## Tohoku Mathematical Journal 2004 March SECOND SERIES VOL. 56, NO. 1

 Tohoku Math. J. 56 (2004), 125-145

Title ON THE FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF INTEGRAL WEIGHT BELONGING TO KOHNEN'S SPACES AND THE CRITICAL VALUES OF ZETA FUNCTIONS

Author Hisashi Kojima and Yasushi Tokuno

(Received April 4, 2002, revised September 22, 2003)
Abstract. The purpose of this paper is to derive a generalization of Kohnen-Zagier's results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces, and to refine our previous results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces. Employing kernel functions, we construct a correspondence \$\varPsi\$ from modular forms of half integral weight \$k+1/2\$ belonging to Kohnen's spaces to modular forms of weight \$2k\$. We explicitly determine the Fourier coefficients of \$\varPsi(f)\$ in terms of those of \$f\$. Moreover, under certain assumptions about \$f\$ concerning the multiplicity one theorem with respect to Hecke operators, we establish an explicit connection between the square of Fourier coefficients of \$f\$ and the critical value of the zeta function associated with the image \$\varPsi(f)\$ of \$f\$ twisted with quadratic characters, which gives a further refinement of our results concerning Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces.

2000 Mathematics Subject Classification. Primary 11F 37; Secondary 11F30, 11F67.

Key words and phrases. Modular forms of half integral weight, Fourier coefficients of modular forms, special value of zeta function.