A Weak Multiplicity-one Theorem for Siegel Modular Forms

In a recent paper by Breulmann and Kohnen [BK99], the authors obtain a weak multiplicity-one result on (integral weight) Siegel-Hecke cuspidal eigenforms of degree 2, showing that such forms are completely determined by their coefficients on matrices of the form mS, where S is primitive and m is square-free. To show this, they twist Andrianov’s identity relating the Maaß-Koecher series and the spinor zeta function of an eigenform [An74] by a Größencharacter. This allows them to then use Imai’s converse theorem for degree 2 forms [Im80] and thereby obtain their result. In this note, we use an elementary algebraic argument to reprove and extend their result to Siegel modular forms of arbitrary degree n and arbitrary level which are only assumed to be eigenforms for the operators T (p) (but not necessarily for the full Hecke algebra). We first show that such an eigenform must have primitive matrices in the support of its Fourier development. Then it is immediate from the explicit action of the Hecke operators on Fourier coefficients that if two such forms have the same eigenvalues for all T (p) and the same coefficients on primitive matrices then their difference must be zero. Since, moreover, the assumption of coinciding eigenvalues can be derived from the above stated assumption of Breulmann and Kohnen, we recover their result for n = 2. Note that Andrianov’s identity and Imai’s converse theorem are currently only known for n = 2 and level 1, so the analytic approach used in [BK99] cannot at this time be extended to general n.