Transformation law and trace formula on theta series under Siegel modular group

摘要

The purpose of this paper is to discuss symplectic transformation laws on theta series and give an explicit formula for trace of the symplectic operator. Theta series plays an important role in number theory and also in the theory of modular forms. In1978, Shimura established a close relation between Jacobi forms and theta series. In 1985, Eichler and Zagier developed systematically the theory of Jacobi forms. Later, Skoruppa and Zagier studied the trace formula for Jacobi forms. Li studied the trace formula for Jacobi forms of general degree. In his formula, the trace of the operator on the theta series is of importance. But he did not compute the trace formula explicitly. The computation of the transformation law of theta series is interest though it is not easy to perform. Many mathematicians contributed greatly to this problem. However, the definition of theta series here is slightly different from the usual symplectic theta series they studied. The purpose of this paper is to give some symplectic transformation laws on theta series and to obtain the trace formula of symplectic operator.