On the period of the Ikeda lift for U(m, m)

Abstract

In [18], the second and the third named authors constructed the Ikeda type lift for the exceptional group E 7,3 from an elliptic modular cusp form. In this paper, we prove an explicit formula for the period or the Petersson norm of the Ikeda type lift in terms of the product of the special values of the symmetric square L-function of the elliptic modular form. There are similar works done by the first author with his collaborator, but new technical inputs are required and developed to overcome some difficulties coming from the hugeness of E 7,3 .

Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts

“…The reader will note that we use the classical and adelic language somewhat inconsistenly reserving the first one for the elliptic modular form φ while using the second one for its Ikeda lift I φ . This is however the convention used by both Ikeda [22] and Katsurada [26] and we chose not to alter it here in order to make the references which we use more readily applicable to our situation.…”

“…Remark 5.2. In [26] an automorphic normalization of the standard Lfunction is used, which we temporarily denote by L 0 (s). However, here and in the rest of the paper we use Shimura's normalization.…”

“…The translation is given by L(s) = L 0 (s − n + 1/2). In particular in the normalization given in [26] in (5.1) one has factors of the form L D K 0 (s + k + m − i + 1/2, BC(f ) ⊗ ψ) which in our normalization equals L D K (s + k + m − n − i + 1, BC(f ) ⊗ ψ). We also note that our use of letters n and m is reversed from that in [26].…”

“…The translation is given by L(s) = L 0 (s − n + 1/2). In particular in the normalization given in [26] in (5.1) one has factors of the form…”

“…Given these arithmetic results and an inner product formula due to Katsurada [26] we construct the aforementioned congruence in section 6. More precisely, we show that up to some technical hypotheses I φ is congruent to an orthogonal automorphic form f ′ on U(n, n)(A Q ) modulo ℓ b where (1.1)…”