Congruence primes for Ikeda lifts and the Ikeda ideal

Abstract

Abstract. Let f be a newform of level 1 and weight 2κ − n for κ and n positive even integers. In this paper we study congruence primes for the Ikeda lift of f . In particular, we consider a conjecture of Katsurada stating that primes dividing certain L-values of f are congruence primes for the Ikeda lift. Instead of focusing on a congruence to a single eigenform, we deduce a lower bound on the number of all congruences between the Ikeda lift of f and forms not lying in the space spanned by Ikeda lifts.

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

“…The paper also discusses the application of the result to the Bloch-Kato conjecture for φ. Our current result can be viewed as partly complementary to that of [9] in the sense that it constructs congruences for the Ikeda lift of φ ∈ S k (1) to U(n, n) for odd values of n, while [9] construct congruences for the Ikeda lift of φ to Sp 2n for even values of n.…”

“…In[9], Keaton and the first-named author construct congruences for the (symplectic) Ikeda lift on the group Sp 2n of an elliptic modular form φ ∈ S k (1) when k and n are even. These are controlled by the L-value L alg ((k + n)/2, φ) n/2−1 j=1…”

Period of the adelic Ikeda lift for U(m, m)

“…As a result, in [13], we characterized prime ideals giving congruence between the DII lift and non-DII lift. (See also [5].) Klosin [17] gave the congruence between the Hermitian Maass lift and non-Hermitian Maass lift using the period relation in [10].…”

Congruence Primes of the Kim–Ramakrishnan–Shahidi Lift

“…Böecherer, Dummigan, and Schulze-Pillot [4] proved the period relation for the Yoshida lift and gave a similar result on the congruence between the Yoshida lift and non-Yoshida lift. Katsurada and Kawamura [26] proved Ikeda's conjecture on the period of the Duke-Imamoglu-Ikeda lift proposed in [23], and by using this period relation Katsurada proved Problem B for the Duke-Imamoglu-Ikeda lift in [25] (see also [8]). Based on the conjectural period relation in [23], Ibukiyama, Katsurada, Poor, and Yuen [18] proposed a conjecture on the congruence of the Ikeda-Miyawaki lift and tested it numerically.…”