title] (Site not responding. Last check: )
		The simplest class of intertwining operators consists of the algebra (which we denote R(V)) of operators intertwining a given representation G with itself.
		In fact a standard usage of the term "Hecke algebra" (though not precisely the one used in modular form theory) - aka the Hecke-Iwahori algebra, is this group H(G;K) where G is a p-adic group and K is its "Iwahori" subgroup, a compact open subgroup slightly smaller than the above maximal compact.
		In this case the Hecke algebra is almost abelian (in fact its large abelian subgroup is itself often called Hecke algebra),and its basis is indexed by a slight extension of the Weyl group..
	www.ma.utexas.edu /~benzvi/math/Langlands2 (1321 words)

	Hecke Operators (Site not responding. Last check: )
		A Brandt module M is equipped with a family of linear Hecke operators which act on it.
		Basis(M), but the Hecke operators of the ambient module may also be applied to elements of submodules.
		The system of Hecke operators are computed by default using the theta series which define the classical Brandt matrices.
	www.math.niu.edu /help/math/magmahelp/text1073.html (141 words)

	Hecke and Atkin-Lehner Operators
		The Hecke operator T_n of index n induced on the abelian variety A by virtue of its morphism to a modular symbols abelian variety.
		Finally we compute Hecke operators on the quotient of a simple factor of J_0(65) by a finite subgroup.
		The factored characteristic polynomial of the Hecke operator T_n acting on the abelian variety A. This can be faster than first computing T_n, then computing the characteristic polynomial, and factoring, because we can take into account information about the decomposition of A, in order to avoid factoring.
	www.math.lsu.edu /magma/text1332.htm (517 words)

	Hecke operator - Definition up Erdmond.Com
		is a certain kind of 'averaging' operator that plays a significant role in the structure of vector_spaces of modular forms (and more general automorphic_representations).
		These operators can be realised in a number of contexts; the simplest meaning is combinatorial, namely as taking for a given integer ''n'' some function ''f''(Λ) defined on lattices to :Σ ''f''(Λ′) with the sum taken over all the Λ′ that are subgroups of Λ of index ''n''.
		The theory of Hecke operators on modular forms is often said to have been founded by Mordell in a paper on the special cusp_form of Ramanujan, ahead of the general theory given by Erich_Hecke.
	www.erdmond.com /Hecke_operator.html (426 words)

	The Proof of Fermat's Last Theorem
		There are certain operators called Hecke operators, after Erich Hecke, on spaces of modular forms, and for the subspace S(N) in particular, since they preserve the weight of a form.
		Hecke operators can be defined concretely in various ways.
		S(N) is a normalized eigenform of all Hecke operators, it can in fact be shown that the coefficients in the Fourier expansion are all algebraic numbers and that they generate a finite extension K of Q.
	cgd.best.vwh.net /home/flt/flt08.htm (1543 words)

	Hecke (print-only)
		Schoeneberg describes Hecke's contributions to a number of topics which he lists as follows: Hilbert modular functions, Dedekind zeta functions, arithmetical notions and methods, elliptic modular forms of level N, algebraic functions, Dirichlet series with functional equation, Hecke-operators T
		Hecke's best work was in analytic number theory where he continued work of Riemann, Dedekind and Heinrich Weber.
		Hecke's contributions in number theory are discussed in detail in [2].
	www-groups.dcs.st-and.ac.uk /~history/Printonly/Hecke.html (774 words)

	Alex Ghitza - Research
		Modular forms and their generalizations are naturally endowed with a collection of commuting linear operators, called Hecke operators.
		Theta operator for Siegel modular forms, at the 2005 AMS Summer Institute on Algebraic Geometry, University of Washington, Seattle.
		A numerical exploration of mod p Hecke eigensystems, at the CMS Summer Meeting, University of Waterloo.
	www.colby.edu /personal/a/aghitza/research.html (830 words)

	42.9 Hecke operators on modular abelian varieties
		A Hecke operator is defined by given a modular abelian variety and an index.
		SAGE can compute the characteristic polynomial, and the action of the Hecke operator on various homology groups.
		Return the characteristic polynomial of this Hecke operator in the given variable.
	modular.math.washington.edu /sage/doc/html/ref/module-sage.modular.abvar.hecke-operator.html (119 words)

	Ell-E99.html
		Exercise: Draw pictures of fundamental domains for the Hecke groups of level 2, 4 and 8.
		Draw pictures of fundamental domains for the Hecke groups of level 2, 4 and 8.
		The existens of a basis of simultanous eigenvectors of selfadjoint operators is treated in general in Gantmacher: Matrix Theory, Chap.
	home.imf.au.dk /matjph/Ell-E99.html (0 words)

	The n-Category Café
		Using ‘crackpot matrices’ to describe Hecke operators between flag representations.
		James Dolan on two applications of Hecke operators: showing that any doubly transitive permutation representation is the direct sum of two irreducible representations, and getting ahold of the irreducible representations of n!
		The trick is to chop away at the fat ones using Hecke operators.
	golem.ph.utexas.edu /category (5491 words)

	Schedule
		Congruences for Hecke traces of singular moduli modulo powers of 2
		Gaussian hypergeometric functions and traces of Hecke operators
		The number of Siegel Hecke eigensystems (mod p)
	mars.vnet.wnec.edu /~jbeineke/automorphic/pages/schedule.htm (220 words)

	Combinatorics of traces of Hecke operators -- Frechette et al., 10.1073/pnas.0407223101 -- Proceedings of the National ...
		Combinatorics of traces of Hecke operators -- Frechette et al., 10.1073/pnas.0407223101 -- Proceedings of the National Academy of Sciences
		of the traces of the nth Hecke operators on the spaces of weight
		in general that Hecke traces are explicit rational linear combinations
	www.pnas.org /cgi/content/abstract/0407223101v1 (186 words)

	RMC - Vol. 17 Num. 1, Min Ho Lee (Site not responding. Last check: )
		Lee, Hecke operators on de Rham cohomology, Rev. Mat.
		We introduce Hecke operators on de Rham cohomology of compact oriented manifolds.
		When the manifold is a quotient of a Hermitian symmetric domain, we prove that certain types of such operators are compatible with the usual Hecke operators on automorphic forms.
	www.mat.ucm.es /serv/revmat/vol17-1/vol17-1e.html (50 words)

	Matches for: (Site not responding. Last check: )
		The original trace formula for Hecke operators was given by Selberg in 1956.
		This includes detailed discussions of modular forms, Hecke operators, adeles and ideles, structure theory for $\operatorname{GL}_2(\mathbf{A})$, strong approximation, integration on locally compact groups, the Poisson summation formula, adelic zeta functions, basic representation theory for locally compact groups, the unitary representations of $\operatorname{GL}_2(\mathbf{R})$, and the connection between classical cusp forms and their adelic counterparts on $\operatorname{GL}_2(\mathbf{A})$.
		This leads to an expression for the trace of a Hecke operator, which is then computed explicitly.
	www.mathaware.org /bookstore?fn=20&arg1=survseries&item=SURV-133 (254 words)

	Sharon M. Frechette
		Number Theory: Specifically modular forms of integral and half-integral weight, the Shimura correspondence, special values of L-functions, the connection between modular forms and elliptic curves, and Gaussian hypergeometric functions as related to traces of Hecke operators.
		"The Combinatorics of Traces of Hecke Operators," (with Ken Ono and Matthew Papanikolas), Proc.
		I was one of the co-organizers for the 18th Annual Workshop on Automorphic Forms and Related Topics, which was held March 21-24, 2004 at the University of California, Santa Barbara.
	mathcs.holycross.edu /~sfrechet/index.html (225 words)

	PlanetMath: algebraic number theory
		Read the entry on the arithmetic of elliptic curves for a full account of this beautiful theory.
		Definition of modular form and the Hecke algebra of Hecke operators.
		See Also: modular form, Hecke operator, bibliography for number theory, the arithmetic of elliptic curves, topics on ideal class groups and discriminants, examples of ring of integers of a number field, number field, theory of algebraic and transcendental numbers, theory of rational and irrational numbers, norm and trace of algebraic number, Hecke algebra
	www.planetmath.org /encyclopedia/AlgebraicNumberTheory.html (953 words)

	UNH Department of Mathematics & Statistics
		Abstract: We study the space of rational functions in one variable, and show that it is a union of eigenspaces of some natural operators called Hecke operators.
		The m\\\'th Hecke operator picks out every m\\\'th Taylor series coefficient of a rational function.
		It turns out the the simultaneous Hecke eigenfunctions are those rational functions whose numerators are called Fekete polynomials - basically polynomials that are generating functions for the values of a Dirichlet character mod L, where L is the degree of the polynomial.
	www.math.unh.edu /people/seminar.php?abstract=376 (169 words)

	St. Petersburg Mathematical Society
		Abstract: The explicit formulas for the transformation of theta-functions of integral positive definite quadratic forms under the action of regular Hecke operators, obtained in the author's earlier paper (1996), are converted to transformation formulas for the theta-functions with rational characteristics (the theta-series) viewed as Siegel modular forms.
		As applications, sequences of invariant subspaces and eigenfunctions for all regular Hecke operators on spaces of theta-series are constructed.
		Keywords: Hecke operators, Siegel modular forms, theta functions of quadratic forms, zeta functions of modular forms
	www.ams.org /spmj/2004-15-05/S1061-0022-04-00828-3/home.html (316 words)

	[No title] (Site not responding. Last check: )
		Normal operators are diagonalizable (for a proof, refer to Math H110).
		The Diamond Operator and the Decomposition of $S_k(\G_1(N))$} \begin{thm} $\G_1(N)$ is a normal subgroup of $\G_0(N)$ and we have $\G_0(N)/\G_1(N) \simeq (\Z/n\Z)^*$.
		The Hecke Operators on $S_k(\G_1(N))$} For $n \geq 1$, we have the operation on $S_k(\G_1(N))$ of the $n$th Hecke operator $T_n$.
	www.isc.tamu.edu /~mann/STILLER/STEIN/2 (16723 words)

	Lynne Walling's Home Page (Site not responding. Last check: )
		Hecke eigenforms and representation numbers of quadratic forms (Pacific J. of Math., 1991)
		Hecke eigenforms and representation numbers of arbitrary rank lattices (Pacific J. of Math., 1992)
		Hecke operators on Hilbert-Siegel modular forms (with S. Caulk; to appear)
	euclid.colorado.edu /~walling (235 words)

	Hecke Operator
		For a fixed integer k and any positive integer n, the Hecke operator
		For n a prime p, the operator collapses to
		Apostol, T. "The Hecke Operators." §6.7 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed.
	users.skynet.be /fa956617/math/topics/HeckeOperator.html (120 words)

	MIT Topology Seminar
		This discussion results in a formula describing the effect of Hecke operators on Hopkins-Kuhn-Ravenel characters.
		The resulting Hecke operators are described by the same formula as the twisted Hecke operators in generalized Moonshine.
		String operations parameterized by the homology of the mapping class groups.
	www-math.mit.edu /topology (1615 words)

	Trace identities of twisted Hecke operators on the spaces of cusp forms of half-integral weight, Masaru Ueda
		Let $R_{\psi}$ be a twisting operator for a quadratic primitive character $\psi$ and $\tilde{T}(n^2)$ the $n^2$-th Hecke operator of half-integral weight.
		When $\psi$ has an odd conductor, we already found trace identities between twisted Hecke operators $R_{\psi} \tilde{T}(n^2)$ of half-integral weight and certain Hecke operators of integral weight for almost all cases (cf.
		In this paper, the restriction is removed and we give similar trace identities for every quadratic primitive character $\psi$, including the case that $\psi$ has an even conductor.
	www.projecteuclid.org /Dienst/UI/1.0/Display/euclid.pja/1116442330 (260 words)

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