On a correspondence between p-adic Siegel–Eisenstein series and genus theta series

Some congruences for Saito-Kurokawa lifts

“…It can be expected that similar results hold for other types of modular forms. In fact, Kikuta and the second named author [8] obtained an example using Siegel-Eisenstein series, and Böcherer and the second named author [3] established this kind of general results including Yoshida lifting (weight 2, level p) as a corollary. The third purpose of this paper is to give an example of such a congruence between half-integral weight modular forms belonging to the plus spaces.…”

“…In [8], an explicit description ofG (2) 2 is given in terms of a genus theta series associated with quaternary quadratic forms of level p and discriminant p 2 . This also implies the case k = 2 of Theorem 1.…”

On the restriction of the Hermitian Eisenstein series and its applications

“…Remark 5.6. p-adic Siegel cusp forms: Combining the results from [8] and [11] we immediately see that for a prime p ≡ 3 (mod 4) and k m := 2 + (p − 1)p m the sequence f k m converges p-adically to a true modular form f for Γ 0 (p). Based on some numerical evidence we conjecture that f is a cusp form.…”

On p-adic quaternionic Eisenstein series

“…More precisely, they coincide with the genus theta series (cf. [4], [11]). In these cases (Siegel, Hermitian cases), the p-adic Eisenstein series is algebraic.…”

“…For example, we showed that a p-adic limit of a Siegel Eisenstein series becomes a "real" Siegel modular form (cf. [4]). The same result has also been proved for Hermitian modular forms (e.g., [11]).…”