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Hasse–Weil Zeta Function
In mathematics, the Hasse–Weil zeta function attached to an algebraic variety ''V'' defined over an algebraic number field ''K'' is a meromorphic function on the complex plane defined in terms of the number of points on the variety after reducing modulo each prime number ''p''. It is a global ''L''-function defined as an Euler product of local zeta functions. Hasse–Weil ''L''-functions form one of the two major classes of global ''L''-functions, alongside the ''L''-functions associated to automorphic representations. Conjecturally, these two types of global ''L''-functions are actually two descriptions of the same type of global ''L''-function; this would be a vast generalisation of the Taniyama-Weil conjecture, itself an important result in number theory. For an elliptic curve over a number field ''K'', the Hasse–Weil zeta function is conjecturally related to the group of rational points of the elliptic curve over ''K'' by the Birch and Swinnerton-Dyer conjecture. Definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field . In this case, one speaks of a rational function and a rational fraction ''over ''. The values of the variables may be taken in any field containing . Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is . The set of rational functions over a field is a field, the field of fractions of the ring of the polynomial functions over . Definitions A function f is called a rational function if it can be written in the form : f(x) = \frac where P and Q are polynomial functions of x and Q is not the zero function. The domain of f is the set of all values of x for which the denominator Q(x) is not zero. How ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-defined
In mathematics, a well-defined expression or unambiguous expression is an expression (mathematics), expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well defined (and thus not a function). The term ''well-defined'' can also be used to indicate that a logical expression is unambiguous or uncontradictory. A function that is not well defined is not the same as a function that is undefined (mathematics), undefined. For example, if f(x)=\frac, then even though f(0) is undefined, this does not mean that the function is ''not'' well defined; rather, 0 is not in the Domain of a function, domain of f. Example Let A_0,A_1 be sets, let A = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Product
In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a''''n'' as ''n'' increases without bound. The product is said to '' converge'' when the limit exists and is not zero. Otherwise the product is said to ''diverge''. A limit of zero is treated specially in order to obtain results analogous to those for infinite sums. Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence ''a''''n'' as ''n'' increases without bound must be 1, while the converse is in general not true. The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète ( Viète's formula, the first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, ''holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which deals wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Series
In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ''ζ(s)'' is the Dirichlet series of the constant unit function ''u(n)'', namely: \zeta(s) = \sum_^\infty \frac = \sum_^\infty \frac = D(u, s), where ''D(u, s)'' denotes the Dirichlet series of ''u(n)''. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet. Combinatorial importance Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian product ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Néron Model
In algebraic geometry, the Néron model (or Néron minimal model, or minimal model) for an abelian variety ''AK'' defined over the field of fractions ''K'' of a Dedekind domain ''R'' is the "push-forward" of ''AK'' from Spec(''K'') to Spec(''R''), in other words the "best possible" group scheme ''AR'' defined over ''R'' corresponding to ''AK''. They were introduced by for abelian varieties over the quotient field of a Dedekind domain ''R'' with perfect residue fields, and extended this construction to semiabelian varieties over all Dedekind domains. Definition Suppose that ''R'' is a Dedekind domain with field of fractions ''K'', and suppose that ''AK'' is a smooth separated scheme over ''K'' (such as an abelian variety). Then a Néron model of ''AK'' is defined to be a smooth morphism, smooth Separated morphism, separated scheme ''AR'' over ''R'' with fiber ''AK'' that is universal in the following sense. :If ''X'' is a smooth separated scheme over ''R'' then any ''K''-mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise '' Éléments de géométrie algébrique'' (EGA); one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Schemes elaborate the fundamental idea that an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod n, integers mod p when p is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set that is a fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost All
In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite set, finite, countable set, countable, or null set, null. In contrast, "almost no" means "a negligible quantity"; that is, "almost no elements of X" means "a negligible quantity of elements of X". Meanings in different areas of mathematics Prevalent meaning Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) except for finite set, finitely many". This use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) except for countable set, countably many". Examples: * Almost all positive integers are greater than 1012. * Almost all prime numbers are odd (2 is the only ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
