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Topic: Modular discriminant

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	PlanetMath: discriminant
		The discriminant of a given polynomial is a number, calculated from the coefficients of that polynomial, that vanishes if and only if that polynomial has one or more multiple roots.
		Proposition 1 The discriminant of a polynomial is the resultant of the polynomial and its first derivative.
		This is version 13 of discriminant, born on 2002-03-07, modified 2006-10-08.
	planetmath.org /encyclopedia/Discriminant.html (517 words)

	Weierstrass's elliptic functions - Wikipedia, the free encyclopedia
		This is studied in its own right, as a cusp form, in modular form theory (that is, as a function of the period lattice).
		The discriminant is a modular form of weight 12.
		That is, under the action of the modular group, it transforms as
	en.wikipedia.org /wiki/Weierstrass's_elliptic_functions (1249 words)

	NationMaster - Encyclopedia: Epsilon theorem (Site not responding. Last check: 2007-11-03)
		In number theory, Serre's epsilon conjecture stated a property of Galois representations associated with modular forms which was proven by Ken Ribet in the summer of 1986, in in a significant step towards the proof of Fermat's Last Theorem.
		In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as HΓ where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modular group of integral 2×2 matrices.
		In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group — equivalently for a Galois group G and a group ring R[G] of G with respect to some ring R, it is some R[G]-module M. In that general sense...
	www.nationmaster.com /encyclopedia/Epsilon-theorem (895 words)

	Modular form - TheBestLinks.com - Modular forms, Algebraic geometry, Complex analysis, Elliptic curve, ... (Site not responding. Last check: 2007-11-03)
		A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition.
		Modular functions can be thought of as functions on the moduli space of isomorphism classes of complex elliptic curves.
		The crucial conceptual link between modular forms and number theory are furnished by the theory of Hecke operators, which also gives the link between the theory of modular forms and representation theory.
	www.thebestlinks.com /Modular_forms.html (1243 words)

	PlanetMath: modular discriminant
		function (delta function or modular discriminant) is defined to be
		See Also: elliptic function, j-invariant, Weierstrass sigma function, discriminant, discriminant
		This is version 2 of modular discriminant, born on 2003-08-25, modified 2003-08-26.
	planetmath.org /encyclopedia/ModularDiscriminant.html (104 words)

	Elliptic and Modular Functions
		The discriminant of the elliptic curve corresponding to the complex lattice L_z spanned by 1 and z is given by Delta(z) = g_2(z)^3 - 27 g_3(z)^2 = (((2pi i)^6)/(1728)) (E_4(q)^3 - E_6(q)^2) If we take the normalization (Delta)(z) = (1 /((2pi i)^6)) Delta(z) the q-expansion has rational coefficients.
		For a binary quadratic form F = ax^2 + bxy + cy^2 with negative discriminant, this returns the elliptic j-invariant of F. This is the j-invariant of tau where tau = (- b + Sqrt(b^2 - 4ac)) / (2a).
		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.math.niu.edu /help/math/magmahelp/text554.html (1242 words)

	Cusp form (Site not responding. Last check: 2007-11-03)
		In number theory, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion
		Taking the quotient by the modular group, say, this limit corresponds to a cusp of a modular curve (in the sense of a point added for compactification).
		which represents (up to a normalising constant) the discriminant of the cubic on the RHS of the Weierstrass equation of an elliptic curve; and the 24-th power of the Dedekind eta function.
	www.worldhistory.com /wiki/C/Cusp-form.htm (439 words)

	[No title]
		An old conjecture holds that the number N_n(X) of degree n number fields with discriminant less than X is asymptotic to c_n X when X grows and n is fixed.
		There's a slight wrinkle: the weight 1 Hilbert modular form f attached to E[3] by Langlands-Tunnell might not be ordinary, and thus one cannot necessarily find a higher-weight modular lift of E[3] whose restriction to decomposition groups at 3 we can control.
		When E is a quadratic field of discriminant 5 or 8, we produce the desired pairs of Hilbert modular forms explicitly and show how they can be used to compute the group of Néron components of a Hilbert-Blumenthal abelian variety with real multiplication by E..dvi version.pdf version
	www.math.princeton.edu /~ellenber/papers.html (1955 words)

	Présentation du séminaire de cryptographie (Site not responding. Last check: 2007-11-03)
		The computations are done in MAGMA, using modular symbols and numerical analysis.
		Recently, we have computed such polynomials for the mod 13 representation associated to Delta, the discriminant modular form of weight 12.
		In that case, we work with the modular curve X_1(13), which is of genus 2, and the polynomials have degrees 14 and 168.
	www.math.univ-rennes1.fr /crypto/2004-05/bosma.html (89 words)

	Modular toolkit for Data Processing (MDP)
		Modular toolkit for Data Processing (MDP) is a Python data processing framework.
		Modular toolkit for Data Processing (MDP) is a data processing framework written in Python.
		From the user's perspective, MDP consists of a collection of trainable supervised and unsupervised algorithms or other data processing units (nodes) that can be combined into data processing flows.
	mdp-toolkit.sourceforge.net (837 words)

	CiteULike: A comparison of neural networks and linear scoring models in the credit environment (Site not responding. Last check: 2007-11-03)
		analysis consumer credit discriminant generic linear loans logistic network neural regression union
		The purpose of the present paper is to explore the ability of neural networks such as multilayer perceptrons and modular neural networks, and traditional techniques such as linear discriminant analysis and logistic regression, in building credit scoring models in the credit union environment.
		Although we found significant differences in the results for the three credit unions, our modular neural network could not accommodate these differences, indicating that more innovative architectures might be necessary for building effective generic models.
	www.citeulike.org /user/QFRMC/article/650155 (418 words)

	The Proof of Fermat's Last Theorem
		Specifically, it cannot be modular in the sense that there exists a modular form which gives rise to the same Galois representation.
		To show that E is modular, we have to show this representation is modular in a suitable sense.
		This is done mainly by working with the theory of representations as much as possible, without specific reference to the curve E. The proof uses a concept called "deformation", which suggests intuitively what goes on in the process of lifting.
	www.mbay.net /~cgd/flt/flt08.htm (1543 words)

	Glossary of terms for Fermat's Last Theorem
		A complete algebraic variety which is an algebraic curve that is essentially the quotient space of the upper half of the complex plane by the action of a subgroup of finite index of the modular group.
		An elliptic curve E for which there is a modular curve X of a certain kind and a surjective map X -> E. Such an elliptic curve is said to have a "parameterization by modular functions".
		There are equivalent definitions, the simplest of which is that there exists a modular form whose L-function is the same as that of E. The Taniyama-Shimura conjecture states that every elliptic curve is modular.
	cgd.best.vwh.net /home/flt/flt10.htm (2633 words)

	[No title]
		On the other hand, in the modular case, where p P ïGï, the transfer is never zero nor surjective, see Section 11.5 in [12].
		The discriminant D n is a sum of orbit sums of mono mials x E with partition l(E) = {0,.
		REMARK REMARK REMARK REMARK REMARK: Note that this description is valid in both, the nonmodular and the modular situation (except, of course, the calculation of the height: this is always maximal in the non modular case).
	hopf.math.purdue.edu /Neusel/bertin2.txt (2946 words)

	Elliptic and Modular Invariants (Site not responding. Last check: 2007-11-03)
		As such, it is possible to apply modular and elliptic functions to the form, interpreting this as an element of the upper half plane.
		Note that the lattice L is the half-integral lattice such that integral representations f(x, y) = n are in bijection with vectors (x, y) of norm n, which will be a rational number.
		For a binary quadratic form f = ax^2 + bxy + cy^2 with negative discriminant, return the j--invariant of f, equal to the j--invariant of tau = (- b + Sqrt(b^2 - 4ac))/2a.
	wwwmaths.anu.edu.au /research.groups/aat/htmlhelp/text735.htm (316 words)

	SVCL - Probabilistic Kernels
		The generative route to classifier design is, in many ways, more appealing than the discriminant one: it can take advantage of any prior knowledge about the structure of the classification problem (e.g.
		On the other hand, estimating densities is a harder problem than determining classification boundaries directly and discriminant methods have better generalization guarantees, which usually translate into better practical classification performance.
		This is achieved by making the kernels, used in discriminant learning, functions of probabilistic models for the class densities.
	www.svcl.ucsd.edu /projects/klk (664 words)

	A. Zaharescu (Site not responding. Last check: 2007-11-03)
		Abstract: The values of the famous j-invariant at quadratic irrationalities in the upper half plane are known as singular moduli and are of particular interest in number theory.
		Title: Quotients in \delta geometry and modular forms (cont.) Abstract: If X is an algebraic variety and R is an equivalence relation on X which is a Zariski dense countable union of subvarieties then the quotient X/R does not exist, in any reasonable sense, as an object of usual algebraic geometry.
		Mimicking the classical case, we describe the algebraic-geometric approach to Siegel modular forms (mod p), and a technique that simplifies the study considerably.
	www.math.uiuc.edu /~zaharesc/seminar.html (2066 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Our first contribution is an extension of Encarnacion's modular GCD algorithm to the case n>1 without converting to a single field extension.
		We have a complete implementation of the modular GCD algorithm using it.
		Our fourth contribution is a generalization of the reduced discriminant to the case n>1.
	www.math.fsu.edu /~aluffi/archive/paper155.abs.html (172 words)

	Citebase - Serre's conjecture over F_9 (Site not responding. Last check: 2007-11-03)
		Our main tool is an idea of Taylor, which reduces the problem to that of exhibiting points on a Hilbert modular surface which are defined over a solvable extension of Q, and which satisfy certain reduction properties.
		As a corollary, we show that Hilbert-Blumenthal abelian surfaces over Q with good ordinary reduction at 3 and 5 are modular.
		We prove that any abelian surface defined over \Q of GL -type having quaternionic multiplication and good reduction at 3 is modular.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0107147 (311 words)

	Open Questions: Elliptic Curves and Modular Forms
		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.
		Although the discriminant of a defining polynomial isn't an invariant of an elliptic curve, it is close.
		The most important fact about the minimal discriminant is that the primes which divide it are precisely the ones at which the curve has bad reduction.
	www.openquestions.com /oq-ma017.htm (18524 words)

	Discriminants of Hecke algebras (Site not responding. Last check: 2007-11-03)
		Abstract: Using an implementation of the modular symbols algorithm described in Cremona's book Algorithms for modular elliptic curves I computed, for each prime N between 2 and 577, an integer D
		The discriminant of T, denoted disc(T), is the product of the discriminants of the number fields E
		be the discriminant of the characteristic polynomial of T
	modular.fas.harvard.edu /Tables/discriminants/disc (353 words)

	Class Polynomials (Site not responding. Last check: 2007-11-03)
		Class polynomials are invariants of elliptic curves with complex multiplication by an imaginary quadratic order of discriminant D. As such the Hilbert class polynomials can be interpreted as defining a subscheme or divisor on the modular curve X(1) isomorphic to PP^1, while the Weber variants define a subscheme of a modular curve of higher level.
		Given a negative discriminant D congruent to 1 modulo 8, returns the Weber class polynomial, defined as the minimal polynomial of f(tau), where Z[tau] is an imaginary quadratic order of discriminant D and f is a particular normalized Weber function generating the same class field as j(tau).
		A root f(tau) of the Weber class polynomial is an integral unit generating the ring class field related to the corresponding root j(tau) of the Hilbert class polynomial by the expression
	www.math.niu.edu /help/math/magmahelp/text1046.html (306 words)

	Functions
		Modular polynomial greatest common divisor and cofactors.
		C <- MUPGCD(p,A,B) Modular univariate polynomial greatest common divisor.
		MUPEGC(p,A,B; C,U,V) Modular univariate polynomial extended greatest common divisor.
	www.mcs.drexel.edu /~krandick/saclib/node51.html (681 words)

	Algorithmic Number Theory: Tables and Links
		Elliptic curves of large rank and small conductor (arXiv preprint; joint work with Mark Watkins; to appear in the proceedings of ANTS-VI (2004)): Elliptic curves over Q of given rank r up to 11 of minimal conductor or discriminant known; these are new records for each r in [6,11].
		modular forms database, containing some of the same information and also modular forms of other levels and/or characters
		Class polynomials for Weber functions associated with CM elliptic curves of fundamental discriminant down to -422500
	www.math.harvard.edu /~elkies/compnt.html (842 words)

	A Modular GCD algorithm over Number Fields presented with Multiple Extensions (Site not responding. Last check: 2007-11-03)
		The topic of this paper is an algorithm for computing the gcd of two polynomials with coefficients in an algebraic number field.
		For the case Q[x] one can prove that by showing that the true gcd g, when reduced mod p, is a factor of both f1 mod p and f2 mod p, and hence g mod p divides the modular gcd (the gcd of f1 mod p and f2 mod p).
		In the previous version of this paper (FSU preprint 02-03), available as dvi, postscript file, or pdf file, we also give a generalization of the reduced discriminant, which is not used by our algorithm, but has other useful applications.
	www.math.fsu.edu /~hoeij/papers/comments/issac2002.html (326 words)

	Computer Algebra Group at SFU - Meetings, Colloquia (Site not responding. Last check: 2007-11-03)
		For this we generalize the reduced discriminant of Bradford (1989) to n>1.
		We have completed a Maple implementation of Browns' algorithm for the modular GCD algorithm for the multivariate case assuming it does anyway.
		In the talk, I will first show how the modular GCD algorithm works for Q[x] using rational reconstruction, and also without integer reconstruction, by working through one example.
	www.cecm.sfu.ca /CAG/abstracts/21012002.html (310 words)

	Science Math Number Theory Elliptic Curves and Modular Forms Software (Site not responding. Last check: 2007-11-03)
		Modular Forms Software - HECKE can be used to compute basis of q-expansions and Hecke operators on fairly general spaces of modular forms.
		Modular Quaternion Groups - Fundamental domains for Shimura curves, written in GAP.
		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.
	www.iper1.com /iper1-odp/scat/id/Science/Math/Number_Theory/Elliptic_Curves_and_Modular_Forms/Software (245 words)

	Sasa Radomirovic - Math 356 - Number Theory - Summer 2004
		Recall that Q, R, C, and the congruence classes of integers modulo a prime number are fields, while the integers are not.
		The condition that the discriminant is nonzero makes sure that the curve is "nonsingluar".
		This conjecture (and since 1999 a theorem) is known as the Taniyama-Shimura conjecture.
	www.math.rutgers.edu /~sasar/Math356?cd617.txt (987 words)

	The Number Theory Room
		The secondary goal is to provide a number-theoretic explanation for the structure of the interior of the Mandelbrot Set.
		The real part of the Modular Discriminant (Dedekind eta to the 24'th), as a function of the square of the nome q=exp (i pi tau).
		The imaginary part of the Modular Discriminant (Dedekind eta to the 24'th), as a function of the square of the nome q=exp (i pi tau).
	linas.org /art-gallery/numberetic/numberetic.html (1008 words)

	Ramanujan's Constant And Its Cousins
		The modular function involved in the first pair of examples is known as the j-function.
		However, these discriminants also happened to be multiples of 3.
		We pointed out earlier that there is a connection between the modular functions we have mentioned and what is called the Monster group.
	www.geocities.com /titus_piezas/Ramanujan_a.htm (2972 words)

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