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Topic: Modular curves

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	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)

	Directory - Science: Math: Number Theory: Elliptic Curves and Modular Forms: Tables
		Elliptic Curves with Unusual Torsion · cached · Two tables: the smallest conductor observed for a given rank and torsion, and the smallest conductor observed among curves of rank zero with a given Sha and torsion.
		Mordell Curves · cached · Minimal known positive and negative k for Mordell curves (y^2=x^3+k) of given rank, by Tom Womack.
		Iwasawa Invariants of Elliptic Curves · cached · For each curve (labelled as in Cremona) the mu and lambda-invariants are listed for the primes between 2 and 17.
	www.incywincy.com /default?p=400123 (445 words)

	Modular Curves
		A package for computing with modular curves has been developed by David Kohel.
		Modular curves in Magma are a special type of plane curve.
		Creation of a modular curve of specified level rom a database.
	magma.maths.usyd.edu.au /magma/Features/node234.html (110 words)

	Elliptic Curves and Modular Forms Science, Directory (Site not responding. Last check: 2007-11-07)
		Modularity of Elliptic Curves and Beyond Workshop, MSRI Berkeley, 6-10 December 1999.
		Modular Curves X_0(N) A series of lectures by Bas Edixhoven at the ICTP Summer school on rational torsion of elliptic curves over number fields.
		Modular Forms and Hecke Operators Notes by William A. Stein of a course by Ken Ribet.
	www.indiapolicyinstitute.org /aW5kXzIwMDI5NQ==.aspx (711 words)

	Enrique González Jiménez (Site not responding. Last check: 2007-11-07)
		"Modular Hyperelliptic Curves" Ph.D.thesis (in Spanish), January 2002.
		"Modular Curves of Genus 2" with Josep González.
		"Finiteness Results for Modular Curves of Genus at Least 2", with Matthew Baker, Josep González and Bjorn Poonen.
	www.maths.nottingham.ac.uk /personal/pmzeg1 (65 words)

	Elliptic curves (Site not responding. Last check: 2007-11-07)
		Modularity of a family of elliptic curves, by Fred Diamond, Kenneth Kramer.
		Modular Symbols and the computation of Modular Elliptic Curves, John Cremona.
		Beppo Levi and the arithmetic of elliptic curves, by Norbert Schappacher.
	www.fermigier.com /fermigier/elliptic.html.en (746 words)

	Directory - Science: Math: Number Theory: Elliptic Curves and Modular Forms: Software
		The calculation of properties of elliptic curves such as their torsion groups, rank, and point counting on elliptic curves over finite fields is a highly non-trivial process.
		Modular Forms Software · cached · HECKE can be used to compute basis of q-expansions and Hecke operators on fairly general spaces of modular forms.
		Periods of Hilbert Modular Forms · cached · A package of PARI programs (v.2.1.1 or higher) for calculations described in "Periods of Hilbert modular forms and rational points on elliptic curves" by H. Darmon and A. Logan.
	www.incywincy.com /default?p=405239 (244 words)

	Open Directory - Science: Math: Number Theory: Elliptic Curves and Modular Forms (Site not responding. Last check: 2007-11-07)
		Modular Forms and Hecke Operators - Notes by William A. Stein of a course by Ken Ribet.
		Modular Forms Course - Notes of a 1996 Berkeley course of Ken Ribet's on modular forms and Hecke operators.
		Modularity of Elliptic Curves and Beyond - Workshop, MSRI Berkeley, 6-10 December 1999.
	dmoz.org /Science/Math/Number_Theory/Elliptic_Curves_and_Modular_Forms (793 words)

	Science Math Number Theory Elliptic Curves and Modular Forms Tables (Site not responding. Last check: 2007-11-07)
		Elliptic Curves with Unusual Torsion - Two tables: the smallest conductor observed for a given rank and torsion, and the smallest conductor observed among curves of rank zero with a given Sha and torsion.
		Iwasawa Invariants of Elliptic Curves - For each curve (labelled as in Cremona) the mu and lambda-invariants are listed for the primes between 2 and 17.
		MODI - Interactive Modular Forms Data Base - Data about modular forms which are computed on demand or taken from a data base of precomputed items, maintained by Nils-Peter Skoruppa.
	www.iper1.com /iper1-odp/scat/id/Science/Math/Number_Theory/Elliptic_Curves_and_Modular_Forms/Tables (478 words)

	Ernst Kani's homepage
		Endomorphisms of Jacobians of modular curves (revised) Aug. 2005, 12pp.
		The existence of curves of genus 2 with elliptic differentials, J.
		Curves of genus 2 with elliptic differentials and the Height Conjecture for elliptic curves.
	www.mast.queensu.ca /~kani (1562 words)

	ALGORITHMS FOR MODULAR ELLIPTIC CURVES: INTRODUCTION (Site not responding. Last check: 2007-11-07)
		First, we describe in detail an algorithm based on modular symbols for computing modular elliptic curves: that is, one-dimensional factors of the Jacobian of the modular curve $X_0(N)$, which are attached to certain cusp forms for the congruence subgroup $\Gamma_0(N)$.
		As with the modular symbol algorithms, we have rewritten all the elliptic curve algorithms in C++.
		Also concerning the order and naming of the curves: the convention we normally use is that in each isogeny class the first curve is the strong Weil curve whose period lattice is exactly that of the corresponding newform for $\Gamma_0(N)$, such as 11A1 for example.
	www.maths.nott.ac.uk /personal/jec/book/chapter1.html (3376 words)

	A Mathematical Lie
		Now, we have already seen that an elliptic curve as a complex torus is essentially determined by the period lattice of the [Weierstrass] p-function that parameterizes the curve.
		This fact was already clear in the 19th century - the moduli of elliptic curves are expressible in terms of modular forms of the parameter tau in the upper half plane.
		This is from "Topics in Elliptic Curves and Modular Forms," by J. William Hoffman, Sept. 28, 2001, p.
	log24.com /log03/1130.htm (1789 words)

	Knapp, A.W.: Elliptic Curves. (MN-40).
		An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group.
		Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties.
		Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match.
	pup.princeton.edu /titles/5272.html (272 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.
		When E is a curve of known rank r and C(E) is computed, then if C(E) is divided by all of the other factors which can be computed, what is left has been found to be the square of an integer in all cases.
		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)

	Steven Galbraith Thesis (Site not responding. Last check: 2007-11-07)
		The use of the canonical embedding to obtain equations for modular curves as in Chapter 2 (including some of the examples given in the thesis) is also described in:
		Josep Gonzalez, Equations of hyperelliptic modular curves, Ann.
		To obtain the modular form data for computations such as these I recommend consulting William Stein's tables.
	www.isg.rhul.ac.uk /~sdg/thesis.html (469 words)

	L-functions and elliptic curves
		The L-functions of elliptic curves and modular forms we are about to discuss provide one important source of examples.
		In the introduction to the discussion of elliptic curves, we mentioned that global questions can often be studied by looking at them locally, i.
		Now that the Taniyama-Shimura conjecture is known for "semistable" curves E, we know that such E have a parameterization by modular functions, so the Hasse-Weil conjecture holds for them also.
	www.mbay.net /~cgd/flt/flt06.htm (2077 words)

	KJPublications
		February 2000 ``Congruent Numbers, Elliptic Curves and Modular Forms'', given in the Department of Mathematical Sciences of Clemson University in Clemson, South Carolina.
		February 2000 ``Congruent Numbers, Elliptic Curves and Modular Forms'', given in the Mathematics Department of University of North Texas in Denton, Texas.
		March 2000 ``Congruent Numbers, Elliptic Curves and Modular Forms'', given in the Mathematics Department of University of Missouri in Columbia, Missouri.
	www.math.clemson.edu /~kevja/PROFESSIONAL/InvitedLectures.html (422 words)

	Software Science, Directory (Site not responding. Last check: 2007-11-07)
		Periods of Hilbert Modular Forms A package of PARI programs (v.2.1.1 or higher) for calculations described in "Periods of Hilbert modular forms and rational points on elliptic curves" by H. Darmon and A. Logan.
		Code to Compute Heights on Elliptic Curves Magma implementation of Silverman's algorithm to compute the canonical height on an elliptic curve over a number field or function field, by Martine Girard.
		Modular Forms Software HECKE can be used to compute basis of q-expansions and Hecke operators on fairly general spaces of modular forms.
	www.morrisarearedcross.org /bWFfNDA1MjM5.aspx (175 words)

	Equations For Modular Curves - Galbraith (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Abstract: The primary topic of this thesis is the construction of explicit projective equations for the modular curves X 0 (N).
		In particular, equations are given for all curves X 0 (N) having genus 2 g 5.
		Equations are also given for all X + 0 (p) having genus 2 or 3, and for the genus 4 and 5 curves X + 0 (p) when p 251.
	citeseer.ist.psu.edu /galbraith96equations.html (852 words)

	Matt Baker - Abstracts (Site not responding. Last check: 2007-11-07)
		We prove that for each g≥2, the set of new modular curves over Q of genus g is finite and computable.
		Similar finiteness results are proved for new modular curves of bounded gonality, for new modular curves whose jacobian is a quotient of J_0(N) new
		In particular, we find all new modular curves of genus 2 explicitly, and construct what might be the complete list of all new modular hyperelliptic curves of all genera.
	www.math.gatech.edu /~mbaker/FINITENESS.html (209 words)

	Amazon.com: Books: Algorithms for Modular Elliptic Curves Full Canadian Binding (Site not responding. Last check: 2007-11-07)
		Elliptic curves are of central and growing importance in computational number theory, with numerous applications in such areas as cryptography, primality testing and factorisation.
		First, the author describes in detail the construction of modular elliptic curves, giving an explicit algorithm for their computation using modular symbols.
		Secondly a collection of algorithms for the arithmetic of elliptic curves is presented; some of these have not appeared in book form before.
	www.amazon.com /exec/obidos/tg/detail/-/0521598206?v=glance (532 words)

	Finiteness Results for Modular Curves of Genus at Least 2 (ResearchIndex)
		Abstract: A curve X over Q is modular if it is dominated by X 1 (N) for some N ; if in addition the image of its jacobian in J 1 (N) is contained in the new subvariety of J 1 (N), then X is called a new modular curve.
		We prove that for each g 2, the set of new modular curves over Q of genus g is finite and computable.
		Similar finiteness results are proved for new modular curves of bounded gonality,...
	citeseer.ist.psu.edu /605644.html (296 words)

	Elliptic Curves and Cryptology
		Modular forms in one variable, Lecture Notes in Mathematics 476, Springer-Verlag, 1975.
		Modular forms and Fermat's Last Theorem, Springer-Verlag, 1997.
		This program uses 2-descent (via 2-isogeny if possible) to determine the rank of an elliptic curve E over Q, list a set of points which generate E(Q) modulo 2E(Q), and finally search for further points on the curve.
	www.geocities.com /CapeCanaveral/Launchpad/9160/biblio_ell.html (1215 words)

	Cyclotomic points on modular curves (Site not responding. Last check: 2007-11-07)
		For which primes p does there exist an elliptic curve E over
		When p=2,3,5 the corresponding moduli space has genus zero and infinitely many examples exist.
		Recent work of L. Merel, combined with computations he enlisted me to do, suggest that these are the only primes p for which such elliptic curves exist.
	modular.fas.harvard.edu /job/Prop/node12.html (176 words)

	Kolyvagin Seminar (Site not responding. Last check: 2007-11-07)
		Background: Introduction to modular curves as quotients of upper half plane, the j and j_N functions generate their field of functions, the modular equation, modular curves as moduli spaces, Weil parametrizations of elliptic curves, and the definition of Heegner points.
		Background: An introduction to the basics of modular curve and modular form theory, including modular forms parametrizing elliptic curves, the Hecke correspondence and the Eichler-Shimura congruence.
		Background: L functions of elliptic curves, their functional equations, and the sign of said functional equations.
	math.berkeley.edu /~osserman/semold (846 words)

	Seekmeup.Directory - Science: Math: Number Theory: Elliptic Curves and Modular Forms: Software (Site not responding. Last check: 2007-11-07)
		Code to Compute Heights on Elliptic Curves - Magma implementation of Silverman's algorithm to compute the canonical height on an elliptic curve over a number field or function field, by Martine Girard.
		Modular Forms Software - HECKE can be used to compute basis of q-expansions and Hecke operators on fairly general spaces of modular forms.
		Periods of Hilbert Modular Forms - A package of PARI programs (v.2.1.1 or higher) for calculations described in "Periods of Hilbert modular forms and rational points on elliptic curves" by H. Darmon and A. Logan.
	seekmeup.com /directory/Science/Math/Number_Theory/Elliptic_Curves_and_Modular_Forms/Software (688 words)

	On the Hasse-Witt invariants of modular curves, Pilar Bayer, Josep González
		We briefly discuss the relationship between several characterizations of the Hasse--Witt invariant of curves in characteristic $p$ with the goal of computing its value in concrete instances.
		We study its asymptotic behaviour when dealing with the geometric fibres of curves of genus $\geq 2$ defined over the rationals.
		Numerical evidence gathered for several modular curves supports certain conjectural distribution laws.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1047565284 (71 words)

	Elliptic Curves
		Koblitz, Introduction to elliptic curves and modular forms.
		Arithmétique des courbes elliptiques et théorie d'Iwasawa by B. Perrin-Riou (studies elliptic curves with CM using Iwasawa theory).
		Connell's Handbook of elliptic curves is an ambitious project and still uncomplete.
	www.rzuser.uni-heidelberg.de /~hb3/elleng.html (928 words)

	Description (Site not responding. Last check: 2007-11-07)
		Several themes drawn from number theory of the last 35 years will be considered, including those relating to elliptic curves, modular forms and representations of Galois groups.
		Modular elliptic curves and Wiles' proof that sufficiently many elliptic curves are modular
		Prerequisites: For the first third of the semester only the usual first year graduate courses will be assumed.
	www.math.rutgers.edu /~tunnell/Description.html (290 words)

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