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Serre's Modularity Conjecture
In mathematics, Serre's modularity conjecture, introduced by , states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form. The conjecture in the level 1 case was proved by Chandrashekhar Khare in 2005, and a proof of the full conjecture was completed jointly by Khare and Jean-Pierre Wintenberger in 2008. Formulation The conjecture concerns the absolute Galois group G_\mathbb of the rational number field \mathbb. Let \rho be an absolutely irreducible, continuous, two-dimensional representation of G_\mathbb over a finite field F = \mathbb_. : \rho \colon G_\mathbb \rightarrow \mathrm_2(F). Additionally, assume \rho is odd, meaning the image of complex conjugation has determinant -1. To any normalized modular eigenform : f = q+a_2q^2+a_3q^3+\cdots of level of a modular form, level N=N(\rho) , weight of a modular form, weight k= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and Algebraic function field, function fields. These properties, such as whether a ring (mathematics), ring admits unique factorization, the behavior of ideal (ring theory), ideals, and the Galois groups of field (mathematics), fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weight Of A Modular Form
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group \mathrm_2(\mathbb Z) \subset \mathrm_2(\mathbb R). Every modular form is attached to a Galois representation. The term "modular form", as a systematic description, is usually attributed to Erich Hecke. The importance of modular forms across multiple field of mathematics has been humorously represent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Forms
In mathematics, a modular form is a holomorphic function on the Upper half-plane#Complex plane, complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the Group action (mathematics), group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group \mathrm_2(\mathbb Z) \subset \mathrm_2(\mathbb R). Every modular form is attached to a Galois representation. The term "modular form", as a systematic description, is usually attributed to Erich Hecke. The importance of modular forms across m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ken Ribet
Kenneth Alan Ribet (; born June 28, 1948) is an American mathematician working in algebraic number theory and algebraic geometry. He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Fermat's Last Theorem, as well as for his service as President of the American Mathematical Society from 2017 to 2019. He is currently a professor of mathematics at the University of California, Berkeley. Early life and education Kenneth Ribet was born in Brooklyn, New York to parents David Ribet and Pearl Ribet, both Jewish, on June 28, 1948. As a student at Far Rockaway High School, Ribet was on a competitive mathematics team, but his first field of study was chemistry. Ribet earned his bachelor's degree and master's degree from Brown University in 1969. In 1973, Ribet received his Ph.D. from Harvard University under the supervision of John Tate. Career After receiving his doctoral degree, Ribet taught at Princeton University for three ye ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiles's Proof Of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Sir Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to prove using previous knowledge by almost all living mathematicians at the time. Wiles first announced his proof on 23 June 1993 at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations". However, in September 1993 the proof was found to contain an error. One year later on 19 September 1994, in what he would call "the most important moment of isworking life", Wiles stumbled upon a revelation that allowed him to correct the proof to the satisfaction of the mathematical community. The corrected proof was published in 1995. Wiles's proof uses many techniques from algebraic geometry and number theory and has many ramificatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luis Dieulefait
Luis is a given name. It is the Spanish form of the originally Germanic name or . Other Iberian Romance languages have comparable forms: (with an accent mark on the i) in Portuguese and Galician, in Aragonese and Catalan, while is archaic in Portugal, but common in Brazil. Origins The Germanic name (and its variants) is usually said to be composed of the words for "fame" () and "warrior" () and hence may be translated to ''famous warrior'' or "famous in battle". According to Dutch onomatologists however, it is more likely that the first stem was , meaning fame, which would give the meaning 'warrior for the gods' (or: 'warrior who captured stability') for the full name.J. van der Schaar, ''Woordenboek van voornamen'' (Prisma Voornamenboek), 4e druk 1990; see also thLodewijs in the Dutch given names database Modern forms of the name are the German name Ludwig and the Dutch form Lodewijk. and the other Iberian forms more closely resemble the French name Louis, a deriva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Conductor
In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by as an expression appearing in the functional equation of an Artin L-function. Local Artin conductors Suppose that ''L'' is a finite Galois extension of the local field ''K'', with Galois group ''G''. If \chi is a character of ''G'', then the Artin conductor of \chi is the number :f(\chi)=\sum_\frac(\chi(1)-\chi(G_i)) where ''G''''i'' is the ''i''-th ramification group (in lower numbering), of order ''g''''i'', and χ(''G''''i'') is the average value of \chi on ''G''''i''.Serre (1967) p.158 By a result of Artin, the local conductor is an integer.Serre (1967) p.159 Heuristically, the Artin conductor measures how far the action of the higher ramification groups is from being trivial. In particular, if χ is unramified, then its Artin conductor is zero. Thus if ''L'' is unramified over ''K'', then the Artin conductors of all χ are zero. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modularity Theorem
In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001. Before that, the statement was known as the Taniyama–Shimura conjecture, Taniyama–Shimura–Weil conjecture, or the modularity conjecture for elliptic curves. Statement The theorem states that any elliptic curve over \Q can be obtained via a rational map with integer coefficients from the classical modular curve for some integer ; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases and have been known since antiquity to have infinitely many solutions.Singh, pp. 18–20 The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of ''Arithmetica''. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently, the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, Wiles's proof of Fermat's Last Theorem, the first success ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also ''is prime to'' or ''is coprime with'' . The numbers 8 and 9 are coprime, despite the fact that neither—considered individually—is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing When the integers and are coprime, the standard way of expressing this fact in mathematical notation is to indicate that their greatest common divisor is one, by the formula or . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed an alte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
