“Standard L-function”的意思、由来-开放百科全书

In mathematics, the term standard L-function refers to a particular type of automorphic L-function described by Robert P. Langlands.

Here, standard refers to the finite-dimensional representation r being the standard representation of the L-group as a matrix group.

Relations to other L-functions

Standard L-functions are thought to be the most general type of L-function. Conjecturally, they include all examples of L-functions, and in particular are expected to coincide with the Selberg class. Furthermore, all L-functions over arbitrary number fields are widely thought to be instances of standard L-functions for the general linear group GL(n) over the rational numbers Q. This makes them a useful testing ground for statements about L-functions, since it sometimes affords structure from the theory of automorphic forms.

Analytic properties

These L-functions were proven to always be entire by Roger Godement and Hervé Jacquet,^[3] with the sole exception of Riemann ζ-function, which arises for n = 1. Another proof was later given by Freydoon Shahidi using the Langlands–Shahidi method. For a broader discussion, see {{harvtxt|Gelbart|Shahidi|1988}}.^[4]

See also

References

1. ^{{citation | last = Langlands | first = R.P. | title = {{mvar|L}}-Functions and Automorphic Representations (ICM report at Helsinki) | url = http://publications.ias.edu/sites/default/files/lfunct-ps.pdf | year = 1978}}.
2. ^{{citation | last = Borel | first = A. | contribution = Automorphic {{mvar|L}}-functions | mr = 546608 | pages = 27–61 | publisher = Amer. Math. Soc. | location = Providence, R.I. | series = Proc. Sympos. Pure Math. | volume = XXXIII | title = Automorphic forms, representations and {{mvar|L}}-functions (Oregon State Univ., Corvallis, Ore., 1977), Part 2 | year = 1979}}.
3. ^{{citation | last1 = Godement | first1 = Roger | author1-link = Roger Godement | last2 = Jacquet | first2 = Hervé | author2-link = Hervé Jacquet | mr = 0342495 | publisher = Springer-Verlag | location = Berlin-New York | series = Lecture Notes in Mathematics | volume = 260 | title = Zeta functions of simple algebras | year = 1972}}.
4. ^{{citation | last1 = Gelbart | first1 = Stephen | author1-link = Stephen Gelbart | last2 = Shahidi | first2 = Freydoon | author2-link = Freydoon Shahidi | isbn = 0-12-279175-4 | mr = 951897 | publisher = Academic Press, Inc. | location = Boston, MA | series = Perspectives in Mathematics | title = Analytic properties of automorphic {{mvar|L}}-functions | volume = 6 | year = 1988}}.