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Martin Eichler
Martin Maximilian Emil Eichler (29 March 1912 – 7 October 1992) was a German number theorist. Eichler received his Ph.D. from the Martin Luther University of Halle-Wittenberg in 1936. Eichler and Goro Shimura developed a method to construct elliptic curves from certain modular forms. The converse notion that every elliptic curve has a corresponding modular form would later be the key to the proof of Fermat's Last Theorem. Selected publications * ''Quadratische Formen und orthogonale Gruppen'', Springer 1952, 1974 * * ''Einführung in die Theorie der algebraischen Zahlen und Funktionen'', Birkhäuser 1963; Eng. trans. 1966''Introduction to the theory of algebraic numbers and functions'' in which a section on modular forms is added; pbk 2014 reprint of 1963 German original * ''Projective varieties and modular forms'' 1971 ( Riemann–Roch theorem); * with Don Zagier: ''The Theory of Jacobi forms'', Birkhäuser 1985; ''Über die Einheiten der Divisionsalgebren'', Ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Germany
Germany,, officially the Federal Republic of Germany, is a country in Central Europe. It is the second most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated between the Baltic and North seas to the north, and the Alps to the south; it covers an area of , with a population of almost 84 million within its 16 constituent states. Germany borders Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, and France, Luxembourg, Belgium, and the Netherlands to the west. The nation's capital and most populous city is Berlin and its financial centre is Frankfurt; the largest urban area is the Ruhr. Various Germanic tribes have inhabited the northern parts of modern Germany since classical antiquity. A region named Germania was documented before AD 100. In 962, the Kingdom of Germany formed the bulk of the Holy Roman Empire. During the 16th ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eichler–Shimura Congruence Relation
In number theory, the Eichler–Shimura congruence relation expresses the local ''L''-function of a modular curve at a prime ''p'' in terms of the eigenvalues of Hecke operators. It was introduced by and generalized by . Roughly speaking, it says that the correspondence on the modular curve inducing the Hecke operator ''T''''p'' is congruent mod ''p'' to the sum of the Frobenius map ''Frob'' and its transpose ''Ver''. In other words, :''T''''p'' = ''Frob'' + ''Ver'' as endomorphisms of the Jacobian ''J''0(''N'')F''p'' of the modular curve ''X''0''N'' over the finite field F''p''. The Eichler–Shimura congruence relation and its generalizations to Shimura varieties play a pivotal role in the Langlands program, by identifying a part of the Hasse–Weil zeta function of a modular curve or a more general modular variety, with the product of Mellin transforms of weight 2 modular forms or a product of analogous automorphic ''L''-functions. References * * * * Goro Shimura ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1992 Deaths
Year 199 ( CXCIX) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was sometimes known as year 952 '' Ab urbe condita''. The denomination 199 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Roman Empire * Mesopotamia is partitioned into two Roman provinces divided by the Euphrates, Mesopotamia and Osroene. * Emperor Septimius Severus lays siege to the city-state Hatra in Central-Mesopotamia, but fails to capture the city despite breaching the walls. * Two new legions, I Parthica and III Parthica, are formed as a permanent garrison. China * Battle of Yijing: Chinese warlord Yuan Shao defeats Gongsun Zan. Korea * Geodeung succeeds Suro of Geumgwan Gaya, as king of the Korean kingdom of Gaya (traditional date). By topic Religion * Pope Zephyrinus succeeds Pope Victor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1912 Births
Year 191 ( CXCI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Apronianus and Bradua (or, less frequently, year 944 '' Ab urbe condita''). The denomination 191 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Parthia * King Vologases IV of Parthia dies after a 44-year reign, and is succeeded by his son Vologases V. China * A coalition of Chinese warlords from the east of Hangu Pass launches a punitive campaign against the warlord Dong Zhuo, who seized control of the central government in 189, and held the figurehead Emperor Xian hostage. After suffering some defeats against the coalition forces, Dong Zhuo forcefully relocates the imperial capital from Luoyang to Chang'an. Before leaving, Dong Zhuo orders his troops to loot the tombs of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Arithmetica
''Acta Arithmetica'' is a scientific journal of mathematics publishing papers on number theory. It was established in 1935 by Salomon Lubelski and Arnold Walfisz. The journal is published by the Institute of Mathematics of the Polish Academy of Sciences The Institute of Mathematics of the Polish Academy of Sciences is a research institute of the Polish Academy of Sciences.Online archives (Library of Science, Issues: 1935–2000) 1935 establishments in Poland [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existence
Existence is the ability of an entity to interact with reality. In philosophy, it refers to the ontological property of being. Etymology The term ''existence'' comes from Old French ''existence'', from Medieval Latin ''existentia/exsistentia'', from Latin ''existere'', to come forth, be manifest, ''ex + sistere'', to stand. Context in philosophy Materialism holds that the only things that exist are matter and energy, that all things are composed of material, that all actions require energy, and that all phenomena (including consciousness) are the result of the interaction of matter. Dialectical materialism does not make a distinction between being and existence, and defines it as the objective reality of various forms of matter. Idealism holds that the only things that exist are thoughts and ideas, while the material world is secondary. In idealism, existence is sometimes contrasted with transcendence, the ability to go beyond the limits of existence. As a form of epi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eichler Order
In mathematics, an Eichler order, named after Martin Eichler, is an order of a quaternion algebra In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ... that is the intersection of two maximal orders. References * * * {{Numtheory-stub Number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eichler Cohomology
Several people are named Eichler: * August W. Eichler (1839–1887), German botanist * Caroline Eichler (1808/9–1843), German inventor, first woman to be awarded a patent (for her leg prosthesis) * Eunice Eichler (1932–2017), New Zealand Salvation Army officer, nurse, midwife and open adoption advocate * Glenn Eichler (born ), American TV comedy writer * Jeremy Eichler (born 1974), American music critic * Joseph Eichler (1900–1974), American residential real estate developer * Martin Eichler (1912–1992), German mathematician * Ralph Eichler (born 1950), Canadian politician Eichler can also refer to: * Eichler system, an early system of plant taxonomy * an ''Eichler home'', a primarily California-based style of midcentury architecture created by Joseph Eichler Joseph Leopold Eichler (June 25, 1900 – July 1, 1974) was a 20th-century post-war American real estate developer known for developing distinctive residential subdivisions of Mid-century modern style tract ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eichler–Shimura Isomorphism
In mathematics, Eichler cohomology (also called parabolic cohomology or cuspidal cohomology) is a cohomology theory for Fuchsian groups, introduced by , that is a variation of group cohomology analogous to the image of the cohomology with compact support in the ordinary cohomology group. The Eichler–Shimura isomorphism, introduced by Eichler for complex cohomology and by for real cohomology, is an isomorphism between an Eichler cohomology group and a space of cusp forms. There are several variations of the Eichler–Shimura isomorphism, because one can use either real or complex coefficients, and can also use either Eichler cohomology or ordinary group cohomology as in . There is also a variation of the Eichler–Shimura isomorphisms using ''l''-adic cohomology instead of real cohomology, which relates the coefficients of cusp forms to eigenvalues of Frobenius acting on these groups. used this to reduce the Ramanujan conjecture to the Weil conjectures that he later proved. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan–Petersson Conjecture
In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients of the cusp form of weight :\Delta(z)= \sum_\tau(n)q^n=q\prod_\left (1-q^n \right)^ = q-24q^2+252q^3- 1472q^4 + 4830q^5-\cdots, where q=e^, satisfies :, \tau(p), \leq 2p^, when is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by , is a generalization to other modular forms or automorphic forms. Ramanujan L-function The Riemann zeta function and the Dirichlet L-function satisfy the Euler product, and due to their completely multiplicative property Are there L-functions other than the Riemann zeta function and the Dirichlet L-functions satisfying the above relations? Indeed, the L-functions of automorphic forms satisfy the Euler product (1) but they do not satisfy (2) because they do not have the completely multiplicative property. However, Ramanujan discovered that the L-function of the mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
