Congruence of Siegel modular forms and special values of their standard zeta functions (Q2481299)

Let \(f\) be an elliptic cusp form of weight \(2k-2\) and \(F\) its Saito-Kurokawa lift. A prime ideal \(P\) in the ring of integers of the field \({\mathbb Q}(f)\) generated by the Hecke-eigenvalues of \(F\) is called a congruence prime for \(F\), if there exists a Siegel modular form \(G\) of weight \(k\) on \(\mathrm{Sp}(2,{\mathbb Z})\) which is orthogonal to \(F\) and has modulo \(P'\) the same Hecke-eigenvalues (where \(P'\) divides \(P\) in a suitable larger number field). The author conjectures that (for large enough residue characteristic) \(P\) is a congruence prime with respect to the orthogonal complement to the Maaß\ space (i.e. \(G\) as above can be found there) iff it divides the numerator of \(\tilde L(k,f)\), the value at \(k\) of a certain nomalization of the standard \(L\)-series attached to \(f\). He can prove this under several additional assumptions on \(P\). One important tool is \textit{S. Böcherer}'s pullback formula for Eisenstein series [Math. Z. 189, 81--110 (1985; Zbl 0558.10022)].