In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives

Motive (algebraic geometry)

In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...

 over global field

Global field

In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...

s. The local L-factor at a finite place v is similarly given by the characteristic polynomial

Characteristic polynomial

In linear algebra, one associates a polynomial to every square matrix: its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace....

 of a Frobenius element at v acting on the v-inertial invariants of the v-adic realization of the motive. For infinite places, Jean-Pierre Serre

Jean-Pierre Serre

Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...

 gave a recipe in for the so-called Gamma factors in terms of the Hodge realization of the motive. It is conjectured that, like other L-functions

L-function

The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...

, that each motivic L-function can be analytically continued

Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

 to a meromorphic function

Meromorphic function

In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...

 on the entire complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

 and satisfies a functional equation

Functional equation

In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...

 relating the L-function L(s, M) of a motive M to , where M^∨ is the dual of the motive M.

Examples

Basic examples include Artin L-functions

Artin L-function

In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin...

 and Hasse–Weil L-functions. It is also known , for example, that a motive can be attached to a newform (i.e. a primitive cusp form

Cusp form

In number theory, a branch of mathematics, a cusp form is a particular kind of modular form, distinguished in the case of modular forms for the modular group by the vanishing in the Fourier series expansion \Sigma a_n q^n...

), hence their L-functions are motivic.

Conjectures

Several conjectures exist concerning motivic L-functions. It is believed that motivic L-functions should all arise as automorphic L-functions

Automorphic L-function

In mathematics, an automorphic L-function is a function L of a complex variable s, associated to an automorphic form π of a reductive group G over a global field and a finite-dimensional comlplex representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a...

, and hence should be part of the Selberg class

Selberg class

In mathematics, the Selberg class S is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called L-functions or zeta functions...

. There are also conjectures concerning the values of these L-functions at integers generalizing those known for the Riemann zeta function, such as Deligne's conjecture on special values of L-functions, the Beilinson conjecture, and the Bloch–Kato conjecture (on special values of L-functions).