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# Topic: Mordell conjecture

###### In the News (Sat 18 May 13)

 Mordell Mordell had to earn the money for his passage to England, and this he did, with some help from his parents, mainly by tutoring his fellow pupils for seven hours a day to earn enough to pay for his passage. Mordell was awarded the second Smith's Prize with his essay, and he went on to publish a long paper on this equation, now sometimes called Mordell's equation, in the Proceedings of the London Mathematical Society. Mordell submitted his subsequent work on indeterminate equations of the third and fourth degree when he became a candidate for a Fellowship at St John's College, but he was not successful. www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Mordell.html   (2196 words)

 Station Information - Mordell conjecture The Mordell conjecture states a basic result regarding the rational number solutions to Diophantine equations. It was eventually proved by Gerd Faltings in 1983, about six decades after the conjecture was made. Suppose we are given an algebraic curve C defined over the rational numbers (that is, C is defined by polynomials with rational coefficients), and suppose further that C is non-singular (though in this case that condition isn't a real restriction). www.stationinformation.com /encyclopedia/m/mo/mordell_conjecture.html   (260 words)

 conjecture In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has been able to prove or disprove. When a conjecture has been proven to be true, it becomes known as a theorem, and joins the realm of mathematical facts. Although many of the most famous conjectures have been tested across an astounding range of numbers, this is no guarantee against a single counterexample, which would immediately disprove the conjecture. www.fact-library.com /conjecture.html   (469 words)

 Mordell conjecture: Definition and Links by Encyclopedian.com - All about Mordell conjecture   (Site not responding. Last check: 2007-11-07) The conjecture, eventually proved by Gerd Faltings[?] after about six decades, states a basic result on rational number solutions to Diophantine equations[?]. From the point of view of number theory, the classification of algebraic curves[?] that matters is into three classes, according to their genus g. Therefore the conjecture took its natural place in the overall picture. www.encyclopedian.com /mo/Mordell-conjecture.html   (210 words)

 Timeline of Fermat's Last Theorem Mordell discovered the connection between the solutions of algebraic equations and topology. Shimura declared his conjecture that an elliptic curve should always be uniformized by a modular curve, but André Weil did not believe Shimura conjecture. Weil had written about the conjecture, modular elliptic curves became know as "Weil curves." After Taniyama's problems became known in the West, the conjecture came to be called erroneously the "Taniyama-Weil" conjecture, and Shimura's name was left out. www.public.iastate.edu /~kchoi/time.htm   (2119 words)

 Mordell-Weil theorem article - Mordell-Weil theorem mathematics abelian variety number field finitely-generated - ...   (Site not responding. Last check: 2007-11-07) In mathematics, the Mordell-Weil theorem states that for an abelian variety A over a number field K, the group A(K) of K-rational points of A is a finitely-generated abelian group. The case with A an elliptic curve E and K the rational number field Q is Mordell's theorem, answering a question apparently posed by Poincaré around 1908; it was proved by Louis Mordell in 1922. The process of infinite descent of Fermat was well known, but Mordell succeeded with in establishing a result on the quotient group www.what-means.com /encyclopedia/Mordell-Weil_theorem   (436 words)

 The Conjecture Of Tate And Voloch On p-Adic Proximity To Torsion - Scanlon (ResearchIndex)   (Site not responding. Last check: 2007-11-07) The Conjecture Of Tate And Voloch On p-Adic Proximity To Torsion (1999) Tate and Voloch have conjectured that the p-adic distance from torsion points of semi-abelian varieties over C p to subvarieties may be uniformly bounded. Conjecture 7.3 is true under the hypothesis that G is de ned over Q alg p. citeseer.lcs.mit.edu /scanlon99conjecture.html   (356 words)

 AWS 1998: Contributed Abstracts   (Site not responding. Last check: 2007-11-07) Though this result is hardly related to the $abc$ conjecture, its proof utilizes the distribution of roots of these polynomials in the usual and $p-$adic complex numbers for $pabc$ and has some flavor of the $abc$ conjecture. The ABC conjecture implies that the equation ax^y+by^x=cz^n has finitely many integer solutions (x,y,z,n) where x,y, and n are greater than 1 and g.c.d.(x,y)=1. A well-known conjecture of Ankeny-Artin-Chowla states that if $L$ is a real quadratic number field with prime discriminant $p$, then the conductor of the fundamental unit of $L$ is not divisible by $p$. math.arizona.edu /~swc/notes/files/98ContribA.html   (455 words)

 Fermat's last theorem   (Site not responding. Last check: 2007-11-07) Using sophisticated tools from algebraic geometry (in particular elliptic curves and modular forms), Galois theory and Hecke algebras, the English mathematician Andrew Wiles, from Princeton University, with help from his former student Richard Taylor, devised a proof of Fermat's last theorem that was published in 1995 in the journal Annals of Mathematics. This latter conjecture proposes a deep connection between elliptic curves and modular forms. Wiles and Taylor were able to establish a special case of the Taniyama-Shimura conjecture sufficient to exclude such counterexamples arising from Fermat's last theorem. www.sciencedaily.com /encyclopedia/fermat_s_last_theorem   (759 words)

 The abc conjecture   (Site not responding. Last check: 2007-11-07) Assuming the Birch and Swinnerton-Dyer conjecture, it is shown in [Go-Sz] that this conjecture is equivalent to the Szpiro conjecture for modular elliptic curves. [Wa2] Walsh, P.G. On a conjecture of Schinzel and Tijdeman. The Wieferich criterion, the ABC conjecture and Shimura's correspondence, Satya Mohit, M.Sc. www.math.unicaen.fr /~nitaj/abc.html   (4284 words)

 Encyclopedia: Mordell   (Site not responding. Last check: 2007-11-07) Louis Joel Mordell (28 January 1888 - 12 March 1972) was a British mathematician, known for pioneering research in number theory. He came in 1906 to Cambridge to take the scholarship examination for entrance to St John's College, and was successful in gaining a place and support. His basic work on Mordell's theorem is from 1921/2, as is the formulation of the Mordell conjecture. www.nationmaster.com /encyclopedia/Mordell   (410 words)

 Mordell conjecture   (Site not responding. Last check: 2007-11-07) In number theory, the Mordell conjecture states a basicresult regarding the rational number solutions to Diophantineequations. Suppose we are given an algebraic curve C defined overthe rational numbers (that is, C is defined by polynomials with rational coefficients), and suppose further thatC is non-singular (though in this case that condition isn't a realrestriction). Case g = 1: no points, or C is an elliptic curve with a finite number of rational points forming an abelian group of quiterestricted structure, or an infinite number of points forming a finitely generated abelian group (Mordell's Theorem, the initial result of the Mordell-Weil theorem). www.therfcc.org /mordell-conjecture-211185.html   (260 words)

 Torsion in Rank 1 Drinfeld Modules and the Uniform Boundedness Conjecture, by Bjorn Poonen   (Site not responding. Last check: 2007-11-07) Torsion in Rank 1 Drinfeld Modules and the Uniform Boundedness Conjecture, by Bjorn Poonen We present three methods for proving these cases of the conjecture, and explain why they fail to prove the conjecture in general. Finally, an application of the Mordell conjecture for characteristic p function fields proves the uniform boundedness for the P-primary part of the torsion for rank 2 Drinfeld Fq[T]-modules over a fixed function field. www.math.uiuc.edu /Algebraic-Number-Theory/0020   (113 words)

 Ernest Schimmerling   (Site not responding. Last check: 2007-11-07) Therefore, the Mordell-Lang conjecture has stood fast as a testament to the applicability of Model Theory to other fields. The conjecture will be traced back to its diophantine roots, and then abstracted to its modern statement. In the process I hope to an idea of the motivation behind the conjecture, and its significance in number theory and algebraic geometry. www.math.cmu.edu /users/eschimme/seminar/baginski1.html   (110 words)

 Erdos conjecture - Encyclopedia, History, Geography and Biography   (Site not responding. Last check: 2007-11-07) The title given to this article lacks diacritics because of certain technical limitations. The prolific mathematician Paul Erdős and his various collaborators made many famous mathematical conjectures, over a wide field of subjects. This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. www.arikah.net /encyclopedia/Erdos_conjecture   (120 words)

 Introduction   (Site not responding. Last check: 2007-11-07) Recently, there have been some major developments, first Faltings proved the Mordell conjecture, then Vojta gave another proof and this latter method was extended by Faltings [F3,4] to prove two outstanding conjectures of Lang. However, the direct transposition of Vojta's conjectures to the case of positive characteristics is false. As mentioned above, the Mordell conjecture was proved by Samuel ([Sa1,2]), extending Grauert's proof for function fields of characteristic zero to characteristic p. www.ma.utexas.edu /users/voloch/surveylatex/node1.html   (726 words)

 ABC Conjecture   (Site not responding. Last check: 2007-11-07) On the conjecture, II, C. Stewart, Kunrui Yu... Relaxations of the ABC conjecture using integer k'th roots... On the height conjecture for algebraic points on curves... www.scienceoxygen.com /math/751.html   (139 words)

 Mordell-Lang plus Bogomolov, by Bjorn Poonen   (Site not responding. Last check: 2007-11-07) We formulate a conjecture for semiabelian varieties A over number fields that includes both the Mordell-Lang conjecture (now proven) and the Bogomolov conjecture. We prove the "Mordellic" (finitely generated) part of the conjecture when A is isogenous to the product of an abelian variety and a torus. The proof makes use of the Mordell-Lang conjecture, the Bogomolov conjecture, and an equidistribution theorem. www.math.uiuc.edu /Algebraic-Number-Theory/0127   (90 words)

 Richard Pink: Recent Preprints The isogeny conjecture for M says that there are only finitely many isomorphism classes of t-motives M' over K, for which there exists a separable isogeny M' → M of degree not divisible by ℘ Abstract: A theorem of Green, Lazarsfeld and Simpson (formerly a conjecture of Beauville and Catanese) states that certain naturally defined subvarieties of the Picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points. This result is then used to prove the Manin-Mumford conjecture in arbitrary characteristic and in full generality. www.math.ethz.ch /~pink/publications.html   (2372 words)

 FreisslerSoft Books Conjecture A Proof of the Q-MacDonald-Morris Conjecture for Bcn (Memoirs of the American Mathematical Society, 516) Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture Polynomial Automorphisms and the Jacobian Conjecture (Progress in Mathematics (Boston, Mass.), Vol. www.freisslersoft.com /co/Book_Conjecture.html   (478 words)

 Model Theory and Algebraic Geometry. An introduction to E. Hrushovski's proof of the geometric Mordell-Lang conjecture ... An introduction to E. Hrushovski's proof of the geometric Mordell-Lang conjecture (Lecture Notes in Mathematics Vol. The authors have collaborated closely to achieve a coherent and self- contained presentation, whereby the completeness of exposition of the chapters varies according to the existence of other good references, but comments and examples are always provided to give the reader some intuitive understanding of the subject. Synopsis This is an introduction to developments in the applications of model theory to algebraic geometry, illustrated by E. Hrushovski's model theoretic proof of the geometric Mordell-Lang conjecture. www.uni-protokolle.de /buecher/isbn/3540648631   (279 words)

 Diophantine Geometry : An Introduction (Graduate Texts in Mathematics) by Marc Hindry [ISBN: 0387989757] - Find Cheap ... This is an introduction to diophantine geometry at the advanced graduate level. The case of elliptic curves and their torsion subgroups is presented as a theorem, but the proof is not given unfortunately. It is intriguing in all of this discussion on the role elliptic curves have furnished as a testing ground for most of the conjectures and results. www.gettextbooks.com /isbn_0387989757.html   (883 words)

 Open Questions: Algebraic Geometry An outline (literally) of some of the conjectures and open problems in this area of arithmetic algebraic geometry. Faltings' proof of Mordell's conjecture is a key result, which deals with important concepts of algebraic geometry, such as abelian varieties and jacobians. Gerd Faltings's 1983 proof of the Mordell conjecture is one of the most important accomplishments in the field of artithmetic algebraic geometry. www.openquestions.com /oq-ma009.htm   (245 words)

 The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics) (Joseph H. Silverman) But now back to this book written in 1986, the most importanr result of chapter 9 is Siegel 's theorem: finiteness of integral points on hyperelliptic curves, with application to the establishment of the Shafarevich conjecture of elliptic curves: finiteness of isomorphism class of elliptic curves with good reduction outside finite set of primes. (Note: the general Shafarevich conjecture lies at the heart of Faltings' original proof of the Mordell conjecture!). While Chapter 10 is an introduction to the Galois cohomology methos of calculating the weak Mordell -Weil group. www.truefresco.com /bookshop/us/product/0387962034.htm   (1342 words)

 Graduate Seminar Abstract 10/12/01   (Site not responding. Last check: 2007-11-07) Abstract: Once the Mordell Conjecture has been proved, it is natural to ask about polynomial equations in more variables. Serge Lang made a conjecture, generalizing the Mordell conjecture, which is still open. I shall give some idea of what this conjecture says, and discuss the Vojta conjectures - which link these questions with Nevanlinna theory - an area in complex analysis. www.math.uic.edu /~marker/gss/ABS101201.html   (111 words)

 Bjorn Poonen   (Site not responding. Last check: 2007-11-07) Let $X$ be a subvariety of an abelian variety $A$ over a number field $k$. The Mordell-Lang conjecture is concerned with the intersection of $X(\overline{k})$ with the division group of a finitely generated subgroup of $A(\overline{k})$. The (generalized) Bogomolov conjecture is concerned with the set of points in $X(\overline{k})$ of small canonical height. math.berkeley.edu /~coleman/Sems/NTS/Sem-Fall98/poonen.html   (78 words)

 References Buium, J. Voloch, Mordell's conjecture in characteristic p: an explicit bound, Compositio Math., to appear. Grauert, Mordells Vermutung über Punkte auf algebraischen Kurven und Funktionenkörper. Voloch, On the conjectures of Mordell and Lang in positive characteristic, Invent. www.ma.utexas.edu /users/voloch/surveylatex/node65.html   (575 words)

 The Absolute Mordell-Lang Conjecture in Positive Characteristic - Scanlon (ResearchIndex)   (Site not responding. Last check: 2007-11-07) Introduction A version of the Mordell-Lang Conjecture in characteristic zero asserts that if G is a semi-abelian variety, # # G is a finitely generated group, and X # G is a subvariety, then the Zariski closure of X # # is a finite union of cosets of algebraic groups. Under the assumption of the adic four exponentials conjecture we give a precise criterion for the existence of modular di erence... 1 Voloch Toward a proof of the Lang conjecture in characterist.. citeseer.csail.mit.edu /425884.html   (290 words)

 Richard Pink: Recent Preprints The isogeny conjecture for M says that there are only finitely many isomorphism classes of t-motives M' over K, for which there exists a separable isogeny M' ---> M of degree not divisible by p For the t-motive associated to a Drinfeld module this was proved by Taguchi. Previously, it had been known only for the group of torsion points of order prime to the characteristic of K. The proofs involve only algebraic geometry, though scheme theory and some arithmetic arguments cannot be avoided. www.math.ethz.ch /~pink/preprints.html   (1657 words)

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