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 (Serre 1970) 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 L(1 − s, M^∨), where M^∨ is the dual of the motive M.^[1]

Examples

Basic examples include Artin L-functions and Hasse–Weil L-functions. It is also known (Scholl 1990), 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,^[2] 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

^ Another common normalization of the L-functions consists in shifting the one used here so that the functional equation relates a value at s with one at w + 1 − s, where w is the weight of the motive.
^ Langlands 1980

References

Deligne, Pierre (1979), "Valeurs de fonctions L et périodes d'intégrales" (PDF), in Borel, Armand; Casselman, William (eds.), Automorphic Forms, Representations, and L-Functions, Proceedings of the Symposium in Pure Mathematics (in French), vol. 33, Providence, RI: AMS, pp. 313–346, ISBN 0-8218-1437-0, MR 0546622, Zbl 0449.10022
Langlands, Robert P. (1980), "L-functions and automorphic representations", Proceedings of the International Congress of Mathematicians (Helsinki, 1978) (PDF), vol. 1, Helsinki: Academia Scientiarum Fennica, pp. 165–175, MR 0562605, archived from the original (PDF) on 2016-03-03, retrieved 2011-05-11 alternate URL
Scholl, Anthony (1990), "Motives for modular forms", Inventiones Mathematicae, 100 (2): 419–430, Bibcode:1990InMat.100..419S, doi:10.1007/BF01231194, MR 1047142, S2CID 17109327
Serre, Jean-Pierre (1970), "Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures)", Séminaire Delange-Pisot-Poitou, 11 (2 (1969–1970) exp. 19): 1–15

[1] Another common normalization of the L-functions consists in shifting the one used here so that the functional equation relates a value at s with one at w + 1 − s, where w is the weight of the motive.

[2] Langlands 1980

[1]

[2]

v t e L-functions in number theory
Analytic examples	Riemann zeta function Dirichlet L-functions L-functions of Hecke characters Automorphic L-functions Selberg class
Algebraic examples	Dedekind zeta functions Artin L-functions Hasse–Weil L-functions Motivic L-functions
Theorems	Analytic class number formula Riemann–von Mangoldt formula Weil conjectures
Analytic conjectures	Riemann hypothesis Generalized Riemann hypothesis Lindelöf hypothesis Ramanujan–Petersson conjecture Artin conjecture
Algebraic conjectures	Birch and Swinnerton-Dyer conjecture Deligne's conjecture Beilinson conjectures Bloch–Kato conjecture Langlands conjecture
p-adic L-functions	Main conjecture of Iwasawa theory Selmer group Euler system