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Topic: Abelian functions

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	Abelian variety at AllExperts
		A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure.
		A morphism of polarized abelian varieties is a morphism A → B of abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the given form on A.
		An abelian scheme, sometimes called an abelian variety, over a base scheme S of relative dimension g is a proper, smooth group scheme over S whose geometric fibers are connected and of dimension g.
	en.allexperts.com /e/a/ab/abelian_variety.htm (1478 words)

	Rudi Weikard (Home Page) (Site not responding. Last check: 2007-10-20)
		Abelian functions are inverses of abelian integrals, i.e., integrals of rational functions of x and an algebraic function of x.
		If the degree of the polynomial is 3 or 4 one has elliptic integrals and elliptic functions while for degree 1 or 2 the integrals may be solved in terms of inverses of trigonometric functions.
		Naturally the study of abelian functions involves the algebraic curve (or Riemann surface) associated with the algebraic function under consideration and this relationship leads back to KdV equations and generalizations thereof.
	www.math.uab.edu /~rudi (407 words)

	NSDL Metadata Record -- Abelian Function -- from MathWorld
		Abelian functions have two variables and four periods, and can be defined by \Theta\left({v,\tau; \matrix{q'\cr q\cr}}\right)= \sum_{\lam...
		Abelian functions are a generalization of elliptic functions, and are also called hyperelliptic functions.
		Baker, H. Abelian Functions: Abel's Theorem and the Allied Theory, Including the Theory of the Theta Functions.
	nsdl.org /mr/696651 (125 words)

	Addition theorem - Wikipedia, the free encyclopedia
		Slightly more generally, as is the case with the trigonometric functions sin and cos, several functions may be involved; this is more apparent than real, in that case, since there cos is an algebraic function of sin (in other words, we usually take their functions both as defined on the unit circle).
		The conclusion of the mathematicians of the time was that the theory of abelian functions essentially exhausted the interesting possibilities: considered as a functional equation to be solved with polynomials, or indeed rational functions or algebraic functions, there were no further types of solution.
		The connected, projective variety examples are indeed exhausted by abelian functions, as is shown by a number of results characterising an abelian variety by rather weak conditions on its group law.
	en.wikipedia.org /wiki/Addition_theorem (339 words)

	Table of contents for Library of Congress control number 95030956
		Table of contents for Abelian functions : Abel's theorem and the allied theory of theta functions / H.F. Baker.
		The fundamental functions on a Riemann surface 3.
		The hyperelliptic case of Riemann's theta functions 12.
	www.loc.gov /catdir/toc/cam027/95030956.html (128 words)

	a directory of all known zeta functions (Site not responding. Last check: 2007-10-20)
		Ruelle explains that Artin-Mazur zeta functions are Weil zeta functions in the case where we have a diffeomorphism on a compact manifold.
		Tamagawa, "On the zeta function of a division algebra", Annals of Mathematics 77 (1963) 387-405.
		"On the poles of topological zeta functions", preprint (2004), 11pp.
	www.secamlocal.ex.ac.uk /~mwatkins/zeta/directoryofzetafunctions.htm (3278 words)

	[No title]
		His treatment of Abelian functions gives expression to the quality of mind that can reach beyond the domain of sense perception, transcend what {appears} to be infinite, recognize the existence of new species of transcendental powers, and {generate} them as higher forms of cognition.
		Thus, to grasp the higher principle we have to think of both functions happening simultaneously: the one on the stage of the plane that is generating the conic sections, and the one on the stage of the sphere, that is generating a lemniscate.
		Where, for example, the circular functions are periodic with respect to the interval 0 to 2Pi, the lemnsicatic functions are periodic with respect to the interval 1 to -1 {and} the interval \/-1 and -\/-1.
	www.wlym.com /antidummies/part54.html (5954 words)

	Abelian variety - Wikipedia, the free encyclopedia
		A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure.
		An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods; equivalently, it is a function in the function field of an abelian variety.
		A morphism of polarized abelian varieties is a morphism A → B of abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the given form on A.
	en.wikipedia.org /wiki/Abelian_variety (1547 words)

	[No title]
		The function \varphi(\alpha) is called an elliptic function and, when extended to the whole complex plane, gives a doubly periodic function.
		Abelian integrals are defined like elliptic integrals by u = \int(0 to v) R(t,\sqrt{f(t)})dt = I(v) where R(x,y) is a rational function of x and y, except that the function f is of a very general type which includes all polynomials.
		The same happened to another paper on Abelian functions which was published in 1848 in the Braunsberg school prospectus.
	math.nist.gov /opsf/personal/weierstrass.html (3054 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)

	a directory of all known zeta functions (Site not responding. Last check: 2007-10-20)
		Ruelle explains that Artin-Mazur zeta functions are Weil zeta functions in the case where we have a diffeomorphism on a compact manifold.
		Tamagawa, "On the zeta function of a division algebra", Annals of Mathematics 77 (1963) 387-405.
		Ruelle defines the Weil zeta function for an algebraic variety over a finite field in terms of the numbers of fixed points of all iterations of the Frobenius map on the extension of the algebraic variety to the algebraic closure of the finite field.
	secamlocal.ex.ac.uk /people/staff/mrwatkin/zeta/directoryofzetafunctions.htm (3385 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-20)
		An analogous construction for elliptic functions is to represent such functions in terms of quotients of theta-functions.
		It is in this way that the concepts and methods of the theory of automorphic functions were applied in the theory of algebraic groups, in which they play an important part in the description of infinite-dimensional representations [10].
		Finally, one must mention the application of automorphic functions to the study of ordinary differential equations in a complex domain [12] and in the construction of solutions of algebraic equations of degrees higher than four.
	eom.springer.de /a/a014170.htm (1232 words)

	zeta functions and L-functions (Site not responding. Last check: 2007-10-20)
		The analogies between function fields and number fields had been known since Dedekind’s time (at least in characteristic zero), but Artin’s work was perhaps the first to take the base field to have positive characteristic as opposed to subfields of the complex numbers.
		Artin also (later) developed a quite general theory of L-functions which, once again by purely algebraic means, defined functions akin to the zeta function for general number fields and for function fields.
		His approach to the ‘Riemann hypothesis for finite fields’ was to use the theory of ‘correspondences’ on an algebraic variety which had been developed by Severi and the ‘Italian school’ of algebraic geometers.
	www.maths.ex.ac.uk /~mwatkins/zeta/directoryofL-functions.htm (1010 words)

	Karl Theodor Wilhelm Weierstrass
		There he attended Gudermann's lectures on elliptic functions, some of the first lectures on this topic to be given, and Gudermann strongly encouraged Weierstrass in his mathematical studies.
		The topics of his lectures included the application of Fourier series and integrals to mathematical physics, an introduction to the theory of analytic functions, the theory of elliptic functions, applications to problems in geometry and mechanics, the foundations of analysis, and the integral calculus.
		The courses were "Introduction to the theory of analytic functions", "Elliptic functions","Abelian functions", and "Calculus of variations or applications of elliptic functions".
	www.stetson.edu /~efriedma/periodictable/html/W.html (835 words)

	Abelian Function -- from Wolfram MathWorld
		Abelian functions have two variables and four periods, and can be defined by
		elliptic functions, and are also called hyperelliptic functions.
		Any Abelian function can be expressed as a ratio of homogenous polynomials of the Riemann theta function (Igusa 1972, Deconinck et al.
	mathworld.wolfram.com /AbelianFunction.html (146 words)

	Several complex variables - Wikipedia, the free encyclopedia
		Many examples of such functions were familiar in nineteenth century mathematics: abelian functions, theta functions, and some hypergeometric series.
		In fact it was the need to put (in particular) the work of Oka on a clearer basis that led quickly to the consistent use of sheaves for the formulation of the theory (with major repercussions for algebraic geometry, in particular from Grauert's work).
		C.L. Siegel was heard to complain that the new theory of functions of several complex variables had few functions in it — meaning that the special function side of the theory was subordinated to sheaves.
	en.wikipedia.org /wiki/Several_complex_variables (766 words)

	Abelian Functions - Cambridge University Press
		Abel's Theorem and the Allied Theory of Theta Functions
		The subject is discussed by Baker in terms of transcendental functions, and in particular theta functions.
		The fundamental functions on a Riemann surface; 3.
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521498775 (248 words)

	zeta functions and L-functions (Site not responding. Last check: 2007-10-20)
		The analogies between function fields and number fields had been known since Dedekind’s time (at least in characteristic zero), but Artin’s work was perhaps the first to take the base field to have positive characteristic as opposed to subfields of the complex numbers.
		Artin also (later) developed a quite general theory of L-functions which, once again by purely algebraic means, defined functions akin to the zeta function for general number fields and for function fields.
		His approach to the ‘Riemann hypothesis for finite fields’ was to use the theory of ‘correspondences’ on an algebraic variety which had been developed by Severi and the ‘Italian school’ of algebraic geometers.
	secamlocal.ex.ac.uk /people/staff/mrwatkin/zeta/directoryofL-functions.htm (1010 words)

	No title
		Because the circle is a compact Abelian group, Fourier theory may be used to approximate the periodic function f by finite sums of characters.
		Classically, the class of almost periodic functions is closed under addition and multiplication, but constructively this is not the case.
		Constructively, the sum of two almost periodic functions need not be almost periodic, so M can not be an integral on the set of all almost periodic functions.
	www.cs.ru.nl /~spitters/almostper.html (2719 words)

	Shimura, G.: Abelian Varieties with Complex Multiplication and Modular 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.
		The third topic explored by Shimura is the various algebraic relations among the periods of abelian integrals.
		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.pupress.princeton.edu /titles/6242.html (300 words)

	University of Toronto Number Theory/Representation Theory Seminar
		Abstract: There are generalizations of the Riemann Zeta function to functions of several variables called the Multiple Zeta functions.
		This is a natural generalization of Dedekind zeta functions and is based on the notion of stable lattices and a new type of cohomology.
		Simply put, non-abelian $L$ functions are defined as integrations of Eisenstein series associated with $L^2$-automorphic forms with integration domain compact subsets obtained by truncating the corresponding fundamental domain using stability.
	www.math.toronto.edu /~jkorman/Num-Rep_seminar/2005.html (892 words)

	weierstrass (Site not responding. Last check: 2007-10-20)
		The topics of his lectures included:- mathematical physics (1856/57), introduction to the theory of analytic functions (where he set out results he had obtained in 1841 but never published), the theory of elliptic functions (his main research topic), and applications to problems in geometry and mechanics.
		In his 1863/64 course on The general theory of analytic functions Weierstrass began to formulate his theory of the real numbers.
		Weierstrass wrote a number of early papers on hyperelliptic integrals, abelian functions and algebraic differential equations.
	library.thinkquest.org /C006364/ENGLISH/mathematician/weierstrass.htm (342 words)

	[No title]
		Instead, Gauss insisted that these phenomena, as in the case of the catenary, must be understood as a unified process, in which the local variations in the position of the plumb bob or compass needle were a function of the characteristic of the principle governing the phenomenon as a whole.
		Gauss invented the idea of a “potential function,” to express the least-action effect of the physical principle over an area or volume, in a similar, but extended manner to that used by Leibniz to express the effect of gravity in producing the curvature of the hanging chain.
		Riemann demonstrated that all elliptical functions, being functions formed by the interaction of two connected principles, are expressed in the complex domain as surfaces with two boundaries (these boundaries are marked in green).
	www.schillerinstitute.org /fid_02-06/2004/044_riemann_dirichlet.html (3892 words)

	Abelian Functions
		Using the al function, we can prove for it to be a solution only by very primitive residual computations and combinatorial tricks, without using any theta functions.
		ABSTRACT: The Weierstrass elliptic pe function together with its derivatives and 1 form a linear basis of the space of meromorphic functions on a elliptic curve which have poles at a fixed point.
		We study the case of a principally polarized abelian variety whose theta divisor is non-singular.
	tmugs.math.metro-u.ac.jp /g-tmu20050526/index.html (1462 words)

	ABEL, N.H.(1802-1829) and GALOIS, E.(1811-1832)
		Abel's researches on elliptic functions arose in exciting and friendly competiton with Jacobi.
		The older Legendre, who had done pioneer work on elliptic functions, was deeply impressed with Abel's discoveries.
		Every student of analysis encounters Abel's integral equation and Abel's theorem on the sum of integrals of algebraic functions that leads to Abelian functions.
	library.thinkquest.org /22584/temh3002.htm?tqskip1=1 (785 words)

	Abelian Functions - Cambridge University Press (Site not responding. Last check: 2007-10-20)
		Abel's Theorem and the Allied Theory of Theta Functions
		Written in 1897, its scope was as broad as it could possibly be, namely to cover the whole of algebraic geometry, and associated theories.
		The subject is discussed by Baker in terms of transcendental functions, and in particular theta functions.
	www.cup.cam.ac.uk /catalogue/catalogue.asp?isbn=0521498775 (248 words)

	Underground Number Theory Seminar (Site not responding. Last check: 2007-10-20)
		I will then show how we are able to realize Hecke's L-functions as certain zeta functions and thereby deduce that they have analytic continuation to the complex plane and satisfy a functional equation.
		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.
	www.its.caltech.edu /~dw/maths/seminarlast.html (1199 words)

	Niels Henrik Abel - Wikipedia, the free encyclopedia
		From Berlin he passed to Freiberg, and here he made his brilliant researches in the theory of functions: elliptic, hyperelliptic, and a new class now known as abelian functions being particularly intensely studied.
		In 1826 Abel moved to Paris, and during a ten-month stay he met the leading mathematicians of France; but he was poorly appreciated, as his work was scarcely known, and his modesty restrained him from proclaiming his researches.
		The adjective "abelian", derived from his name, has become so commonplace in mathematical writing that it is conventionally spelled with a lower-case initial "a" (see abelian group and abelian category; also abelian variety and Abel transform).
	en.wikipedia.org /wiki/Niels_Henrik_Abel (666 words)

	Abelian Functions and the Development of Mathematics » eon
		Abelian Functions and the Development of Mathematics » eon
		When I was a student, abelian functions were, as an effect of the Jacobian tradition, considered the uncontested summit of mathematics and each of us was ambitious to make progress in this field.
		In mathematics, as in other sciences, the same processes can be observed again and again.
	unimodular.net /blog/?p=16 (247 words)

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