Local Theory of Primitive Theta Functions
Tóm tắt
We study primitive theta functions, which were first introduced by Shintani, in a purely local setting. We investigate a metaplectic representation of U(1) acting on the space of local primitive theta functions and give its explicit irreducible decomposition. As a by-product, we give a new proof of epsilon dichotomy for (U(1),U(1)).
Tài liệu tham khảo
Fröhlich, A. and Queyrut, J.: On the functional equation of the Artin L-function for characters of real representations, Invent. Math. 20 (1973), 125-138.
Glauberman, G.I. and Rogawski, J.D.: On theta functions with complex multiplication, J. reine angew. Math. 395 (1989), 68-101.
Harris, M., Kudla, S.S. and Sweet, W.J.: Theta dichotomy for unitary groups, J. Amer. Math. Soc. 9 (1996), 941-1004.
Howe, R.: On the character of Weil's representation, Trans. Amer. Math. Soc. 177 (1973), 287-298.
Kato, Y.: On theta functions occuring in the Fourier expansion of automorphic forms on unitary groups of order 3 (in Japanese), Master thesis (1980), Univ. of Tokyo.
Moen, C.: The dual pair (U(1), U(1)) over a p-adic field, Pacific J. Math. 158 (1991), 365-386.
Mœglin, C. Vignéras, M.-F. and Waldspurger, J.-L.: Correspondences de Howe sur un corps p-adique, Lecture Notes in Math. 1291, Springer-Verlag, New York, 1987.
Prasad, D.: A brief survey on the theta correspondence, Contemp. Math. 210 (1998), 171-193.
Rogawski, J.D.: The multiplicity formula for A-packets, In: R.P. Langlands and D. Ramakrishnan (eds), The Zeta Functions of Picard Modular Surfaces, 1992, pp. 395-419.
Shintani, T.: On automorphic forms on unitary groups of order 3, unpublished manuscript (1979).
Tate, J.: Number theoretic background, Proc. Sympos. Pure Math. 33, part 2 (1979), 3-26.
Weil, A.: Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143-211.
Yang, T.: Eigenfunctions of the Weil representation of unitary groups of one variable, Trans. Amer. Math. Soc. 350 (1998), 2393-2407.