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Modular form - Wikipedia, the free encyclopedia In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. Modular form theory is a special case of the more general theory of automorphic forms, and therefore can now be seen as just the most concrete part of a rich theory of discrete groups. By "holomorphic at the cusp", it is meant that the modular form is holomorphic as en.wikipedia.org /wiki/Modular_forms   (1489 words)

Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-11-05) A modular form is defined by two axes, x and y, but EACH axis has a real and imaginary part. Modular forms come in various shapes and sizes, but each one is built from the same basic ingredients. The ingredients of a modular form are labelled from one to infinity (M1,M2,M3,....) and a particular modular form might contain one lot of ingredient one (M1=1), three lots of ingredient two (M2=3), two lots of ingredient three (M3=2) and so on. mathforum.org /library/drmath/view/51445.html   (888 words)

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)

Elliptic curves   (Site not responding. Last check: 2007-11-05) Galois representations and modular forms, by Kenneth A. Ribet. Modularity of a family of elliptic curves, by Fred Diamond, Kenneth Kramer. Modular Symbols and the computation of Modular Elliptic Curves, John Cremona. www.fermigier.com /fermigier/elliptic.html.en   (746 words)

Modular Forms (L24)   (Site not responding. Last check: 2007-11-05) Modular Forms are classical objects that appear in many areas of mathematics, from number theory to representation theory and mathematical physics. Miyake, Modular forms, Springer, Berlin, 1989 (a standard reference for classical theory of modular forms). Ogg, Modular forms and Dirichlet series, W. Benjamin, New York, 1969 (this used to be a standard introductory text). www.maths.cam.ac.uk /CASM/courses/02-03/descriptions/node33.html   (199 words)

Directory - Science: Math: Number Theory: Elliptic Curves and Modular Forms Modularity of Elliptic Curves and Beyond  · Workshop, MSRI Berkeley, 6-10 December 1999. Modular Forms Course  · cached · Notes of a 1996 Berkeley course of Ken Ribet's on modular forms and Hecke operators. Modular Forms and Hecke Operators  · cached · Notes by William A. Stein of a course by Ken Ribet. www.incywincy.com /default?p=200295   (696 words)

Concrete Forms - Over 100 Form Panel and Filler sizes available - SCI GLobal A strong, durable, modular concrete form system that is designed to withstand concrete loading pressure of 1,000 PSF @ 2:1 FS. Forms are easy to erect and dismantle by hand... In addition to the 24" width Form Panels, 30" and 36" Form Panels are also available....in heights to 10'-0". www.sciglobal.com /formwork/concreteforms1.html   (270 words)

Science Math Number Theory Elliptic Curves and Modular Forms   (Site not responding. Last check: 2007-11-05) 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. www.iper1.com /iper1-odp/scat/id/Science/Math/Number_Theory/Elliptic_Curves_and_Modular_Forms   (771 words)

Science Math Number Theory Elliptic Curves and Modular Forms Tables Deformations of Maass Forms - Tabulated by Stefan Lemurell. 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. Modular Forms Database - Tables computed by William Stein using HECKE, LiDIA, PARI and Magma. www.iper1.com /iper1-odp/scat/id/Science/Math/Number_Theory/Elliptic_Curves_and_Modular_Forms/Tables   (478 words)

Eknath Ghate's Papers   (Site not responding. Last check: 2007-11-05) Congruences between base-change and non-base-change Hilbert modular forms (Cohomology of arithmetic groups, Automorphic forms and $L$-functions, Narosa, 2001) is an expository account of a conjecture of Doi, Hida and Ishii. An introduction to congruences between modular forms (Current trends in number theory, Hindustan Book Agency (2002), 39-58) introduces some of the basic concepts in the theory of congruences between modular forms. On the local behaviour of ordinary modular Galois representations (Progress in Mathematics 224, Birkhauser-Verlag (2004), 105-124) we investigate a question of Greenberg concerning the spitting behaviour of the restriction to a decomposition group at a prime p of the p-adic Galois representation attached to a p-ordinary elliptic modular cusp form. www.math.tifr.res.in /~eghate/math.html   (1044 words)

NIH Modular Research Grant Applications Page The goals of the Modular Budget Research Grant Application initiative are the 1) redefinition of the Research Project Grant as an assistance mechanism, and 2) focusing applicants and peer reviewers on the scientific and technical merit of the proposed research by disengaging them from complex development, evaluation and negotiations of budgets. Applicants using the new 5/2001 version of the PHS 398 must use the modular budget instructions that begin on page 13 of the application instructions available at http://grants.nih.gov/grants/funding/phs398/phs398.html Modular Grants Announced by Dr. Varmus (01/21/1999) - NIH News Release - NIH Announces New Grant Application, Review & Award Procedures: Dr. Varmus approves modular grants for the next millennium. grants.nih.gov /grants/funding/modular/modular.htm   (407 words)

Modular forms This is a one-semester introduction to modular forms and their applications. The first half (or so) of the course will be devoted to basic material (the upper halfplane and the modular group, definition and first properties of modular forms, and Hecke and Atkin-Lehner theory); the second half will be an overview of different applications. The paper's not about modular forms, but instead is about automatic structures in groups, an interesting subject in its own right. www.math.umass.edu /~gunnells/S04/modular/modular.html   (1018 words)

Elliptic Curve Data Algorithms for modular elliptic curves, CUP 1992, second revised edition 1997. The modular degrees for conductors over 12000 were computed using Mark Watkins's program. A table of the degree of the modular parameterizations of each "optimal" or "strong Weil" curve. www.maths.nott.ac.uk /personal/jec/ftp/data/INDEX.html   (1196 words)

bibliography for automorphic and modular forms, L-functions, representations, and number theory   (Site not responding. Last check: 2007-11-05) Bibliography for automorphic and modular forms, L-functions, and representation theory [Garrett 1985] P.B. Garrett, `Integral representations of certain L-functions attached to 1,2, and 3 modular forms', preprint, University of Minnesota, 1985. [Garrett 1990] P.B. Garrett, Holomorphic Hilbert Modular Forms, Wadsworth-Brooks-Cole, 1990. www.math.umn.edu /~garrett/m/b/bib.html   (3638 words)

Amazon.ca: Books: Introductory Lectures on Siegel Modular Forms   (Site not responding. Last check: 2007-11-05) Roughly half a century ago C.L. Siegel discovered a new type of automorphic forms in several variables in connection with his famous work on the analytic theory of quadratic forms. Since then Siegel modular forms have been studied extensively because of their significance in both automorphic functions in several complex variables and number theory. The comprehensive theory of automorphic forms to subgroups of algebraic groups and the recent arithmetical theory of modular forms illustrate these two aspects in an illuminating manner. www.amazon.ca /exec/obidos/ASIN/0521350522   (364 words)

Function Field Modular Forms and Higher Derivations, by Yukiko Uchino, Takakazu Satoh   (Site not responding. Last check: 2007-11-05) In the study of modular forms, Jacobi forms, etc. over the complex numbers, differential operators occasionally play an important role. For function field modular forms whose values lie in a positive characteristic field, differential is not a powerful tool. As an application, we construct the Cohen bracket for function field modular forms. www.math.uiuc.edu /Algebraic-Number-Theory/0084   (125 words)

Modular Forms   (Site not responding. Last check: 2007-11-05) The second, which has a more geometric flavor, will give an introduction to the theory of Hilbert modular forms in two variables (i.e., over real quadratic fields), the geometry of Hilbert modular surfaces, and to Borcherds products and the Borcherds lifting. The third will give an introduction to Siegel modular forms (both scalar- and vector-valued) and present a beautiful application to the theory of curves of finite fields (Harder's conjecture). In each case the theory will be built up from scratch, not presupposing any previous knowledge of modular forms. www.mi.uib.no /~stromme/nordag/eid2004.html   (260 words)

Toric varieties and modular forms   (Site not responding. Last check: 2007-11-05) Toric varieties and modular forms, by Lev A. Borisov and Paul E. Gunnells Let N be a lattice, and let deg be a complex-valued piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on deg, the data (N, deg) determines a holomorphic modular form of weight r on congruence subgroup Gamma_1(l). www.math.uiuc.edu /Algebraic-Number-Theory/0207   (126 words)

Quadratic minima and modular forms, Barry Brent   (Site not responding. Last check: 2007-11-05) We give upper bounds on the size of the gap between the constant term and the next nonzero Fourier coefficient of an entire modular form of given weight for $\flop{L}{-.3}\!_0(2)$. Numerical evidence indicates that a sharper bound holds for the weights $h \equiv 2 \pmod 4$. Our data suggest that, for certain meromorphic modular forms and $p=2$, $3$, the $p$-order of the constant term is related to the base-$p$ expansion of the order of the pole at infinity. projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1047674207   (107 words)

Instructions and Form Files for PHS 398 Please see the Adobe Acrobat or PDF Forms section of the "Help Downloading Files" page for information on using the fillable PDF forms. Form Page 4, formatting of form fields in "% Effort on Project" column has been changed to allow insertion of asterisk. On Form Page 3, Research Grant Table of Contents, the ordering of items in the Research Plan has been revised (12/30/04). grants.nih.gov /grants/funding/phs398/phs398.html   (944 words)

Automorphic Forms and Representations I felt that there was a need for a book which would present the subject in a style which was accessible, yet based on complete proofs, revealing clearly the uniqueness principles which underlie the basic constructions. Since 1990 I have been lecturing on automorphic forms and representation theory at Stanford and the MSRI, and this book is the end result. Its aim is to cover a substantial portion of the theory of automorphic forms on GL(2). math.stanford.edu /~bump/book.html   (363 words)

Math 480a: Senior Seminar in Modular Forms   (Site not responding. Last check: 2007-11-05) Modular forms are mathematical objects which are emerging across the discipline in exciting ways. And recently physicists have begun studying modular forms in connection with string theory (and even fl holes!). There is a surprisingly simple answer to this question via modular forms. www.yale.edu /math480a   (221 words)

Continued Fractions and Modular Forms   (Site not responding. Last check: 2007-11-05) This incursion into the realm of elementary and probabilistic number theory of continued fractions, via modular forms, allows us to study the alternating sum of coefficients of a continued fraction, thus solving the longstanding open problem of their limit law. To this aim, let us recall a few facts about modular forms and Kloosterman sums, since these objects appeared to be the key to the asymptotic analysis. A modular form of weight k is a holomorphic function on H satisfying: algo.inria.fr /seminars/sem99-00/vardi.html   (1676 words)

Open Questions: Elliptic Curves and Modular Forms It follows that all elliptic functions are of the form g(℘(z)) + ℘′(z)h(℘(z)), where g(t) and h(t) are quotients of polynomials in the indeterminate t. In this case, such subsets are called "orbits", because each equivalence class containing some element s consists of all elements of the form gs as g varies over the elements of G. Members of G then become a sort of label for the elements of any particular equivalence class. 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)

14H52: Elliptic Curves Through Riemann surfaces it has connections to topology; through modular forms and zeta functions to analysis. An example of transformation to normal form for an elliptic curve. Two (unstructured) equations equations in three unknowns lead to an elliptic curve (although integer points are not fully known). www.math.niu.edu /~rusin/known-math/index/14H52.html   (750 words)

Abstract for Lloyd Kilford's Thesis   (Site not responding. Last check: 2007-11-05) We establish a sufficient criterion for these slopes to be given by a simple formula, and prove this criterion in several cases. These results on overconvergent modular forms also imply similar results for certain spaces of classical modular forms, which also allows us to prove results about the field over which the Fourier expansions of the normalised classical cuspidal modular eigenforms are defined. The second part of this thesis considers the Hecke algebras attached to certain spaces of classical cuspidal modular forms of prime level. www.its.caltech.edu /~ljpk/abstract/abstract.html   (306 words)

Amazon.com: Books: Modular Forms and Fermat's Last Theorem   (Site not responding. Last check: 2007-11-05) Introduction to Elliptic Curves and Modular Forms (Graduate Texts in Mathematics) by Neal I. Koblitz Representation theory, which lies at the core of Wiles' proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serre's conjectures, Galois deformations, universal deformation rings, Hecke algebras, complete intersections and more, as the reader is led step-by-step through Wiles' proof. Suffice it to say then that this book is excellent, and even a reader interested solely in elliptic curves and modular forms could benefit greatly from the reading of this book. www.amazon.com /exec/obidos/tg/detail/-/0387946098?v=glance   (902 words)

phi_l Here are the coefficients for the classical modular polynomials phi_l, for all prime l 2) "binary compressed" files containing the coefficients in binary form and also exploiting the fact that the coefficients are highly divisable by powers of 2,3,5. The advantage in using this over the decimal ASCII files is that it is computationally intensive to read the files in decimal and convert them into the internal form of most number theory packages, which is binary. www.math.uwaterloo.ca /~mrubinst/modularpolynomials/phi_l.html   (173 words)

Citebase - The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras The Igusa modular forms and ``the simplest'' Lorentzian Kac--Moody algebras With the help of generalized modular forms (such as Siegel and Jacobi forms), we compute the perturbative prepoten... We give variants of lifting construction, which define new classes of modular forms on the Siegel upper half-space of complex dimension 3 with respect to the full paramodular groups (defining moduli of Abelian surfaces with arbitrary polarization). citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:alg-geom/9603010   (1461 words)

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