RÉSUMÉ

This talk is based on joint work with Masao Koike (Kyushu University), Akihiro Munemasa (Kyushu University) and Jiro Sekiguchi (Himeji Institute of Technology). First we give the determination of the finite index subgroups $\Gamma$ of the modular group $SL(2, Z)$ for which the space of modular forms (of integral weights) of $\Gamma$ is isomorphic to a polynomial ring. There are 17 such groups up to the conjugacy in $SL(2, Z)$. One example is $\Gamma = \Gamma (3)$, and this case is related to the space of weight enumerators of ternary self-dual codes and the polynomial invariants by the unitary reflection group (No. 4) of order 24. Then we consider a similar problem for the space of modular forms of fractional weights. We show that if the space of modular forms of half-integral weights (with respect to some multiplier system) of $\Gamma$ is isomorphic to a polynomial ring generated by 2 modular forms of weight $1/2$, then $\Gamma$ must be one of the 191 subgroups up to the conjugacy in $SL(2, Z).$ One example is $\Gamma = \Gamma (4)$, and this case is related to the space of weight enumerators of binary Type II codes, and the polynomial invariants by the unitary reflection group (No. 8) of order 96. Finally, we consider similar problems for the space of modular forms of $1/l$-integral weghts. We show that the space of modular forms of $1/5$-integral weights (with respect to a certain specified multiplier system) of $\Gamma (5)$ is isomorphic to a polynomial ring generated by 2 modular forms of weight $1/5$. Also, we discuiss the connections of this work with the classical work of F. Klein on the icosahedron and with the unitary reflection group (No. 16) of order 600. The main theme here is that, in some cases, by considering the fractional weight modular forms, the ring of modular forms becomes simpler, and we can get better understanding of the space of modular forms of integral weights. We also mention some very recent work by T. Ibukiyama on modular forms of $1/7$-integral weight of $\Gamma (7)$, which was motivated by this theme.

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