Motivic L-function

In mathematics, motivic ''L''-functions are a generalization of Hasse–Weil ''L''-functions to general motives over global fields. The local ''L''-factor at a finite place ''v'' is similarly given by the characteristic polynomial 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 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, that each motivic ''L''-function can be analytically continued to a meromorphic function on the entire complex plane and satisfies a functional equation 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 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), 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, and hence should be part of the Selberg class. 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).

Notes

References

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Zeta and L-functions
Algebraic geometry