Congruence Primes of the Kim–Ramakrishnan–Shahidi Lift

Abstract

For a primitive form f of weight k for SL 2 (Z), let KS(f) be the Kim-Ramakrishnan-Shahidi lift (K-R-S lift) of f to the space of cusp forms of weight det k+1 ⊗Sym k−2 for Sp 2 (Z). Based on some working hypothesis, we propose a conjecture, which relates the ratio KS(f), KS(f) f, f 3 of the periods (Petersson norms) to the symmetric 6-th L-value L(3k − 2, f, Sym 6) of f. From this, we also propose that a prime ideal dividing the (conjectural) algebraic part L(3k−2, f, Sym 6) of L(3k − 2, f, Sym 6) gives a congruence between the K-R-S lift and non-K-R-S lift, and test this conjecture numerically.

On Deligne's conjecture for symmetric sixth $L$-functions of Hilbert modular forms

“…In particular, specializing our main result Theorem 1.3 to F " Q, we show that (1.1) holds for n " 6 assuming κ ě 6. We also refer to [KT16,§ 7] for numerical examples in this case.…”

Structure theorems for vector valued Siegel modular forms of degree 2 and weight detk ⊗Sym(10)

“…By Lemma 7.1, there uniquely exists F 9 ∈ M 9,10 (Γ (2) ) such that F 9 = 0 up to a non-zero constant. By a similar argument in [11] we can explicitly find rational numbers α, β, γ, δ, c…”

“…We recall differential operators defined by Eholzer-Ibukiyama [3] and van Dorp [22]. We use the same notation used in [11].…”

“…(2) ). We can check this equality by computing vector valued modular forms in R 6 (Q) 11 . By Lemma 7.1, there uniquely exists F 9 ∈ M 9,10 (Γ (2) ) such that F 9 = 0 up to a non-zero constant.…”

“…In other words, M 0 Sym(10) (Γ (2) ) is generated by 13 modular forms of determinant weights 6,8,10,10,12,12,14,14,14,16,16,18,20 and they satisfy two fundamental relations. M 1 Sym(10) (Γ (2) ) is generated by 13 modular forms of determinant weights 9,11,13,15,15,15,17,17,17,19,19,21, 23 and they satisfy two fundamental relations.…”