ON SOME RESULTS OF BUMP-CHOIE AND CHOIE-KIM

Title & Authors
ON SOME RESULTS OF BUMP-CHOIE AND CHOIE-KIM
Hundley, Joseph;

Abstract
This paper is motivated by a 2001 paper of Choie and Kim and a 2006 paper of Bump and Choie. The paper of Choie and Kim extends an earlier result of Bol for elliptic modular forms to the setting of Siegel and Jacobi forms. The paper of Bump and Choie provides a representation theoretic interpretation of the phenomenon, and shows how a natural generalization of Choie and Kim's result on Siegel modular forms follows from a natural conjecture regarding ($\small{g}$, K)-modules. In this paper, it is shown that the conjecture of Bump and Choie follows from work of Boe. A second proof which is along the lines of the proof given by Bump and Choie in the genus 2 case is also included, as is a similar treatment of the result of Choie and Kim on Jacobi forms.
Keywords
Siegel modular forms;Jacobi forms;meromorphic automorphic forms;(g;K)-modules;generalized Verma modules;Bol's result;
Language
English
Cited by
References
1.
R. Berndt and R. Schmidt, Elements of the Representation Theory of the Jacobi Group, Progress in Mathematics, 163. Birkhauser Verlag, Basel, 1998.

2.
B. D. Boe, Homomorphisms between generalized Verma modules, Trans. Amer. Math. Soc. 288 (1985), no. 2, 791-799.

3.
B. D. Boe, Homomorphisms between generalized Verma modules, Ph.D. thesis, Yale University, 1982.

4.
G. Bol, Invarianten linearer differential gleichungen, Abh. Math. Sem. Univ. Hamburg 16 (1949), no. 3-4, 1-28.

5.
D. Bump, Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, Cambridge, 1997.

6.
D. Bump and Y.-J. Choie, Derivatives of modular forms of negative weight, Pure Appl. Math. Q. 2 (2006), no. 1, part 1, 111-133.

7.
Y.-J. Choie and H. Kim, An analogy of Bol's result on Jacobi forms and Siegel modular forms, J. Math. Anal. Appl. 257 (2001), no. 1, 79-88.

8.
Harish-Chandra, The characters of semisimple Lie groups, Trans. Amer. Math. Soc. 83 (1956), 98-163.

9.
M. Harris, Special values of zeta functions attached to Siegel modular forms, Ann. Sci. Ecole Norm. Sup. (4) 14 (1981), no. 1, 77-120.