
doi: 10.1007/bf02942038
The author asks the following question: Does a lifting of modular forms preserve congruences for Fourier coefficients? In this paper he gives an affirmative answer in case of the Eisenstein lift. Let \(S^r_k\) denote the space of cusp forms of even weight \(k> 0\) with respect to the modular group \(\Gamma_r\) of degree \(r\). For \(f\in S^r_k\) and \(Z\) in the upper half space \(\mathbb{H}_n\) of degree \(n\geq r\), the Eisenstein lift \([f]^n_r(Z)\) is defined to be the value at \(s= 0\) of the analytic continuation of \[ [f]^n_r(Z, s)= \sum_M \Biggl({\text{det(Im}(M\langle Z\rangle))\over \text{det(Im}(M\langle Z\rangle^*))}\Biggr)^s\cdot f(M\langle Z\rangle^*)\cdot\text{det}(CZ+ D)^{-k}. \] Here, \(M= \left(\begin{smallmatrix} A & B\\ C & D\end{smallmatrix}\right)\) runs through a set \(\Delta_{n,r}\setminus\Gamma_n\) of coset representatives as in Klingen's Eisenstein series, and \(M\langle Z\rangle^*\) is the upper left \(r\times r\) block in \(M\langle Z\rangle= (AZ+ B)(CZ+ D)- 1\). The lift can be extended to a linear map \([\cdot]^n_r: M^r_k\to M^n_k\) of the corresponding spaces of all modular forms, and it maps Hecke eigenforms to Hecke eigenforms. Now suppose that \(n> r\), that \(f\in S_k\) and \(g\in S_k\) are Hecke eigenforms with Fourier coefficients in some algebraic number field \(K\), and that \(f\equiv[g]^n_r\pmod{{\mathfrak p}^a}\) for some prime ideal \({\mathfrak p}\) of \(K\) and some \(a> 0\). Then, for certain values \(m\geq n\) and under some assumptions, it follows that \([f]^m_n\equiv [g]^m_r\pmod{{\mathfrak p}^a}\) or \([f]^m_n\equiv 2[g]^m_r\pmod{{\mathfrak p}^a}\). As a consequence one obtains congruences for special values of standard \(L\)-functions. The assumptions are too complicated to be explained here. But they can be verified in some special instances which yield interesting examples, one of which is the following: For \(k\in\{12,16,18,20,22,26\}\), let \(\Delta_k\) be the unique normalized cusp form in \(S^1_k\), and let \(E_k\in M^1_k\) be the normalized Eisenstein series. Then we have \(\Delta_k\equiv-{B_k\over 2k}\cdot E_k\) modulo the numerator of \({B_k\over 2k}\), and it follows that \([\Delta_k]^m_1\equiv -{B_k\over 2k}\cdot E^{(m)}_k\) for \(1\leq m\leq k-2\). There are examples of so-called stable congruences which hold for all \(m\geq n\).
Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Congruences for modular and \(p\)-adic modular forms
Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Congruences for modular and \(p\)-adic modular forms
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