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Iwasawa Main Conjecture
In mathematics, the main conjecture of Iwasawa theory is a deep relationship between ''p''-adic ''L''-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by . The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields,, CM fields, elliptic curves, and so on. Motivation was partly motivated by an analogy with Weil's description of the zeta function of an algebraic curve over a finite field in terms of eigenvalues of the Frobenius endomorphism on its Jacobian variety. In this analogy, * The action of the Frobenius corresponds to the action of the group Γ. * The Jacobian of a curve corresponds to a module ''X'' over Γ defined in terms of ideal class groups. * The zeta function of a curve over a finite field corresponds to a ''p''-adic ''L''-func ... [...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 of algebraic number theory 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 Diophantin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Endomorphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism maps every element to its -th power. In certain contexts it is an automorphism, but this is not true in general. Definition Let be a commutative ring with prime characteristic (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism ''F'' is defined by :F(r) = r^p for all ''r'' in ''R''. It respects the multiplication of ''R'': :F(rs) = (rs)^p = r^ps^p = F(r)F(s), and is 1 as well. Moreover, it also respects the addition of . The expression can be expanded using the binomial theorem. Because is prime, it divides but not any for ; it therefore will divide the numerator, but not the denominator, of the explicit formula of the binomial coefficients :\frac, if . Ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Elliptic Curve
A modular elliptic curve is an elliptic curve ''E'' that admits a parametrisation ''X''0(''N'') → ''E'' by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular. History and significance In the 1950s and 1960s a connection between elliptic curves and modular forms was conjectured by the Japanese mathematician Goro Shimura based on ideas posed by Yutaka Taniyama. In the West it became well known through a 1967 paper by André Weil. With Weil giving conceptual evidence for it, it is sometimes called the Taniyama–Shimura–Weil conjecture. It states that every rational elliptic curve is modular. On a separate branch of development, in the late 1960s, Yves Hellegouarch came up with the idea of associating solutions (''a'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. A modular function is a function that is invariant with respect to the modular group, but without the condition that be Holomorphic function, holomorphic in the upper half-plane (among other requirements). Instead, modular functions are Meromorphic function, meromorphic (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function). Modular form theory is a special case of the more general theory of automorphic form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eric Urban
Eric Jean-Paul Urban is a professor of mathematics at Columbia University working in number theory and automorphic forms, particularly Iwasawa theory. Career Urban received his PhD in mathematics from Paris-Sud University in 1994 under the supervision of Jacques Tilouine. He is a professor of mathematics at Columbia University. Research Together with Christopher Skinner, Urban proved many cases of Iwasawa–Greenberg main conjectures for a large class of modular forms. As a consequence, for a modular elliptic curve over the rational numbers, they prove that the vanishing of the Hasse–Weil ''L''-function ''L''(''E'', ''s'') of ''E'' at ''s'' = 1 implies that the p-adic Selmer group of ''E'' is infinite. Combined with theorems of Gross- Zagier and Kolyvagin, this gave a conditional proof (on the Tate–Shafarevich conjecture) of the conjecture that ''E'' has infinitely many rational points if and only if ''L''(''E'', 1) = 0, a (weak) form of the Birch– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christopher Skinner
Christopher McLean Skinner (born June 4, 1972) is an American mathematician working in number theory and arithmetic aspects of the Langlands program. He specialises in algebraic number theory. Skinner was a Packard Foundation Fellow from 2001 to 2006, and was named an inaugural fellow of the American Mathematical Society in 2013. In 2015, he was named a Simons Investigator in Mathematics. He was an invited speaker at the International Congress of Mathematicians in Madrid in 2006. Career Skinner graduated from the University of Michigan in 1993. After completing his PhD with Andrew Wiles in 1997, he moved to the University of Michigan as an assistant professor. Since 2006, he has been a Professor of Mathematics at the Princeton University. In joint work with Andrew Wiles, Skinner proved modularity results for residually reducible Galois representations. Together with Eric Urban, he proved many cases of Iwasawa–Greenberg main conjectures for a large class of modular forms. As ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler System
In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper and the work of . Euler systems are named after Leonhard Euler because the factors relating different elements of an Euler system resemble the Euler factors of an Euler product. Euler systems can be used to construct annihilators of ideal class groups or Selmer groups, thus giving bounds on their orders, which in turn has led to deep theorems such as the finiteness of some Tate-Shafarevich groups. This led to Karl Rubin's new proof of the main conjecture of Iwasawa theory, considered simpler than the original proof due to Barry Mazur and Andrew Wiles. Definition Although there are several definitions of special sorts of Euler system, there seems to be no published definition of an Euler system that covers all known cases. But it is poss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thaine's Method
In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by . Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem , to prove that some Tate–Shafarevich groups are finite, and in the proof of Mihăilescu's theorem . Formulation Let p and q be distinct odd primes with q not dividing p-1. Let G^+ be the Galois group of F=\mathbb Q(\zeta_p^+) over \mathbb, let E be its group of units, let C be the subgroup of cyclotomic units, and let Cl^+ be its class group. If \theta\in\mathbb Z ^+/math> annihilates E/CE^q then it annihilates Cl^+/Cl^. References * See in particular Chapter 14 (pp. 91–94) for the use of Thaine's theorem to prove Mihăilescu's theorem Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Rubin
Karl Cooper Rubin (born January 27, 1956) is an American mathematician at University of California, Irvine as Thorp Professor of Mathematics. Between 1997 and 2006, he was a professor at Stanford, and before that worked at Ohio State University between 1987 and 1999. His research interest is in elliptic curves. He was the first mathematician (1986) to show that some elliptic curves over the rationals have finite Tate–Shafarevich groups. It is widely believed that these groups are always finite. Education and career Rubin graduated from Princeton University in 1976, and obtained his Ph.D. from Harvard in 1981. His thesis advisor was Andrew Wiles. He was a Putnam Fellow in 1974, and a Sloan Research Fellow in 1985. In 1988, Rubin received a National Science Foundation Presidential Young Investigator award, and in 1992 won the American Mathematical Society Cole Prize in number theory. In 2012 he became a fellow of the American Mathematical Society. Rubin's parents were mathematic ... [...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 years ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
