Hecke eigenforms and Koecher-Maass series

Hecke eigenforms and Koecher-Maass series

Winfried Kohnen

Mathematisches Institut der Universit¨at Heidelberg Im Neuenheimer Feld 288

69120 Heidelberg Germany

In this short note we briefly want to discuss the question how a Hecke eigenform of genus n is uniquely determined by certain of its Fourier coefficients. More precisely, let f and g be two cuspidal Hecke eigenforms of integral weight k w.r.t. the Siegel modular group Γ_n of genus n and write a(T) resp. b(T) for the Fourier coefficients of f resp. g, where T is a positive definite symmetric half-integral matrix of size n.

If n = 1, of course, one knows by classical Hecke theory that if a(1) = b(1) and a(p) =b(p) for all primes p, then f =g.

If n= 2 one can a prove a weaker statement.

Theorem [2]. Suppose that n= 2 and a(mT) = b(mT) for all primitive matrices T and all squarefree positive integers m. Then f =g.

Let us briefly outline the proof (for details cf. [2]). By the results of Imai [4] it is sufficient to show that

(1) D_f,ϕ(s) =D_g,ϕ(s)

for all Gr¨ossen characters ϕ, where D_f,ϕ(s) resp. D_g,ϕ(s) for Re(s) ≫ 0 denote the associated Koecher-Maass series.

On the other hand, using results of Andrianov [1] relating eigenvalues to Fourier coefficients in the genus 2 case, one can derive a (technically complicated) identity relating these Koecher-Maass series to the corresponding spinor zeta functions Z_f(s) resp. Z_g(s).

Under the conditions posed in the Theorem, it follows from the latter that Df,ϕ(s) is not identically zero if and only if this is true for D_g,ϕ(s), and then the equality

D_f,ϕ(s)

Dg,ϕ(s) = Z_f(2s) Zg(2s)

holds for alls ∈Cas an identity between meromorphic functions. Exploiting the different types of functional equations of the Koecher-Maass series and the spinor zeta functions, one easily deduces (1).

Very recently, Breulmann and myself [3] were able to give a weak generalization of the above result to the case of arbitrary genus n ≥ 3, assuming that the spinor zeta

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functions have good analytic properties. The latter (in particular a functional equation) were conjectured in [1] and certainly are widely believed, but so far have not yet been proved. In [3] under the latter conditions it is shown that if a(mT) = b(mT) for all primitive T and all positive integers m such that vp(m) ≤2ⁿ−2 for all primes p (where vp is the usual p-adic exponent), then f = g. The proof is technically more involved and uses the Spn-converse theorem of Weissauer [5,6] and certain results of ˜Zarkovskaja on a relation between eigenvalues and Fourier coefficients for arbitraryncontained in the paper [7].

References

[1] A.N. Andrianov: Euler products corresponding to Siegel modular forms of genus 2.

Russ. Math. Surv. 29:3, 45-116 (1974)

[2] S. Breulmann and W. Kohnen: Twisted Maass-Koecher series and spinor zeta functions. To appear in Nagoya Math J.

[3] S. Breulmann and W. Kohnen: On Hecke eigenforms of degree n. Preprint 1999 [4] K. Imai: Generalization of Hecke’s correspondence to Siegel modular forms. Amer.

J. Math., vol. 102, no. 5., 903-936 (1980)

[5] T. Sugano: Weissauer’s converse theorem. This volume

[6] R. Weissauer: Siegel modular forms and Dirichlet series. Unpublished manuscript [7] N.A. ˜Zarkovskaja: On the connection of eigenvalues of Hecke operators and the Fourier

coefficients of eigenfunctions of Siegel’s modular forms of genus n. Math. USSR Sb. 25, no. 4, 549-557 (1975)

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