Goro Shimura

Modular Forms of Half Integral Weight

The forms to be discussed are those with the automorphic factor (cz + d)k/2 with a positive odd integer k. The theta function

ARITHMETIC OF UNITARY GROUPS

\theta \left( z \right) = \sum\nolimits_{n = - \infty }^\infty {e^{2\pi in^2 z} }

ON THE TRANSFORMATION FORMULAS OF THETA SERIES

and the Dedekind eta function

Some exact formulas on quaternion unitary groups

\eta \left( z \right) = e^{\pi iz/12} \prod _{n = 1}^\infty (1 - e^{2\pi inz} )

Generalized Bessel functions on symmetric spaces

are classical examples of such forms. (For some practical reasons, we take \( e^{2\pi in^2 z} \) instead of the usual \( e^{\pi in^2 z} \) in the definition of θ.) In fact, the function θ satisfies