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Iwasawa Theory
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motives. Formulation Iwasawa worked with so-called \Z_p-extensions - infinite extensions of a number field F with Galois group \Gamma isomorphic to the additive group of p-adic integers for some prime ''p''. (These were called \Gamma-extensions in early papers.) Every closed subgroup of \Gamma is of the form \Gamma^, so by Galois theory, a \Z_p-extension F_\infty/F is the same thing as a tower of fields :F=F_0 \subset F_1 \subset F_2 \subset \cdots \subset F_\infty such that \operatorname(F_n/F)\cong \Z/p^n\Z. Iwasawa studied classical Galois modules over F_n ... [...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] |
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 thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Real Number Field
In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polynomial ''P'', all of the roots of ''P'' being real; or that the tensor product algebra of ''F'' with the real field, over Q, is isomorphic to a tensor power of R. For example, quadratic fields ''F'' of degree 2 over Q are either real (and then totally real), or complex, depending on whether the square root of a positive or negative number is adjoined to Q. In the case of cubic fields, a cubic integer polynomial ''P'' irreducible over Q will have at least one real root. If it has one real and two complex roots the corresponding cubic extension of Q defined by adjoining the real root will ''not'' be totally real, although it is a field of real numbers. The totally real number fields play a significant special role in algebraic number the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Main Conjecture Of Iwasawa Theory
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' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Prime
In number theory, a regular prime is a special kind of prime number, defined by Ernst Kummer in 1850 to prove certain cases of Fermat's Last Theorem. Regular primes may be defined via the divisibility of either class numbers or of Bernoulli numbers. The first few regular odd primes are: : 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, ... . History and motivation In 1850, Kummer proved that Fermat's Last Theorem is true for a prime exponent ''p'' if ''p'' is regular. This focused attention on the irregular primes. In 1852, Genocchi was able to prove that the first case of Fermat's Last Theorem is true for an exponent ''p'', if is not an irregular pair. Kummer improved this further in 1857 by showing that for the "first case" of Fermat's Last Theorem (see Sophie Germain's theorem) it is sufficient to establish that either or fails to be an irregular pair. Kumme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet L-function
In mathematics, a Dirichlet ''L''-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet ''L''-function and also denoted ''L''(''s'', ''χ''). These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that is non-zero at ''s'' = 1. Moreover, if ''χ'' is principal, then the corresponding Dirichlet ''L''-function has a simple pole at ''s'' = 1. Otherwise, the ''L''-function is entire. Euler product Since a Dirichlet character ''χ'' is completely multiplicative, its ''L''-function can also be written as an Euler product in the half-plane of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gain in simplicity may outweigh the loss from interpolation error and give better performance ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tomio Kubota
(6 December 1930 – 30 June 2020) was a Japanese mathematician working in number theory. His contributions include works on p-adic L functions and real-analytic automorphic forms. His work on p-adic L-functions, later recognised as an aspect of Iwasawa theory, was done jointly with Leopoldt. He extended the concept of metaplectic group, in a way significant for arithmetic applications. This opened a field for later research on associated Dirichlet series and automorphic forms, and was a major step in the solution of Kummer's conjecture.Jeff Hoffstein, ''Eisenstein Series and Theta Functions on the Metaplectic Group'', pp. 73-78, in Maruti Ram Murty, ''Theta Functions: From the Classical to the Modern'' (1993). Works *''On automorphic functions and the reciprocity law in a number field''. Kinokuniya, Tokyo 1969 *''Notes on analytic theory of numbers''. University of Chicago Press The University of Chicago Press is the largest and one of the oldest university presses in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic L-function
In mathematics, a ''p''-adic zeta function, or more generally a ''p''-adic ''L''-function, is a function analogous to the Riemann zeta function, or more general ''L''-functions, but whose domain and target are ''p-adic'' (where ''p'' is a prime number). For example, the domain could be the ''p''-adic integers Z''p'', a profinite ''p''-group, or a ''p''-adic family of Galois representations, and the image could be the ''p''-adic numbers Q''p'' or its algebraic closure. The source of a ''p''-adic ''L''-function tends to be one of two types. The first source—from which Tomio Kubota and Heinrich-Wolfgang Leopoldt gave the first construction of a ''p''-adic ''L''-function —is via the ''p''-adic interpolation of special values of ''L''-functions. For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a ''p''-adic ''L''-function, the ''p''-adic Riemann zeta function ζ''p''(''s''), whose values at negative odd integers are those o ... [...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 integers , , and satisfy the equation for any integer value of greater than 2. 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, the first successful proof was released in 1994 by Andrew Wiles and for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
