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Congruences for eigenvalues of Hecke operators on Siegel modular forms of degree two. (English) Zbl 0578.10032

Let f be a normalized cuspidal Hecke eigenform of integral weight k with respect to SL(2,\({\mathbb{Z}})\), and [f] be the Eisenstein series associated with f in the sense of Langlands-Klingen which is a modular form of weight k with respect to Sp(2,\({\mathbb{Z}})\). The eigenvalues of the Hecke operators on f generate a totally real algebraic number field \({\mathbb{Q}}(f)\) over \({\mathbb{Q}}\). Suppose a prime \({\mathfrak p}\) of \({\mathbb{Q}}(f)\) satisfies the following two conditions: (i) \({\mathfrak p}^{\alpha}\) \((0<\alpha \in {\mathbb{Z}})\) ”appears” in the denominator of a Fourier coefficient of [f] (which lies in \({\mathbb{Q}}(f))\), (ii) \({\mathfrak p}\) does not divide \(c^*(f)\), an explicitly calculable rational integer determined by f.
Then the main theorem of this paper asserts: The ”multiplicity one condition for weight k” implies the existence of a cuspidal eigenform F of weight k with respect to Sp(2,\({\mathbb{Z}})\) such that \[ N_{{\mathbb{Q}}(F,f)/{\mathbb{Q}}(f)}(\lambda (m,F)-\lambda (m,[f]))\equiv 0\quad (mod {\mathfrak p}^{\alpha}) \] for all \(m\geq 1\). Here \(\lambda\) (m,*) is the eigenvalue of the m-th Hecke operator on *, \({\mathbb{Q}}(F,f)\) the number field generated over \({\mathbb{Q}}\) by \(\{\) \(\lambda\) (m,F), \(\lambda\) (m,f) \(| m\geq 1\}\), and \(N_{{\mathbb{Q}}(F,f)/{\mathbb{Q}}(f)}\) the norm map. The proof is based on the author’s previous result [Math. Ann. 265, 119-135 (1983; Zbl 0505.10013)].
Example: \(\lambda (m,\chi_{22}^{(4)})\equiv \lambda (m,[\Delta_{22}])\) (mod 61\(\cdot 103)\) for all \(m\geq 1\), etc.

MSC:

11F27 Theta series; Weil representation; theta correspondences
11F33 Congruences for modular and \(p\)-adic modular forms

Citations:

Zbl 0505.10013
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References:

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